Supermanifolds and Supergroups: Basic Theory (Mathematics and Its Applications 570)

Supermanifolds and Supergroups

Mathematics and Its Applications

Managing Editor: M. HAZEWINKEL

Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 570

Supermanifolds and Supergroups Basic Theory

Gijs M. Tuynman Universite de Lille i" Lille, France

KLU\VER ACADElVlIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT. LONDON. MOSCOW

eBook ISBN: Print ISBN:

1-4020-2297-2 1-4020-2296-4

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Table of Contents

ix

Preface

I.

1

Ql-graded commutative linear algebra

1. Ql-graded commutative rings and Ql-graded A-modules

2. (M ul ti -) linear maps 3.

4. 5. 6. 7. 8. II.

Direct sums, free Ql-graded A-modules, and quotients Tensor products Exterior powers Algebras and derivations Identifications Isomorphisms

Linear algebra of free graded A-modules

1.

Our kind of Z2-graded algebra A

2. Free graded A-modules 3.

Constructions of free graded A-modules

4. Linear maps and matrices 5. The graded trace and the graded determinant 6. The body of a free graded A-module

55 56 58 62 68 74 80 91

III. Smooth functions and A-manifolds

1.

2 7 14 17 24 28 34 46

92

Topology and smooth functions

2. The structure of smooth functions 3. Derivatives and the inverse function theorem

4. A-manifolds 5. Constructions of A-manifolds v

103 112 124 130

Table of Contents

vi

IV. Bundles

1. Fiber bundles

2. 3. 4. 5. 6. 7. 8. V.

Constructions of fiber bundles Vector bundles and sections Constructions of vector bundles Operations on sections and on vector bundles The pull-back of a section Metrics on vector bundles Batchelor's theorem

The tangent space

1. Derivations and the tangent bundle

2. The tangent map and some standard applications 3. Advanced properties of the tangent map

4. 5. 6. 7. 8.

Integration of vector fields Commuting flows Frobenius' theorem The exterior derivative de Rham cohomology

VI. A-Lie groups

1. A-Lie groups and their A-Lie algebras

2. 3. 4. 5. 6. 7. 8. 9.

The exponential map Convergence and the exponential of matrices Subgroups and sub algebras Homogeneous A-manifolds Pseudo effective actions Covering spaces and simply connected A-Lie groups Invariant vector fields and forms Lie's third theorem

VII. Connections

1. More about vector valued forms

2. Ehresmann connections and FVF connections 3. Connections on principal fiber bundles

4. 5. 6. 7. 8. 9. 10.

The exterior covariant derivative and curvature FVF connections on associated fiber bundles The covariant derivative More on covariant derivatives Forms with values in a vector bundle The covariant derivative revisited Principal fiber bundles versus vector bundles

141 142 151 156 163 173 181 187 196 203 204 211 219 228 236 242 247 260 265 266 277 286 292 298 306 311 315 323 335 336 341 350 354 360 366 374 382 389 395

Table of Contents

vii

References

405

Index of Notation

409

Index

411

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Preface

This book is a self contained introduction to super differential geometry, intended for graduate students in mathematics and theoretical physics and other people who want to leam the basics about supermanifolds. It is self contained in that it only requires standard undergraduate knowledge. However, some knowledge of ordinary (non super) differential geometry will make this text much easier to read. Various versions of super differential geometry exist, some of which are equivalent and some of which are not. The version presented here is equivalent to those that are most widely used: the Hoo supermanifolds of DeWitt and the sheaf theoretic approach to supermanifolds of Kostant and Leites. The approach taken here is based on an index free formalism using a graded commutative ring A containing the usual real numbers as well as so called anticommuting numbers. Starting with a non-standard definition of a differentiable function, valid in the real case, in the complex case and in the super case, the theory is developed as if it were ordinary differential geometry. It is shown that most constructions and theorems in ordinary differential geometry have a natural generalization to the super context. Moreover, even the proofs bear more than a superficial resemblance to their counter parts in ordinary differential geometry. The (equivalent) sheaf-theoretic approach to supermanifolds makes it manifest that the theory is "independent" ofthe choice of A, but at the same time it hides the more geometric nature of the theory. The approach presented here can be seen as a theory with a parameter A. Choosing A = R gives ordinary differential geometry, choosing A = 1\ Roo gives super differential geometry, choosing A = C gives the theory of complex manifolds, etc. Of course, in each of these cases some small but usually superficial changes have to be made, and not all results remain true in all cases (e.g., Batchelor's theorem, which uses partitions of unity, is not valid for (super) complex manifolds). But the main body of the results is not affected by the choice of A. In Chapter I the general theory of graded linear algebra (graded by an arbitrary abelian group) is outlined. This plays the same role in super differential geometry as does linear algebra in ordinary differential geometry and as does commutative algebra in algebraic geometry. Since the basic ring is (in principle) not commutative, we have to make a distinction between left and right linear maps. The isomorphism between these two kinds ix

Preface

x

of maps is given by the operator '1:', which will later be identified with (super) transposition of matrices. In chapter II we specialize to Z2-graded linear algebra and we impose some restrictions on the ring A. The canonical example of A that satisfies all conditions (the ones imposed in chapter II and also other ones imposed later on) is the exterior algebra of an infinite dimensional real vector space: A = t\ Roo. Some of the more important points of this chapter are the following. In section 2 it is shown that any (finitely generated) free graded A-module admits a well defined graded dimension. In section 4 the relation between matrices and linear maps is explained. The reader should really pay attention here, because there are three different natural ways to associate a matrix to a linear map, and these three different ways imply different ways how to multiply a matrix by an element of A (so as to be compatible with the multiplication of the corresponding linear map by the element of A). It is here that we see most clearly the role of the transposition operator introduced in chapter I to relate left and right linear maps. In section 5 the graded trace is defined for any linear map (and thus for any matrix, not only the even ones), as well as its integrated version for even maps, the graded determinant or Berezinian. Finally in section 6 the body map B is introduced, which provides an "isomorphism" between equivalence classes offree graded A-modules and direct sums of two real vector spaces. It is this body map which gives the link between standard linear algebra and Z2-graded linear algebra. The heart of this book lies in chapter III, in which the notion of a supermanifold is developed based on a non-standard definition of differentiable functions. The key idea is expressed by the following formula, valid for functions f of class C 1 on convex domains in Rn:

(1 J'(sx + 1

f(x) - fey) =

(1 - s)y)

dS) .(x -

y) .

If we write this as f(x) - fey) = g(x, y) . (x - y), it is obvious that f is of class C1 if and only if the function 9 is of class Co. Moreover, if a 9 with this property exists, it is also easy to see that 1', the derivative of f, is given by f'(x) = g(x, x). If we now note that the formula f(x) - fey) = g(x, y) . (x - y) does not involve quotients nor limits, we can apply the same definition to super functions, for which there generally do not exist quotients (because of nilpotent elements in A), nor does the natural topology (the DeWitt topology) admit unique limits (being non Hausdorff). Based on this idea, smooth functions (COO) on super domains with p even coordinates and q odd coordinates are defined. It is shown, using the body map B defined in chapter II, that these smooth functions are in bijection with ordinary smooth real-valued functions of p real variables, multiplied by anti symmetric polynomials in q variables. This result is usually taken as the definition of smooth super functions; here it is a consequence of a more general definition, a definition which applies as well to ordinary functions as to super functions. The last two sections of chapter III are devoted to copying the standard definition of manifolds in terms of charts and transition functions to the case in which the transition functions are super smooth functions. In chapter IV the general theory of fiber and vector bundles is developed. The first two sections deal with general fiber bundles and how to construct new ones out of given ones. The next two sections deal with vector bundles and how to generalize the construction

Preface

xi

of new A-modules to the setting of vector bundles. In section 5 the behavior of the operation of taking sections under the various operations one can perform on A-modules is considered. In section 6 the exterior algebra of a (dual) bundle is discussed in more detail, as well as the pull-back of sections. The main purpose of sections 5 and 6 is to provide a rigorous justification for operations everybody performs without thinking twice. In section 7 one finds a proof of part of the Serre-Swann theorem that the module of sections of a vector bundle is a finitely generated projective module over the ring of smooth functions on the base manifold. The proof of this result needs the notion of a metric on a free graded A-module, a notion whose definition is subtly different from what one would expect. These results are not used elsewhere, but they are needed to complete the proofs of statements given in section 5. The last section in chapter IV on Batchelor's theorem merits ample attention. This theorem says that any supermanifold is "isomorphic" to an ordinary vector bundle over an ordinary manifold, or, stated differently, for any smooth supermanifold there exists an atlas in which the transition functions are of the special form: even coordinates depend on even coordinates only, and odd coordinates depend in a linear way on odd coordinates. The proof is "constructive" in that it provides an explicit algorithm to compute such an atlas given an arbitrary atlas. The quotes are needed because this algorithm requires a partition of unity on the underlying ordinary manifold. Chapter V treats the standard machinery of differential geometry. In section 1 the tangent bundle is defined and it is shown that sections of it, called vector fields, are equivalent to derivations of the ring of smooth functions. In section 2 the tangent map is defined, which in turn gives rise to the notions of immersion and embedding. In section 3 the relationship between the tangent map and the derivative of a map are studied in more detail. It turns out that in the super case this is in general not a 1-1 correspondence. Generalizing the notion of the derivative of an A-valued function to vector bundle valued functions, a necessary and sufficient condition is given for a vector bundle to be trivial as a vector bundle. Here one also can find an example of a vector bundle which is trivial as fiber bundle, but not as vector bundle. Sections 4 and 5 then concentrate on the notion of the flow of a vector field and the well known proposition that two vector fields commute if and only if their flows commute. For odd vector fields this amounts to saying that an odd vector field is integrable if and only if its auto commutator is zero. Section 6 treats Frobenius' theorem on integrability of subbundles of the tangent bundle, the notion of integral manifolds and the existence ofleaves for a foliation. In section 7 the calculus of (exterior differential) k-forms is given, including the definition of the Lie derivative and its relation with the flow of a vector field. Finally in section 8 an elementary proof is given of the fact that the de Rham cohomology of a supermanifold is the same as that of the underlying ordinary manifold (its body). Chapter VI treats the basic facts about super Lie groups and their associated super Lie algebras. In section 1 one finds the basic definition of a super Lie group and the construction of the associated super Lie algebra. The exponential map from the super Lie algebra to the super Lie group is defined in section 2. There one also finds the proof that it intertwines a homomorphism of super Lie groups and its induced morphism on the associated super Lie algebras. Section 3 is rather technical and computes the derivative of the exponential map. Section 4 deals with the relationship between Lie subgroups and Lie

xii

Preface

sub algebras, whereas section 5 treats homogeneous supermanifolds. Section 6 is again technical and proves that any smooth action can be transformed into a pseudo effective action. The last section gives a geometric proof that to each finite dimensional super Lie algebra corresponds a super Lie group. Chapter VII is more advanced and discusses the general concept of a connection on a fiber bundle. Sections 1 and 8 are technical and provide the necessary theory of vector valued and vector bundle valued differential forms. In section 2 the general concept of an Ehresmann connection is introduced, as well as the more restrictive notion of FVF connection, which is an Ehresmann connection determined by the fundamental vector fields of the structure group on the typical fiber. FVF connections have nice properties: they are defined on any fiber bundle, they include the standard examples of connections such as the (principal) connection on a principal fiber bundle and linear connections on vector bundles, and they always allow parallel transport. In sections 3 and 4 the particular case of an FVF connection on a principal fiber bundle is studied, which includes the description by a connection I-form, the exterior covariant derivative and a discussion about the curvature 2-form. In section 5 it is shown that any FVF connection can be seen as induced by an FVF connection on a principal fiber bundle. Sections 6 and 7 treat the notion of a covariant derivative on a vector bundle and prove that it is equivalent to an FVF connection. It includes the proof that the covariant derivative measures how far away a (local) section is from being horizontal. In sections 9 and 10 the covariant derivative on a vector bundle is generalized to vector bundle valued differential forms and it is shown how the exterior covariant derivative (on a principal fiber bundle), the ordinary exterior derivative of differential forms and the generalized covariant derivative (on a vector bundle) are intimately related. This book is written in a logical order, meaning that a proof of a statement never uses future results and meaning that related subjects are put together. This is certainly not the most pedagogical way to present the subject, but it avoids the risk of circular arguments. As a consequence, the novice reader should not read this book in a linear order. For a first reading, one can easily skip sections 7 and 8 of chapter I. From chapter IV one should certainly read sections 1-3, but coming back for sections 4-6 (and then only superficially) just before starting to read section 6 of chapter V. The reader who already has a working knowledge of ordinary manifold theory need not read all sections with the same attention and at a first reading (s)he can even skip chapter IV completely. One final word on terminology: in this introduction I have systematically used the adjective super. On the other hand, in the main text I never use this adjective, but rather the prefix A. The reason to do so is that one should regard this theory not as opposed to ordinary differential geometry (super versus non-super), but more as a theory with a parameter A, indicating over which ring it is developed. In preparing chapters I-VI I have relied heavily on the firstthree chapters ofF. Warner's classic "Foundations of Differentiable Manifolds and Lie Groups,", while chapter VII is based on H. Pijls' review article "The Yang-Mills equations." Other sources of inspiration have been the first volume ofM. Spivak's "A Comprehensive Introduction to Differential Geometry" and "Les Tenseurs" ofL. Schwartz. During the years it took me to write this

Preface

xiii

book, I have benefitted from the hospitality of the following three institutions: MSRI (Berkeley, USA), CPT (Marseille, France) and LNCC (Rio de Janeiro, Brazil). Special thanks are due to P. Bongaarts for some excellent suggestions concerning chapter I and to V. Thilliez who helped me with [III. 1.12]. Finally, I am convinced I got the idea for [IV.7.3] from a paper by S. Sternberg, but I can no longer find the source. Lille,january 2004

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Chapter I

m-graded commutative linear algebra

Linear algebra is concerned with the study of vector spaces over the real numbers (or more generally over afield) and linear maps. A standard course on linear algebra more or less starts with the introduction ofthe concept ofa basis. Immediately afterwards one usually restricts attention to finite dimensional vector spaces. Next on the list is the concept of a subspace and with that notion one derives some elementary properties of linear maps. Then one introduces bilinear maps, with a scalar product as the most important example. This gives rise to the notions oforthogonal basis, orthogonal linear map, and orthogonal subspaces, eventually followed by a classification of quadrics. More advanced courses treat the notions ofmultilinear maps, tensor products, and exterior powers. Algebras, and in particular Lie algebras, are usually treated separately. Besides analysis, these concepts in linear algebraform the basis ofdifferential geometry. One could even say that differential geometry is the interplay between analysis and linear algebra. Algebraic geometry is closely related to differential geometry, but hardly relies on analysis; it is mainly concerned with algebraic structures. For that it needs a generalization of linear algebra in which a vector space over a field is replaced by a module over a commutative ring with unit. Commutative algebra is the theory which plays in algebraic geometry the same role as linear algebra does in differential geometry. In commutative algebra the notion of basis more or less disappears, but subspaces, tensor products, and exterior powers can still be defined. In supergeometry one replaces the field of real numbers, not by a commutative ring, but by a graded commutative ring. Since such a ring is not commutative, commutative algebra does not apply. In this context, graded means Z2-graded, i.e., the ring and all modules are a direct sum oftwo subspaces, the even and odd parts. In this first chapter we look at an even more general situation. We denote by Q( an arbitrary abelian group and we denote by A an arbitrary Q(-f(raded commutative ring with unit 1 =1= 0 (i.e., a ring which splits as a direct sum ofsubspaces indexed by Q( and satisfying conditions how these subspaces commute). We will show that all concepts of linear algebra that are important

2

Chapter 1. 2t-graded commutative linear algebra

for differential geometry can be generalized to Qi-graded commutative linear algebra, i.e., to the theory of Qi-graded A-modules.

1.

Qt-GRADED COMMUTATIVE RINGS AND Qt-GRADED A-MODULES

In thisfirst section we give the definitions ofthe principal objects ofthis book: Qi-graded commutative algebras and Qi-graded A-modules. Qi-graded A-modules are a special kind of Qi-graded A-bimodules, afact that will greatly facilitate constructions of new Qi-graded A-modules, one of which is discussed in this section: the Qi-graded submodule.

1.1 Definition. Given abelian groups G I , ... , G k and H, a map ¢ : G I x ... is called k-additive if for all i and for all 9i, (Ii E G i we have: ¢(91, ... , 9i-l, 9i

X

G k -+ H

+ (Ii, 9i+1' ... , 9k) =

¢(91, ... ,9i-I,9i,9i+1,··· ,9k)

+ ¢(91, ... ,9i-I,!Ii,9i+1, ... ,9k)

.

1.2 Definition. Let G be an abelian group and let G i , i E I be a family of subgroups. One writes G = EBiEI G i if and only if for each element 9 E G there exist unique 9i E G i , only finitely many of them non-zero, such that 9 = I: iE I9i; it is called the (unique) decomposition of 9 into Gi-components.

1.3 Definitions . • Let A be a ring. A left module over the ring A (or a left A-module) is an abelian group E equipped with a map m L : A x E -+ E that is bi -additive and satisfies

mL(a, mL(b, e)) = mL(ab, e) . This map is called left multiplication by elements ofA, and (as is usual) we will omit the symbol m£ifno confusion is possible andjust write ae or a· e for mL( a, e). If A contains a unit 1 -I=- 0, we also require that 1· e = e for all e E E. In a similar way, a right A-module is an abelian group E equipped with a map (right multiplication) mR : E x A -+ E that is bi-additive and satisfies mR( mR( e, a), b) = mR( e, ab). And as before, if no confusion is possible we will just write ea or e· a for mR(e, a). As forleft A-modules, if A contains a unit 1 -I=- 0, we require that e· 1 = e for all e E E. Since A is in general not commutative, the notions of left and right A-modules do not coincide . • An A-bimodule is an abelian group E which is at the same time a left and a right A-module such that the left and right actions commute, i.e., for all a, b E A, e E E: mL(a, mR(e, b)) = mR(mL(a, e), b), which can also be written as (ae)b = a(eb) . • A subset F of a left/right A-module E is called a submodule if F is a subgroup with respect to the additive structure of E such that AF c F. It follows that F, with the induced multiplication of A, is itself a left/right A-module.

§ 1. 2t-graded commutative rings and 2t-graded A-modules

3

1.4 Convention. Throughout this book we will denote by Q( an arbitrary abelian group.

1.5 Definition. An Q(-graded ring is aring A with the additional property (the Q(-grading) that there exists a family of subgroups A", a E Q( (subgroups with respect to the additive (abelian) structure of the ring A) such that:

(i) A = EB"E21 A" and (ii) Va, (3 E Q( : A" . Ap

c A,,+p.

The elements of A" are called homogeneous elements of parity a. For homogeneous elements the parity map c : U"E21A" ---- Q( is defined by c(A,,) = a. Note that the parity map is not defined on the whole of A. In analogy with the special case Q( = Z2, elements of parity 0 will always be called even.

1.6 Remarks . • There is an ambiguity in the definition of the parity of the zero element However, we will not avoid this ambiguity since it comes in useful to say that o has every parity one wishes. Purists might want to exclude 0 from having a parity, but then in a lot of proofs one has to treat the zero element as a special case . • Using the parity map, condition (ii) above can be stated as: if a, b E A are homogeneous, then c(ab) = c(a) +c(b).

o E A.

1.7 Lemma. If an Q(-graded ring A has a unit 1 =1= 0, then c(l)

= o.

Proof Let 1 = I:"E21 1" be the decomposition of the unit into homogeneous components, and let a E A be any homogeneous element. From the equation a = 1 . a, the unique decomposition into homogeneous components and the fact that c(l" . a) = a + c(a) we deduce that 10 . a = a. Since multiplication is bi-additive, it follows that 10 . a = a for all a E A. In the same way one proves that a . 10 = a for all a E A. But units, if they exist, are unique and hence 1 = 10 . IqEDI

1.8 Definition. An Q(-graded commutative ring is an Q(-graded ring A together with a symmetric bi-additive map U J : Q( x Q( ---- Z2 ~ {O, I} such that

The above property is called the Q(-graded commutativity of A.

1.9 Remark. A more general definition ofQ(-graded commutativity is possible if A is an Q(-graded k-algebra [6.1] with k = R or k = C. In that case one can define Q(-graded commutativity by the condition

aEA",bEAp

===}

a.b=p(a,{3)·b.a,

4

Chapter 1. 2t-graded commutative linear algebra

where p : Q( x Q( ----., k* = k \ {a} is a "bi -additi ve" map satisfying p( 0:, (3) . p(f3, 0:) = 1 (we put bi-additive in quotes because the abelian operation in k* is multiplication). For instance, ifQ( = Z3 x Z3 ~ {( i, j) I a ::; i, j < 3} and k = C, we can take for p the map p((i,j), (p, q)) = >.iq-jPwith >. a third root of unity. On the other hand, for k = R or k = C and Q( cyclic, the only possibilities for p are of the form p( 0:, (3) = (-1) (a:lp) with (0:1f3) E Z2.

1.10 Convention. Throughout this text we will denote by A an arbitrary Q(-graded commutative ring with unit 1 -I=- a. From time to time we will impose additional restriction on A, but those restrictions will always be stated clearly.

1.11 Definition. An Q(-graded left (respectively right) A-module is a left (respectively right) A-module E together with a family of subgroups Ea:,O: E Q( (subgroups with respect to the additive (abelian) structure of E) satisfying:

(i) E = EBa:E21 Ea: and (ii) Aa: Ep C Ea:+p (respectively Ea: Ap C Ea:+p). The following definitions and remarks are as for Q(-graded rings. The elements of Ea: are called homogeneous elements of parity 0:. The parity map c : Ua:E21 Va: ----., Q( is defined by c( Ea:) = 0:. Note that the parity map is not defined on the whole of E and that the parity of the zero element a E E is ambiguous. Using the parity map, condition (ii) can be stated as: for homogeneous a E A and e E E, c(ae) = c(a) + c(e). Elements of parity 0 will be called even.

1.12 Example. Let A be an Q(-graded ring and n E N a natural number, then we can make An into an Q(-graded A-bimodule. Addition and left/right multiplication by A is component wise, while the Q(-grading is given by (An)a: = (Aa:)n. Forn = a we formally define AD = {a}, which is trivially an Q(-graded A-bimodule.

1.13 Definition. A subset F of an Q(-graded left/right A-module E is called an Q(-graded submodule if F is a submodule of the left/right A-module E such that F together with the subsets Fa: = F n Ea: is itself an Q(-graded left/right A-module.

1.14 Lemma. Let F be a submodule ofan Q(-graded left/right A-module E. Then F is an Q(-graded submodule if and only ifVe E E : e E F ~ ea: E F, where e = I:a:E21 ea: is the unique decomposition ofe into homogeneous components in the Q(-graded left/right A-module E.

§ 1. 2t-graded commutative rings and 2t-graded A-modules

5

1.15 Lemma. Let A be an Qi-graded commutative ring and let E = EBo: E 2l Eo: be an Qi-graded left/right A-module. Then there exists a unique rightlleft action of A on E turning E into an Qi-graded right/left A-module with the same subsets Eo: such that the left and right actions ofA are related by the fonnula (1.16)

Proof Let us suppose that mL has been given, then if mR exists, bi-additivity implies that it must be given by mR(e, a)

== mR(L e{3, (3E2l

Lao:) = o:E2l

L

mR(e{3, ao:) =

o:,{3E2l

L

(-I)(o:l{3)mL(ao:, e(3) ,

o:,{3E2l

where e = I:{3E2l e{3 and a = I:O: E 2l ao: are the unique decompositions of these elements into homogeneous components. From this formula, the uniqueness of mR is clear. On the other hand, bi-additivity of mL proves that mR defined by this formula is also bi-additive. It thus remains to show that mR is a right action, i.e., mR(mR(e, a), b) = mR(e, ab). Since this relation is 3-additive in the variables (e, a, b) E E x A x A, it suffices to show it for homogeneous elements: mR(mR(e, a), b)

=

(-I)(c(e)lc(a))mR(mL(a, e), b)

=

(-1) (c(e)lc(a)) (_I)(c(e)+c(a)lc(b))mL(b,

mL(a, e))

= (-1) (c(e)lc(a)+c(b)) (-1) (c(a)lc(b)) mL (ba, = (_I)(c(e)lc(a)+c(b))mL(ab,

e)

e) = mR(e, ab) .

IqEDI

1.17 Remark. The left and right actions of A on the Qi-graded A-module An are related to each other by (1.16).

1.18 Lemma. With the assumptions as in [l.15J, the left and right actions of A on E commute, i.e., Va, bE A, Ve E E : a(eb) = (ae)b. Proof The relation a(eb) = (ae)b is 3-additive in (a, e, b) and it thus suffices to show it for homogeneous elements: a(eb)

= a((_I)(c(e)lc(b))be) = (_I)(c(e)lc(b))(ab)e = (-1) (c(e)lc(b)) (-1) (c(a)lc(b)) (ba)e = (_1)(c(e)+c(a)lc(b))b(ae) = (_I)(c(e)+c(a)lc(b))( _1)(c(ae)lc(b)) (ae)b

= (ae)b

because c:(ae)

= c:(a) + c:(e).

IqEDI

Chapter 1.

6

2t-graded

commutative linear algebra

1.19 Definitions. By an Qi-graded A-module (without the adjectives left or right) we will always mean an Qi-graded A-bimodule for which the left and right actions of A are related by (1.16). Note that [ 1.15] implies that every Qi-graded left A-module and every Qi-graded right A-module can be turned into such an Qi-graded A-module. A subset F of an Qi-graded A-module E is called an Qi-graded submodule of E if it is an Qi-graded submodule of the left or right A-module E. It follows from [1.20] that this is a sensible definition and that such an F is automatically an Qi-graded A-module itself.

1.20 Lemma. Let E be an Qi-graded A-module and let F be a submodule of either the Qi-graded left or the Qi-graded right A-module E. If F is an Qi-J?raded submodule for this structure, it is an Qi-graded submodule for the opposite structure. It thus is itself an Qi-graded A-module.

Proof Suppose F is an Qi-graded submodule for the Qi-graded left A-module E. For a E A and f E F denote by a = I:aE21 aa and f = I: pE 21 f 13 their decomposition in homogeneous components. Then fa = I:a,pE21fpaa = I:a,/3E21(-I)(a lp )aafp. This belongs to F because each fp belongs to F [1.14]. Hence F is also a submodule of the IQEDI right A-module E. The conclusion then follows again from [1.14].

1.21 Guiding principle. The guiding principle for Qi-graded linear algebra is that in any formula in which we interchange two homogeneous Qi-graded objects, a sign (-1) (alP) appears. This (additional) sign is already visible in the definition of Qi-graded commutativity and the relation between the left and right actions of A on an Qi-graded A-module. In order to adhere to this principle, we are occasionally led to change notation (e.g., [2.12]). Its advantages will be mostly notational: additional signs in equations will be "obvious" from the order in which one writes the separate terms.

1.22 Definition. Let E be a left or right A-module, let S be any subset of E and let B be any subset of A. For a left A-module we define the subset SpanB(S) C E by n

SpanB(S)

= (~~:>iSi In E N,a i

E B,Si E S};

i=1 for aright A-module onejust replaces I:~=1 aisi in this definition by I:~=1 sia i . In case B = A we will drop the subscript B and speak of Span(S). This will occur by far the most frequently, but we will occasionally need proper subsets B of A. The subset Span(S) is obviously a submodule of E, usually called the submodule generated by S. One special case should be mentioned separately. Suppose Fi , i E I is a family of submodules of E, then the submodule Span (UiEI F i ) is usually denoted as I:iEI Fi and called the sum ofthe submodules F i . This notation is justified because obviously n

Span(UiEIFi)

= {L k=1

fik

In EN, ik

E I,fik E Fik } .

§2. (Multi-) linear maps

7

Note that I:iEI Fi is not called the direct sum [3.1] of the submodules Fi ; that is in general a completely different A-module.

1.23 Nota Bene. If E is an S2t-graded A-module, the notation SpanB( S) is ambiguous because we have to specify whether we see E as a right or as a left A-module. If, in either view, SpanB( S) is an S2t-graded submodule, then it follows from [1.20] that it is an S2t-graded submodule of the S2t-graded A-module E. However, if the submodule SpanB (S) is not an S2t-graded submodule, then we have to specify whether we view E as a right or as a left A-module [1.26].

1.24 Lemma. If E is an S2t-graded A-module and if F i , i E I is a family of S2t-graded submodules of E, then I:iEI Fi is an S2t-graded submodule of E.

1.25 Lemma. If E is an S2t-graded A-module and if SeE consists of homogeneous elements only, then Span(S) is an S2t-graded submodule of E. Proof According to [1.14] we have to prove that the homogeneous components of an element e = I:~=l aisi E Span(S) are itself in Span (S). Let a i = I:a: a~ be the decomposition of a i E A into homogeneous components, then by assumption a~si is homogeneous. Regrouping the terms in the (finite) sum e = I:i a: a~si according to parity immediately gives the desired result. ' IqEDI

1.26 Counter example. To show that the condition of homogeneity is not superfluous, consider the S2t-graded A-module E = A2 and an element s = (/-L, v) where /-L and v are two non-zero homogeneous elements of different parity. In this case the submodule F = Span ( { s }) = { (a/-L, av) I a E A} of E seen as left A-module is not an S2t-graded submodule because F n A; = {a}. Note also that, had we interpreted E as a right A-module, the submodule F would (for generic A) have been different. This shows that for general generating subsets S we have indeed to specify whether we use the left or the right A-module approach.

2. (MULTI-)

LINEAR MAPS

After the introduction of S2t- graded A-modules in §1, we introduce in this section the notion of a k-linear map, of which the more elementary notion of linear map is a special case. We show that the set ofal! k-linear maps MaPs(E1 , ... , E k ; F) is not in general an S2t-graded A-module, but that there is a natural subset Homs(E 1 , ... , E k ; F) which is an S2t-graded A-module. Two other main points of this section are that left k-linear maps

Chapter 1.

8

2t-graded

commutative linear algebra

should be seen as operating on the right rather than on the left, and that, despite their difference, there exists a natural isomorphism 'I' between left and right linear morphisms. This section ends with the definition ofdual maps.

2.1 Definition. Given Q(-graded A-modules E 1 , •.. , E k , and F, then a k-additive map ¢ : E1 x ... X Ek ----., F is said to be left k-linear if\li, \lei E Ei and \la E A we have: (i)

(ii)

¢(e1, ... , ei-1, eia, ei+1,···, ek) = ¢(e1, ... , ei, aei+1, ei+2,· .. , ek) ¢(ae1,e2, ... ,ek) =a¢(e1, ... ,ek).

The map ¢ is called right k-linear if condition (ii) is replaced by condition (ii)': (ii)'

¢(e1, ... ,ek-1,eka) =¢(e1, ... ,ek-1,ek)a

According to common usage, a I-linear map will just be called linear, bilinear stands for 2-linear and trilinear for 3-linear. We denote the set of all left k-linear maps by MapdE1, ... , E k ; F) and the set of all right k-linear maps by MaPR(E 1, ... , E k ; F). Since we will often discuss properties of left and right k-linear maps at the same time, we will use the notation Maps to denote, during the whole discussion concerned, either MaPL or MaPR. This should be interpreted as that the discussion is valid for both MaPL and MaPR. A map ¢ E MaPs(E1, ... , E k ; F) is called ofparity a E Q( if \I{3i E Q(: ¢((E1){3" ... , (Ek ){3k) C F CX +{31+"'+{3k

Note that a k-linear map ¢ is of parity a if and only if ¢( e1, ... , ek) is homogeneous when all entries are homogeneous and such that c:(¢(e1, ... , ek)) = a + I:~=1 c:(ei). By MaPs(E1, ... ,Ek;F)cx C MaPs(E1, ... ,Ek;F) we denote the subset of all maps of fixed parity a. In the set MaPs(E1' ... ' Ek; F) we define an addition by

We also define a multiplication by elements in A; for left k-linear maps the right multiplication is defined by (¢a)(e1' ... ' ek) = (¢(e1, ... ' ek))a , and for right k-linear maps the left multiplication is defined by

2.2 Lemma. With the above definitions, the set MapdE1, ... , E k ; F) becomes a right A-module and MaPR(E 1, ... , E k ; F) becomes a left A-module. Proof What we have to show is in the first place that addition and multiplication are well defined, i.e., that the result is again left/right k-linear. When that has been done, the IqEDI axioms of a left/right A-module have to be verified. All this is left to the reader.

§2. (Multi -) linear maps

9

2.3 Lemma. For a,,6

E Qi and

2.4 Definition. A k-linearmap

Homs(E 1 , ... , E k; F)

=

L

MaPs(E1"'" E k; F)a C MaPs(E1,"" E k; F) .

aE2l In the context ofmorphisms, we will also use the name Horns (E 1, ... , E k; F)a for the set MaPs(El"'" E k ; F)a (two names for the sameobjectl). Note that for Chomo)morphisms we use the same notation as for k-linear maps: Horns denotes either HomL or HomR, but never both at the same time in a discussion. In case all Qi-graded A-modules Ei coincide, say with the Qi-graded A-module E, we denote Horns (E1, ... , Ek; F) by Horns (Ek; F). Two special cases have an alternative notation. Homs(E; E) is denoted as Ends(E); its elements are called endomorphisms of E. HomL(E; A) is denoted as *E and is called the left dual of E; HomR(E; A) is denoted as E* and is called the right dual of E.

2.5 Lemma. The left/right A-module Homs( E 1 , .•• , E k; F) together with its subsets Homs( E 1, ... , E k; F)a is an Qi-graded A-module. If the abelian group Qi is finite, then Homs(E1, ... , Ek; F) = MaPs(E 1, ... , Ek; F). Proof We give the prooffor the right linear case; the left linear case is analogous. From the definition of parity it is obvious that the subsets MaPR(E1, ... , E k ; F)a are additive subgroups that satisfy the condition

It thus remains to show that each element

a+ 2::7=1 c:( ei) are all different. Hence for each a separately <Pa (e1, ... , ek) = 0 (because F is an Qi-graded A-module). Since each <Pa is k-additive, <Pa(e1, ... , ek) = 0 for all possible choices of the ei, not necessarily homogeneous. It follows that <Pa = 0, i.e., any decomposition into homogeneous components is unique. To show the second part of the lemma, choose

=

L P"""PkE2l

(
a+p'+"'+Pk

Chapter 1.

10

2t-graded

commutative linear algebra

Since ¢ is k-additive, it follows immediately that ¢ = LaE21 ¢a, which is a finite sum by hypothesis; comparison with the condition for parity a also shows immediately that c:( ¢a) = a. It thus remains to show that ¢a is k-linear. Since taking homogeneous parts is additive, the k-additivity will be obvious. To verify conditions (i) and (ii)', we note that these are additive in a E A. We thus may assume that a is homogeneous. But for homogeneous a, these relations are obvious after a reparametrization of the appropriate dummy summation variables fk IQEDI

2.6 Counter example. The finiteness condition on Q( is not superfluous because a linear map could be an infinite sum of non-zero homogeneous components, in which case it is not a morphism in our sense. The following example shows that this can indeed happen. Consider the Z-graded commutative algebra A defined by Ao = R and An = {O} for n -I=- 0 with the trivial bilinear map (min) = O. In this case Z-graded A-modules are nothing more than real vector spaces. Let us consider next the full exterior algebra E = 1\ X == EB~=o I\n X where X is an infinite dimensional real vector space; it is a Zgraded A-module with grading En = {O}for n < 0 and En = I\n X for n ~ O. In X we choose an infinite set of independent elements {Xi E X liE N } and we define the linear map ¢ : E ---. Eby its restrictions to En: e E En = I\n X =? ¢(e) = e 1\ Xl 1\ ... 1\ X n . Restricted to each En the map ¢ augments the parity by n, but since none of these restrictions is the zero map, there are infinitely many non-zero maps of different parities involved in the definition of ¢. Hence ¢ is not a finite sum of homogeneous maps, i.e., ¢ tj. Homs(E; E).

2.7 Lemma. For ¢ E HomL(E I , ... , Ek; F), ei E Ei and a E A we have the relation (a¢) (el, ... , ek) = ¢( el, ... , ek-l, ek a ). (f ¢ is right k-linear instead of left k-linear, we have the relation (¢a) (el' ... ,ek) = ¢( ael, e2, ... ,ek)' Proof We prove the left linear case, the right linear case being similar. The relation is additive in all its entries, i.e., in el, ... , ek and a but also in ¢. We thus may assume that all entries are homogeneous. It follows thatc:(¢(el, e2, ... , ed) = c:(¢)+c:(el)+" ·+c:(ek). We then compute: (a¢)(el, ... , ek)

= = =

(-1)(e(a)lc(¢))(¢a)(el, ... , ek)

=

(_1)(c(a)le(¢))(¢(el,"" ek))a

(-1) (c(a)lc(¢)) (-1) (c(a)lc(¢)+c(e,)+"'+C(ek)) a( ¢( el, ... ,ek)) (-1) (c(a)lc(e,)+ ... +c(ek)) ¢( ael, e2, ... , ek)

2.8 Corollary. For ¢ : EI x ... X Ek ---. F we have: ¢ E HomL(E I , ... , E k ; F)o if and only if¢ E HomR(E I , ... ,Ek ; F)o. We are thus allowed to drop the subscript and to write Hom(E I , ... , E k ; F)ofor both sets.

§2. (Multi-) linear maps

11

2.9 Definitions. Given two Qt-graded A-modules E and F, a map ¢ E Homs(E; F) is said to be invertible if there exists a map 'l/J E Horns (F; E) such that ¢ 0 'l/J = id( F) and 'l/J 0 ¢ = id(E). An even invertible map ¢ E Hom(E; F)o is called an isomorphism between E and F. If there exists an isomorphism between E and F, the two Qt-graded A-modules E and F are called isomorphic, denoted by E 9! F. An even invertible endomorphism of an Qt-graded A-module E is called an automorphism ofE; the set of all automorphisms of E is denoted by Aut(E). We also introduce the notion of identification as being synonym to even linear map. However, use of the word identification will usually mean that the even linear map concerned will not be noted in the sequel. For instance, if ¢ : E -+ F is an injective even linear map, we may identify E with its image and write E c F, forgetting about the actual map¢.

2.10 Remarks .• If a morphism ¢ E Horns(E; F) is bijective, it is elementary to show that its inverse is also a morphism, proving that invertible morphisms are the same as bijecti ve morphisms . • If we equip the Qt-graded A-module Ends (E) with composition of maps as multiplication, it becomes an Qt-graded ring; its subset Aut(E) becomes a group. This follows immediately from [2.3].

2.11 Discussion. For right k-linear maps [2.7] leads to the following series of relations:

= (a¢)(el,"" ek) = ¢(ael,e2, ... ,ek) ¢(ela,e2, ... ,ek) = ¢(el,ae2,e3, ... ,ek) a(¢(el,"" ek)) (¢a)(el, ... ,ek)

¢(el, ... ,ek-2,ek-l a ,ek) = ¢(el, ... ,ek-l,aek) ¢(el, ... , ek-l, ek a ) = (¢(el"'" ek))a . These relations can be summarized by saying that for multiplication by a E A, the position of commas and parentheses is of no importance. The technical way to say the same is the statement that the evaluation map HomR(E1 , ... , E k ; F) x El X ... X Ek -+ F is (k + 1 )-linear and even. Moreover, one easily verifies that the composition operator o : HomR(F; G) x HomR(E; F) -+ HomR(E; G), (¢, 'l/J) I--> ¢ 0 'l/J is an even bilinear map. For left k-linear maps the situation might seem to be not so nice. However, a small change in notation yields a similar result.

2.12 Notation. For ¢ E HomL(E 1 , ... , E k ; F) and ei E Ei we introduce an alternative notation for the evaluation of ¢ in the vectors ei by

Chapter 1.

12

2t-graded

commutative linear algebra

In this notation we interpret ¢ as an operator on the right rather than on the left. It should be read as substitution of the vectors el, ... , ek in the map ¢, or as contraction of the vectors el, ... , ek with the map ¢, a notation well known in differential geometry. The choice of the name contraction is explained in [11.5.2] (which refers to [4.10]). In case k = 1, we will also use the notation ~ell¢))

==

~(e)¢

== ¢(e) .

Parallel to the contraction/evaluation operator ~ we introduce an alternative notation for the composition of two left linear homomorphisms by

Aficionados of categories might say that <> is the composition operator in the opposite category; we will use it just as a different notation for the composition.

2.13 Discussion. With the alternative notation as introduced above, we obtain for left k-linear maps the following series of relations: a(~(el"'"

ek)¢) = ~(ael' e2,.··, ek)¢ ~(ela, e2, ... , ek)¢ = ~(el' ae2, e3, ... , ek)¢

~(el,

... , ek-2, ek-la, ek)¢ = ~(el,"" ek-l, aek)¢ ~(el, ... , ek-l, ek a )¢ = ~(el"'" ek)(a¢) ~(el,"" ek)(¢a) = (~(el"'" ek)¢)a .

Thus, here again we see that for multiplication by a E A, we can ignore the position of commas and parentheses (and of course the symbol ~). As for right k-linear maps we can state this by saying that the evaluation map ~ : El x ... X Ek x HomL(E l , ... , E k ; F) is an even (k + 1)-linear map. In the same vein, the alternative composition operator <> leads to the statement that <> : HomL(E; F) x HomL(F; G) ---- HomL(E; G) is an even bilinear map. Using the contraction operator ~ and the composition operator <> at the same time allows us to write for left linear maps: (2.14)

or

((ell¢<>'l/J)) = UeII¢»I!'l/J))·

As for right linear maps, evaluation of a composite of left linear maps becomes a mere question of parentheses (and the formal evaluation operator~, ifused). The definite advantage of our alternative notation for left linear maps is that we can adhere systematically to the guiding principle [ 1.21] that interchanging two objects gives an additional sign ( -1) (alP). With the alternative notation we avoid notational interchangings that do not involve additional signs (such as (¢a) (e) = ¢( e)a and (¢a) o'l/J = (¢ 0 'l/J)a for left linear maps).

§2. (Multi-) linear maps

13

2.15 DiscussionlDefinition. So far we have insisted on the difference between left linear and right linear morphisms, and we will continue to do so. Nevertheless there exists a natural isomorphism 'r : HomL(E; F) ----- HomR(E; F) between these two sets of maps. It is defined by the equation

(2.16) "',PE21

"',PE21

where ¢ = I: PE 21 ¢p and e = I:"'E21 e", denote the decomposition of ¢ and e into their homogeneous parts. One advantage of interpreting left linear maps as acting on the right now becomes obvious: going from the left hand side to the right hand side of (2.16), we have to interchange ¢ and e, which "explains" the sign ( -1) ("'IP) . Several verifications have to be made, the first of which is to show that 'r¢ is indeed right linear. Since (2.16) is obviously additive in e, it is sufficient to show the relation ('r¢)(ea) = ('r¢)(e)a for homogeneous e and a. If c:(a) = a and c:(e) = "(, and thus c:(ea) = a + "(, we compute:

('r¢)(ea) = I)-I)("'+'Ylp) ((eall¢p} = (I)-I)hlp)((ell¢p)))a = ('r¢)(e)a, pE21

PE21

where we have used that in the Qt-graded A-module HomL(E; F) the right and left multiplication are related to each other by (1.16). In a similar way one proves that'r itself is left linear. Since an explicit expression for its inverse is given by (( e II 'r-I'ljJ))

=

L

(-1) ("'113) 'ljJp( e",) ,

"',PE21 and since 'r obviously preserves parity, we conclude that it indeed is an isomorphism.

2.17 Lemma.

IN and'ljJ are homogeneous, 'r( ¢ <> 'ljJ) =

(-1) ("(¢)I,,(1/») ('r'ljJ)

0

('r¢).

2.18 Definition. We will call the operator 'r the transposition operator, and we will call 'r¢ the transpose of ¢. These names will be justified in §II.4. However, the reader is warned that 'r is not its own inverse: 'r is defined on left linear maps and its inverse on right linear maps!

2.19 Discussion. If E is an Qt-graded A-module, there exist canonical isomorphisms cI> s : Homs(A; E) ----- E given by ¢ !----> ¢(I). The inverse of cI> R is given by cI>RI (e) : a!----> ea. We thus find for homogeneous ¢ and a the relation

( cI>L(¢). a = ((III¢}' a = (-I)(¢la)((all¢)) = ('r¢)(a) . We conclude that cI>RI 0

= 'ljJ

0

¢ .

Similarly we define for any ¢ E HomL(E; F) the left dual map *¢ E HomR(*F; *E) by the formula \:j'ljJ E *F: (*¢)('ljJ) = ¢o1/!. One should note that taking the dual map switches sides: the dual of a left linear map is right linear and vice versa.

2.21 Proposition. (i) If ¢ is surjective, its dual map is injective. (ii) (¢o'ljJ)* =¢*o'ljJ*and*(¢o'ljJ) =*¢o*'ljJ. (iii) The map HomR(E; F) ----. HomdF*; E*), ¢ f-> ¢* is even and linear, as is the map HomL(E; F) ----. HomR(*F; *E), ¢ f-> *¢. (iv) \:j¢ E HomL(E; F) : 'r(('r¢)*) = 'ro*¢o'r-I. Proof Properties (i), (ii), and (iii) are elementary. For (iv) one has to realize which transposition operators are involved. In 'ro *¢ 0 'I-Ion the right hand side they represent the sequence F* ----. *F ----. *E ----. E*. In 'r( ('r¢ )*) on the left hand side the first one represents the switch HomL(F*; E*) ----. HomR(F*; E*) and the second one the switch HomdE; F) ----. HomR(E; F). Once one has this, the proof is elementary. IqEDI

2.22 Remark. In [2.21-iiJ we see again the advantage of the notation 0 for composition of left linear maps: we do not have to change the order of ¢ and 'ljJ in these formula;.

3.

DIRECT SUMS, FREE Qt-GRADED A-MODULES, AND QUOTIENTS

In the previous sections we have seen the construction ofthe Qi-graded A-modules submodule and morphisms; in this section we provide three new constructions of Qi-graded A-modules. In the first place the free Qi-graded A-module F( G, c) on a set G of homogeneous generators whose parity is given by c. The next construction is the direct sum of a family of Qi-graded A-modules. The third construction is that of the quotient ofan Qi-graded A-module by an Qi-waded sub module.

3.1 Construction (direct sums). If E i , i E I is a collection of Qi-graded A-modules, we define their direct sum E = EBiEI Ei as the subset of the direct product DiEI Ei

§3. Direct sums, free 2t-graded A-modules, aud quotients

15

consisting of those vectors (ei)iEI with ei = a except for finitely many indices i E I (recall that the direct sum of real vector spaces is defined exactly in this way). By defining a componentwise addition and (left) multiplication by elements of A, E becomes a left A-module. Finally we define the subsets En, a E Qt by En = En TIiEI(Ei)n. We leave it to the reader to verify that with these definitions E becomes an Qt-graded A-module. For each i E I we define maps 7ri : E ---- Ei and si : Ei ---- E by 7ri( (ej )jEr) = ei and (si(e))i = e, (si(e))j = a for j -I=- i. It follows immediately thatthe 7ri are sUljective even linear maps and that the Si are injective even linear maps, related by 7ri 0 Si = id(E i ). We will usually denote a general element (ei)iEI E E by EBiElei instead of by (ei)iEI, just to stress that it is not an arbitrary element of the direct product, but one with only finitely many non-zero entries. In case the index set I has a finite number n of elements, we will write El EB ... EB En for E = EB~=1 E i , and an arbitrary element will be denoted by el EB ... EB en' If the A-vector spaces Ei are all equal to a given one, Vi : Ei = P, the direct sum EB~=1 Ei is also denoted as pk. It is indeed the k-th power of F because for a finite index set I the direct sum equals the direct product. And if we define formally pO = {a}, then the equality p k EB pi = p k+i holds for all k, f!. E N.

3.2 Remark. One might ask why we do not define direct products of Qt-graded A-modules. There are several reasons. In the first place, ifboth the index set I and the abelian group Qt are infinite, one can easily find examples in which the direct product is not an Qt-graded A-module, the failure being that not every element can be written as a finite sum of homogeneous elements. In the second place, we never need infinite direct products. And in the third place, a direct product of finitely many Qt-graded A-modules is the same as the direct sum of these spaces.

3.3 Definition. If Pi, i E I is a family of Qt-graded submodules of a given Qt-graded A-module E, we can consider the map EBi Pi ---- E defined by (fi)iEI I--> I:i Ii- Note that this map is well defined because there are only finitely many f/s non-zero; its image is I:iEI Pi' One easily verifies that this map is even and linear by definition of Qt-graded submodules. Officially EBi Pi is never a submodule of E (but I:iEI Pi is); nevertheless, we will write E = EBi Pi whenever this map is an isomorphism onto E. As for real vector spaces, this is the case if and only if every element e E E can be written in a unique way as e = I:iEI fi with fi E Pi and only finitely many of them non-zero. If I contains two elements, we will write E = PI EB P2 . The Qt-graded submodules PI and P2 will be called supplements to each other.

3.4 Construction (free Qt-graded A-modules). Let c : G ____ Qt be a map from an abstract set G to Qt, and define G n C Gby G n = c 1 (a). We define the space P(G,c) as the set of all maps f : G ---- A with the property that f (g) = a for all 9 E G except finitely many. In P(G,c) we define an addition by (f + I')(g) = f(g) + I'(g), and a (left) multiplication by elements of A by (aJ)(g) = af(g). In this way P(G, c) becomes a left

Chapter 1. 2t-graded commutative linear algebra

16

A-module. One usually identifies each element 9 E G with the map ¢g : G -+ A defined by ¢g(g) = 1 and ¢g(h) = 0 for h -=I- g. It follows that each 1 E F(G, e) can be written in a unique way as 1 = I:9Ec Ig¢g == I:gEc f9 . 9 where Ig is defined as Ig = I(g) and where the sum is actually a finite sum by definition of F( G, e). To make F( G, e) into an Qt-graded A-module, we define F(G, e)a, (Y E Qt by

F(G,e)a = {I

E

F(G,e) I \:/g

E

G: I(g)

E

Aa-c(g)}.

1 = I:9Ec Ig . 9

has parity (Y if and only if the coefficient f9 has parity (Y e(g). In particular the element (map) ¢g has parity e(¢g) = e(g), justifying the use of the symbol e for the abstract map e : G -+ Qt. Decomposing the coefficients of an arbitrary element 1 E F(G, e) into homogeneous parts, it follows immediately that F(G,e) = EBaE21 F(G,e)a; since by construction AaF(G,e)(3 is contained in F(G, e)a+(3, we conclude that F(G, e) is an Qt-graded A-module. The Qt-graded A-module F(G, e) is usually called the free Qt-graded A-module on (homogeneous) generators G with parity e. Using the notion of Span, we can summarize the construction of F(G,e) by saying F(G,e) = Span(G). Using the notion of direct sums, we can write F(G, e) = EB9Ec F( {g}, eg), where F( {g}, eg) is the free Qt-graded A-module on the single generator 9 of parity eg = e(g). In words,

3.5 Nota Bene. We have seen that each element 1 E F(G,e) admits a unique decomposition 1 = I:9Ec Ig . 9 with Ig E A. Using the induced right action of A, it follows that there also exists a unique representation with the coefficients on the right of the ¢g, i.e., 1 = I:gEc 9 . I'g. In general the coefficients Ig and I'g are different; only if e(g) = 0 can we be sure that Ig = I'g. For any free Qt-graded A-module on a single homogeneous generator F({g},eg) we can define the map ¢ : A -+ F( {g}, eg) by a !----> ago This is a bijective linear map ofparity eg. It is an isomorphism if and only if eg = O. It follows that for eg -=I- 0 we cannot identify (in the naive and official sense of the word) the Qt-graded A-module F({g},eg) with A because left and right multiplication in the Qt-graded A-modules A and F( {g}, eg) are not related in the same way due to the difference in parity between 1 E A and 9 E F ( {g}, e g).

3.6 Corollary. Let F(G i , ei), i E I be afamily offree Qt-graded A-modules on generators G i (i -=I- j =? Gi n G j = ¢). Then EBiEI F( G i , ei) 9'! F(UiEIG i , e), where e is defined as elC i = ei.

3.7 Construction (quotients). Let E be an Qt-graded A-module and let F be an Qt-graded submodule. The quotient G = E / F with canonical projection 7r : E -+ G is defined in the sense of abelian groups, i.e., 7r(e) = 7r(e') {:} e - e' E F. As for abelian groups, the element 7r( e) E G will also be denoted as e mod F. We claim that G can be equipped with the structure of an Qt-graded A-module. Addition and (left) multiplication

§4. Tensor products

17

by elements in A are defined by 7r( e) + 7r( e') = 7r( e + e'), a7r( e) = 7r( ae). The subgroups en are defined by 7r(e) E en {:} 31 E F : e - 1 E En. It follows immediately that with this grading 7r is an even morphism. The only tricky point in proving that G is an S2t-graded A-module, is in the proof that the decomposition in homogeneous components is unique. Therefore, let us suppose I:nE217r(e n ) = owith 7r(e n ) E en (and of course only finitely many of them non-zero, which implies I:nE21 e a E F). By definition of en and the projection 7r we may assume that en E En. Since these en are homogeneous and F is an S2t-graded submodule, we have by [1.14] that en belongs to F, i.e., 7r( en) = O. This proves that the decomposition into homogeneous components is unique.

3.8 Lemma. Let E and H be S2t-graded A-modules, Fan S2t-graded submodule of E and c/J : E ---4 H a linear map that vanishes on F, i.e., c/J( F) = {O}. Then there exists a unique induced map

07r. Ifc/J has parity a, then so has ; ifc/J is a homomorphism, i.e., c/J E Homs(E; H), then so is
3.9 Proposition. Let E and F be S2t-graded A-modules and let c/J E Homs(E; F) be homogeneous. Then thefollowing assertions hold (see also [Il.3.I2J). (i) ker(c/J) and im(c/J) are S2t-gradedsubmodules ofE and F respectively. (ii) Ifc/J is even, there exists a canonical isomorphism im( c/J) ~ E / ker( c/J). (iii) If c/J is even and if ker( c/J) admits a supplement, there exists an isomorphism E ~ ker( c/J) EB im( c/J) which is completely determined by the choice ofthe supplement. Proof • For any linear map it is immediate that ker( c/J) and im( c/J) are submodules, so we only have to check the grading. Therefore, let 1 = c/J( e) E im( c/J), and decompose e into homogeneous components 1 = c/J(I:n en) = I:n c/J(e n ). Since c/J is homogeneous, the c/J( en) are also homogeneous. It follows that the homogeneous components of 1 E im( c/J) are again in im( c/J), i.e., im( c/J) is a graded submodule. For ker( c/J) the reasoning is the same: if e E ker( c/J) then 0 = I: n c/J( en). Since all c/J( en) have different parities, they must be zero, i.e., the en are in ker( c/J). The result then follows . • If c/J is even, consider the induced (even!) map

4.

TENSOR PRODUCTS

In this section we introduce the construction of a tensor product of S2t-graded A-modules. We show thatforming tensor products is associative in a very nice way and we prove the

Chapter 1.

18

2t-graded

commutative linear algebra

principal property of tensor products: transforming k-linear maps into linear maps; in other words, we prove that the tensor product is the solution ofa universal problem. We finish with the construction of the permutation operator 9't on multiple tensor products.

4.1 Construction (tensor product). Let E and F be Qt-graded A-modules and consider the set G = (U"E21E" \ {O}) x (U(3E21F(3 \ {O}) c E x F, i.e., G is the product of all non-zero homogeneous elements in E and in F. On G we define a parity map c : G -+ Qt by c(e, I) = c(e) + c(f), which is well defined because e and f are by assumption homogeneous. We thus can consider the free Qt-graded A-module F(G, c) and we recall that we have identified the abstract elements 9 E G with the elements ¢g E F (G, c). With this in mind, we define the subset S of F(G, c) as the union of two subsets: S = Sa U Sm with Sa

=

{¢(e+e''!) - ¢(e,!) - ¢(e''!),

Sm

= { ¢(ae,!) - a¢(e,!),

¢ee,!) F(3 }

¢(e'!+f') -

a, {3 E Qt, e, e' E E", f,

f'

E

¢(ea,!) - ¢(e,a!)

¢(e,!')

I

I

a,{3,,,( E Qt,a E A",e E E(3,f E F-y}.

By construction, all the elements of S are homogeneous and hence Span(S) is an Qt-graded submodule [1.25]. With this Qt-graded submodule we then define the Qt-graded A-module E QSi F as the quotient E QSi F = F( G, c) / Span( S). This Qt-graded A-module is called the tensor product of E and F. The construction of the tensor product is not complete without the definition of the map

X:ExF-+EQSiF: x(e, I)

=

L

Jr(¢(ec.,!{3)) .

",(3E21

4.2 Lemma. The map X : E x F

-+

E QSi F is even and bilinear.

Proof If e and f are homogeneous, the sum over a and (3 in the definition of X contains only one term. It follows immediately that c(x(e, I)) = c(Jr(¢(e,!))) = c(¢(e,!)) = c(e) + c(f), i.e., X is even. The bi-additivity follows easily from the definition of S. Let us show for instance the additivity in e :

",(3E21

= x(e, I) + x(e', I) ,

",(3E21

",(3E21

19

§4. Tensor products

Once we know that X is bi-additive, we notice that conditions (i) and (ii) of left bilinear maps are additive in all the coordinates ei but also in a. It thus suffices to show these relations for homogeneous ei and a. Let us show for instance condition (ii) for homogeneous a, e and J:

x(ae, f)

= 1f(¢(ae,J)) = 1f(a¢(e,J)) = a1f(¢(e,j)) = ax(e, f) ,

where we used that ¢(ae,j) - a¢(e,J) is in 8 m C Span(8) = ker 1f.

4.3 Proposition. Given any Qi-graded A-module H and any 'ljJ E Maps(E, Fi H), there exists a unique W E MaPs(E Q9 Fi H) such that 'ljJ = W0 X. If'ljJ has parity a, then so has W.

Proof Let us consider the case of right bilinear maps. Suppose'ljJ E MaPR(E, Fi H) g g has been given. Define the map":;jj: F(G,e) ---., H by ":;jj('£qEG ¢gC ) = '£qEG 'ljJ(g)c , where we have used that GeE x F. This map is by construction right linear but, more important, it vanishes on Span(S) just because 'ljJ is right bilinear. By [3.8] there exists an induced map W E Map R (E Q9 Fi H) such that ":;jj = W0 1f. If we now consider W0 X E MaPR(E, F; H) and apply it to homogeneous e and J, we find

(wo x)(e, f)

= w(1f(¢(e,n)) = ":;jj(¢(e,J)) = 'ljJ(e, f) .

Since both W0 X and'ljJ are bilinear, it follows that W0 X = 'ljJ, showing existence. To show that W is unique, let WI and W2 be two such maps. We then have two maps \iii E MapR(F( G, e) i H) defined by \iii = Wi 0 1f. We then compute:

\ii1(¢g)

= WI(1f(¢g)) = (WI oX)(g) = 'ljJ(g) = \ii 2(¢g)

,

or in other words, \iiI = \ii2' It then follows from the uniqueness in [3.8] that Wi = W2. Finally, the parity claim follows from the analogous parity claim in [3.8]. For left biling ear maps, one only has to replace the definition of":;jj by the formula (( '£gEG c ¢g II ":;jj)) = g '£9EG c ~(g )'ljJ, which is left linear by construction. IqEDI

4.4 Lemma. Given Qi-graded A-modules E and F, there exists up to isomorphism a unique pair (H == E ® F, X) with the property mentioned in [4.3J

Proof Suppose (H, x) is a second pair with this property. Since X is bilinear and since (H, X) has this property, there exists a ~ : H ---., H with X = ill 0 X. In the same way, interchanging (H, X) and (H, X), there exists a cI> : H ---., H such that X = 0 X. Moreover, since X and X are even, it follows that cI> and ill also are even. Now denote W = 0 ~ : H ---., H to obtain the equality X = W0 X. Ifwe view the X on the left of this equation as arbitrary and apply the property with the pair (H, X), we see that W is the unique linear map given by [4.3]. Since the identity is also a solution, it follows that W = 0 ill = id(H). In exactly the same way one proves that ~ 0 cI> = id(H). We thus conclude that cI> and ~ are isomorphisms between Hand H. This finishes the proof IqEDI (modulo some small details that are left to the reader).

Chapter 1.

20

2t-graded

commutative linear algebra

4.5 Discussion. The tensor product symbol @ is not only used to denote the S2t-graded A-module E @ F, it is also used to replace the map X : Ex F -+ E @ F in the following way: e @ j == X( e, f) .

There is no symbol, not on the left nor on the right, to tell us that it is either right or left bilinear. But this is justified because the map X is even, and thus is left and right bilinear at the same time. Since F(G, c:) in [4.1] is generated by the elements ¢(e,J) with e and j homogeneous, it follows that the elements e @ j generate E @ F (but note that due to the quotient, a decomposition as linear combination of this kind of elements is not necessarily unique). It follows immediately that any linear map on E @ F is completely determined by its values on elements of the form e @ j with e and j homogeneous. Finally note that for homogeneous e E E and j E F we have c:(e @ f) = c:(e) + c:(f).

4.6 Lemma. Given three S2t-graded A-modules E, F and G, there exists a canonical

isomorphism (identification) between (E@F) @G and E@ (F@G) mapping (e@f) @g toe@(f@g). Proof Let us first distinguish the various maps X (or @) that intervene in the construction of these spaces: Xl ,2 : E x F -+ E @ F, Xl2 ,3 : (E @ F) x G -+ (E @ F) @ G, X2,3 : F x G -+ F @ G, and Xl ,23 : E x (F @ G) -+ E @ (F @ G). Next consider the trilinear map ¢ E Map(E, F, G; E @ (F @ G))o defined by

¢(e,j,g) = Xl ,23(e,X 2,3(f,g)). For a fixed element 9 E G we define the map ¢g E MapdE, F; E @ (F @ G)) by ¢g (e, f) = ¢( e, j, g) (Nota Bene. In general ¢g is not right bilinear). Thus there exists a unique induced linear map ¢g E MapdE@F;E @ (F@G)) such that ¢g = ¢g 0 Xl ,2' Defining ¢( h, g) = ¢g(h), one can easily show that ¢ E MaPL(E@F, G; E@(F@G)) is even. Thus there exists a unique induced even linear map cI> : (E@F) @G -+ E@ (F@G) with the property that ¢ = cI> 0 Xl2 ,3' or in other words,

cI>(X l2 ,3 (X l ,2 (e, f), g))

= Xl,23 (e, X2,3 (f, g)) == ¢( e, j, g) .

In exactly the same way, starting with the trilinear map 'ljJ E Map(E, F, G; (E@F) @G)o defined by 'ljJ(e,j,g) = Xl2 ,3(Xl,2(e,f),g), one shows the existence ofa unique even linear map W : E @ (F @ G) -+ (E @ F) @ G with the property

w(X l ,23(e,X 2,3(f,g))) = Xl2 ,3(X l ,2(e,f),g) == 'ljJ(e,j,g). We now claim that cI> and W are inverse to each other. Therefore we note that the map @ F) @ G)o is a map that satisfies (w 0 cI>)('ljJ(e, j, g)) = 'ljJ(e, j, g). Exactly as we showed the uniqueness of the maps cI> and W, one can show the uniqueness of a map that is the identity on elements of the form 'ljJ( e, j, g). But since the identity map satisfies this property, the uniqueness proves that W 0 cI> = id((E @ F) @ G). In exactly IQEDI the same way one proves cI> 0 W = id(E @ (F @ G)).

W 0 cI> E End((E

§4. Tensor products

21

4.7 Discussion. Using the identification given in [4.6] we are thus allowed to say that the operation of taking the tensor product is associative. It follows that we can write E l @ ... @ Ek without using parentheses, and that we can speak of elements el @ ... @ ek in this multiple tensor product. Note that (by an easy induction argument) the elements of the form el @ .. , @ ek with ei E Ei homogeneous (and then c:(el @ ... @ ek) = c:(el) + ... + c:(ek)) generate the tensor product E l @ .. , @ E k . It follows that a linear map defined on E l @ ... @ Ek is completely known once we know its values on the elements el @ ... @ ek.

4.8 Proposition. Given Qi-graded A-modules E l , ... , E k, there exists, up to isomorphism, a unique Qi-graded A-module H = E l @ ... @ Ek and an even k-linear map X E Hom(E l , ... , Ek; H)o, x(el, ... , ek) = el @ ... @ ek, with thefollowing property. Given any Qi-graded A-module F and any ¢ E MaPs(El, ... , E k ; F), there exists a unique E MaPs(H; F) such that ¢ = 0 x. If¢ has parity a, then so has .

Proof The proof of the uniqueness is a word by word copy of the same proof in the case k = 2, replacing bilinear by k-linear. To prove that this unique space is E l @ ... @ Eb it suffices to show that it has the announced property. The easiest way to do this is by induction on k. The principle of such a proof has been used in the proof of [4.6], where IqEDI essentially the case k = 3 has been shown.

4.9 Corollary. Given Qi-graded A-modules E l , .,. , Ek, and F, the map ¢ !----> ¢ 0 X from MaPs(E l @ ... @ E k ; F) to MaPs(El"'" E k ; F) is a bijection. Restricted to Homs(E l @ ... @ E k ; F) it is an isomorphism onto Homs(E l , ... , Ek; F).

4.10 Examples.• The map ~ : El x ... X Ek x HomL(E l , .. . , E k ; F) -+ F given by (el, ... ,ek,¢) !----> ~(el, ... ,ek)¢ is (k + I)-linear and even and thus induces an even linear map E l @ ... @ E k @ HomL( E l , ... , E k ; F) -+ F. • The evaluation map HomR(E; F) x E -+ F given by (¢, e) !----> ¢(e) is even and bilinear and thus induces an even linear map HomR(E; F) @ E -+ F which maps ¢ @ e to ¢(e). • The composition map <> : HomL(E; F) x HomL(F; G) -+ HomL(E; G) given by (¢, 'l/J) !----> ¢ <> 'l/J == 'l/J 0 ¢ is even and bilinear. We thus have an induced even liner map HomL(E; F) @ HomL(F; G) -+ HomL(E; G). • Left multiplication by elements of A in an Qi-graded A-module E, mL : A x E -+ E is bi -additive and, by definition of multiplication, left bilinear. It thus induces a linear map mL : A@E -+ E. Since mL(1 @e) = e, the map mL is surjective. Now any f E A@E can be written (in a non unique way) as f = Li ai @ ei for homogeneous ai E A and ei E E. But then we have, using the bilinearity of @, f = Li I . ai @ ei = I @ Li aiei. It follows that m L is also injective, i.e., m L : A@ E -+ E is an isomorphism. In the same way, right multiplication mR : E x A -+ E induces an isomorphism mR : E @ A -+ E.

Chapter 1.

22

2t-graded

commutative linear algebra

In the sequel we will always identify A Q9 E and E Q9 A with E by the isomorphisms m L and mR'

4.11 NotationIDefinition. If E l , ... , Ek are Qt-graded A-modules, we will usually denote their tensor product El Q9 .•• Q9 Ek as ®~=l E i . Implicit in this notation is the order: ®7=k Ei will denote the tensor product Ek Q9 ..• Q9 E l . If all Ei coincide with E, the k-fold tensor product is denoted as ®k E, for which the above mentioned order problem does not exist. Obviously, if k = 1, we do not take a tensor product and ®l E = E. We also formally define ®o E = A. With these definitions the equality (®k E) Q9 (®£ E) = ®k+£ E holds for all k, C E N (for k or C = 0, use the isomorphism ms [4.10], see also [5.8]).

4.12 Construction. For any two Qt-graded A-modules we let R : E x F ----., F the map defined by

R(e, f) =

L

Q9

E be

(-1)(<>1f3) ff3 Q9 e<> .

<>,f3E21

We leave it to the reader to check that R is an even bilinear map. The sign, which is in agreement with our guiding principle [1.21], will be crucial. It thus induces a linear map 9't : E Q9 F ----., F Q9 E. With a slight abuse of notation, denoting the analogous map from F Q9 E ----., E Q9 F also by 9't, it is immediate that 9't 0 9't applied to e Q9 f yields e Q9 f, and thus 9't 0 9't = id, proving that it is an isomorphism. This canonical isomorphism 9't : E Q9 F ----., F Q9 E is called the interchanging map of E and F. For homogeneous elements e and f it has the property

4.13 Discussion. We have already said that we will denote the two interchanging maps

E Q9 F ----., F Q9 E and F Q9 E ----., E Q9 F both by 9't. However, we will employ the symbol 9't in an even wider context. In a multiple tensor product G Q9 E Q9 F Q9 H it is easy to construct an isomorphism onto G Q9 F Q9 E Q9 H such that an element 9 Q9 e Q9 f Q9 h is mapped to 9 Q9 9't( e Q9 f) Q9 h. By abuse of notation we will denote this isomorphism also by 9't; we will say that it is the (interchanging) map that interchanges the neighbors E and F in such a multiple tensor product. Let us denote by 9't(ii+l) the map that interchanges the i-th and i + I-st place in a multiple tensor product El Q9 • •• Q9 Ek. It is well known that if (J' E fh is a permutation of k elements, we can write it as a product of neighbor interchanges (ii + 1) (which permutes the elements i and i + 1). Taking the corresponding product of isomorphisms 9't(ii+l) gives us an isomorphism 9't0' : El Q9 ••• Q9 Ek ----., EO'-l(l) Q9 •.• Q9 EO'-l(k)' Obviously the target space EO'-l(l) Q9 ••. Q9 EO'-l(k) is completely determined by (J' and does not depend upon the way we write (J' as a product of neighbor interchanges. However, the map 9't0'

§S. Exterior powers

23

itself, which is a product of maps 9't(ii+l)' might quite well depend upon the way we write as a product of neighbor interchanges. We are thus faced with the problem: how do we investigate whether 9't" does or does not depend upon the way we write (J as a product of neighbor interchanges? Another way to pose this problem is to ask whether the maps 9't(ii+l) generate an action of6k. Phrased this way the problem is solved in the theory ofCoxeter groups [Bo, Ch IV, §1-2J. Since 6k is a Coxeter group, the isomorphisms 9't(ii+l), 1 :::; i < k generate an action of6k if and only if they satisfy the relations

(J

(9't(ii+1) (9't(ii+l)

0

0

9't(ii+ 1) =

id

1:::; i < k ,

9't(jJ+l)) 2 =

id

l:::;i
9't(i+ 1i+ 2 ) 3

= id

1:::;i:::;k-2.

Since the maps 9't(ii+ 1) are essentially all the same and of square id, the only relation that is not completely trivial in our case is the relation

We leave it to the reader to verify that our interchanging maps do indeed satisfy this relation. (In mathematical physics this relation is known as the Yang-Baxter equation.) We can summarize this discussion by saying that for each permutation (J E 6k we have a well defined isomorphism 9't" : El QSi ... QSi Ek -+ E,,-l(l) QSi ... QSi E,,-l(k); if all ei are homogeneous it is given by

where the sign depends on (J and the parities of the ei. In case all the Ei are the same S2t-graded A-module E, we obtain a representation 9't : 6 k -+ Aut(QSik E).

4.14 Example. The fact that left and right multiplication are related to each other by (1.16) can be stated as the relati ons or where we interchange the factors E and

[4.lOJ.

A in the tensor product, and where ms is as in

Chapter 1.

24

5.

2t-graded

commutative linear algebra

EXTERIOR POWERS

In this section the concepts ofQi-graded skew-symmetric k-linear maps and exterior products are introduced. The k-th exterior power I\k E of an Qi-graded A-module E is defined as a quotient of the klold tensor product ®k E; no attempt is made to identify it with an Qi-graded submodule of®k E. After we have proved the principal property ofthe exterior power I\k E: transforming Qi-graded skew-symmetric k-linear maps into linear maps, we introduce the wedge product oftwo exterior powers I\k E and 1\£ E, which takes values in I\k+£ E. This allows us to show that 1\ E = EB~o I\k E, called the exterior algebra of E, can be given a natural structure of a Z x Qi-graded commutative ring. We thus have completed afull circle: starting with an Qi-graded commutative ring A, we have created a Z x Qi-graded commutative ring 1\ E via the intermediate ofan Qi-graded A-module E.

5.1 Definition. A k-linear map ¢ : Ek ----., F, where E and F are Qi-graded A-modules, is called Qi-graded skew-symmetric iffor all homogeneous ei E E and allj = 1, ... , k - 1 we have:

i.e., the value of ¢ changes sign if we interchange two neighboring homogeneous entries, except if both entries are odd. We will denote the set of all left/right k-linear Qi-graded skew-symmetric maps from Ek to F by Mapsk(Ek; F). We also define Homsk(Ek; F) as Homsk (Ek; F) = MaPsk (Ek; F) n Horns (Ek; F); we leave it to the reader to verify that it is an Qi-graded submodule of the Qi-graded A-module Horns (Ek; F). The above definition of Qi-graded skew-symmetry is the usual one, but we can give a more intrinsic definition by using the interchanging maps D't(jJ+l) E Aut(®k E). A k-linear map ¢ is Qi-graded skew-symmetric if and only if for all j = 1, ... , k - 1 the induced linear map cI> : ®k E ----., F satisfies or equivalently

cI> 0 (id + D't(jj+l)) = 0 .

Since the D't(jj+l) generate the action ofthe permutation group 6k on ®k E, this condition is equivalent to the condition that for all 0' E 6k one has cI> oD'tO' = (-1)0'cI>, where (_1)0' denotes the signature of the permutation 0'. If we replace this relation by cI> = cI> 0 D't0" the map is called Qi-graded symmetric. However, the subject of Qi-graded symmetric maps will not be pursued in this book.

5.2 Construction (k-th exterior power). Let E be an Qi-graded A-module, then we define the Qi-graded submodule N k of ®k E as being generated by the subset Tk defined as

§5. Exterior powers

25

Since Tk satisfies the assumptions of [1.25], Nk = Span(Tk) is an S2t-graded submodule of®k E. With these preparations we define theS2t-graded A-module I\k E as the quotient

N E = ®k E / Nk It is called the k-th exterior power of the S2t-graded A-module E. Associated to this exterior power we define an even k-linear map w : Ek ---., I\k E by w = 7r 0 X, i.e., as the composition of the tensor product map X : Ek ---., ®k E with the canonical projection 7r:

®k E ---.,

N E.

5.3 Nota Bene. The above definitions and constructions have no direct meaning if k < 2. For k = 1 we let the definition ofS2t-graded skew-symmetry be an empty condition, i.e., we define Mapsk(El; F) = MaPs(E; F). This is compatible with the definition of skew-symmetric maps as satisfying o9't" = (-1)"cI> for (J E 6k, because 61 = {id} and thus the condition of skew-symmetry reduces to the empty condition = cI>. We E = E and also define Nl = {O} which, together with ®1 E = E [4.11], implies w

= id(E) : E1

---.,

N E.

N

N

For k = 0 we define No = {O}, and thus E = A, because ®o E = A [4.11]. We also define formally MaPsk (EO; F) = MaPs(A; F), but no natural justification for this definition can be found and neither do we define a map w : EO ---., 1\0 E. We thus have by definition the equalities MaPsk(EO; F) = Maps(N E; F) as well as Homsk(EO; F) = Homs(N E; F).

5.4 Proposition. Let E be an S2t-graded A-module and k :2: 1, then w : Ek ---., I\k E is S2t-graded skew-symmetric. Moreover, given any S2t-graded A-module F and any map ¢ E MaPsk (Ek; F), there exists a unique map cI> E Maps (I\k E; F) such that ¢ = cI> 0 w. If ¢ has parity a, then so has cI>. Proof Let n : Hom(®k E; I\k E)o be the by w induced linear map. In order to show the S2t-graded skew-symmetry, we have to show that no (id + 9't(jj+1)) = O. But n is the canonical projection 7r : ®k E ---., ®k E/Nk = I\k E. We thus have to show that id + 9't(jj+1) maps ®k E into Nk. But this is immediate from the definition of the generating subset Tk . If ¢ : Ek ---., F is an S2t-graded skew-symmetric map, then the induced linear map 2 : ®k E ---., F satisfies the relations 2 0 (id+ 9't(jj +1)) = O. Butthis says that 2 vanishes

=

on Tk, hence on Nk and hence induces a unique linear map cI> : I\k E ®k E / Nk ---., F. Since both operations ¢ to 2 to cI> preserve parity, the parity claim follows immediately. IqEDI

5.5 Corollary. Given S2t-graded A-modules E and F and k :2: 1, we have a bijection !----> ¢ 0 w from Maps (N E; F) to Mapsk (Ek; F). Restricted to Horns (N E; F) it is an isomorphism onto Homsk(Ek; F). For k = 0 we refer to [5.3J.

¢

Chapter 1. 2t-graded commutative linear algebra

26

5.6 Lemma. Given an Qi-graded A-module E and k ::::: 1, then, up to isomorphism, the Qi-graded A-module I\k E is the unique Qi-graded A-module enjoying the property of

[5.41. Proof The proof is an exact copy of the proof of [4.4].

5.7 Notation. According to standard usage, one uses the wedge product symbol 1\ to replace the map w in the following way. Given k elements ei E V, one writes

It follows that the Qi-graded A-module I\k E is generated by the elements el 1\ ... 1\ ek = 1l"(el Q9 ..• Q9 ek), where the ei run through E (one might even assume the ei to be homogeneous [4.7]).

5.8 Construction. For kC > 0 we define the map cI> : ®k E x ®£ E ---+ ®k+£ E by (K, L) I--> K Q9 L. If either k or C is 0, we let cI> be left/right multiplication by elements of A (remember that ®o E = A). In this way is defined for all k, C E N. Note that the induced linear map [4.3] is the canonical map which identifies (®k E) Q9 (®£ E) with ®k+£ E (and thus the maps ms are the special cases k£ = 0). We now note that both (Nk, ®£ E) and (®k E, N£) are contained in NkH. Hence there exists a unique map 1\ : I\k Ex 1\£ E ---+ I\k+£ E, called the wedge product and also denoted by a wedge, such that the following diagram is commutative:

NExNE We leave it to the reader to verify that this map is even and bilinear (but beware: the notion of Qi-graded skew-symmetry does not apply). Because of the associativity of the tensor product and, more precisely, because we may use the same tensor product symbol throughout, we are justified in the use of the wedge product symbol throughout:

just because the wedge product is induced by the tensor product. In the particular cases k or Cis 0 the definition of together with the fact that the projection 1l" : ®o E ---+ 1\0 E is the identity map id : A ---+ A immediately gives, for a E A, the equalities

§S. Exterior powers

27

5.9 Proposition. Given an Qi-graded A-module E, k, C E N, and K E I\k E and L E 1\£ E homogeneous, then

Proof Since the wedge products I\k Ex

N

N

E ----+ I\k+£ E and Ex I\k E ----+ I\k+£ E are in particular bi-additive, it suffices to show this equality for elements of the form K = eli'" . ·/\ek and L = h /\ ... /\ h, where the ei, fj E E are homogeneous. The result

now follows immediately if we realize that c:(K) = 2::7=1 c:( ei), that c:(L) = 2::~=1 c:(Jj)' and that interchanging two neighboring homogeneous elements from ei /\ fj to fj /\ ei IqEDI introduces the sign (_1)1+(c(e i )lc(fj)).

5.10 Definition. The exterior algebra 1\ E of an Qi-graded A-module E is defined as the direct sum 1\ E = EB~=o N E. We equip 1\ E with a Z x Qi-grading by defining for (k,a) E Z x Qi: (N E)a, if k ~ 0 ( 1\ E) k a = { (, ) {O} , otherwise. We also extend the wedge product to a map /\ : 1\ E x 1\ E ----+ 1\ E, as the unique even E, k, C E N, reproduces the already bilinear map which, when restricted to I\k E x defined wedge product. Since taking wedge products is associative (taking tensor products is), 1\ E equipped with the wedge product as multiplication becomes a ring. We finally define the symmetric bi-additive map U J : (Z x Qi) x (Z x Qi) ----+ Z2 (abuse of notation because the same symbol is used for the map Qi x Qi ----+ Z2) by

N

((k,a)I(C,,6))

= (kC

mod 2)

+ (al,6)

.

It is now an immediate consequence of [5.9] that 1\ E is a Z x Qi-graded commutative ring. Anticipating definition [6.1] of an Qi-graded A-algebra, the fact that 1\ E also is an Qi-graded A-module turns it into a Z x Qi-graded commutative A-algebra, explaining the algebra part in the name exterior algebra.

5.11 Remarks .• In terms of the more general definition [1.9] ofQi-graded commutativity, the symmetric bi-additive map on Z x Qi becomes the product of the functions p for Z, given by p( k, C) = (-ll£, and for Qi, given by p( a,,6) = (-1) (alP). In other words, the function p for a product grading is the product of the separate functions p. • In the particular case Qi = Z2 (which is the case we will use exclusively starting in chapter II), some authors define a (single) Z2-grading on the exterior algebra 1\ E by

(k mod 2)+p=a

(k mod 2)+p+1=a

However, such a Z2-grading is not compatible with [5.9] in the sense that there does not exist a symmetric bi-additive map on Z2 reproducing the sign (_I)(c(K)lc(L))+k£.

28

6.

Chapter 1. 2t-graded commutative linear algebra

ALGEBRAS AND DERrV ATIONS

In the previous sections we have introduced constructions of new Qi-graded A-modules out of given Qi-graded A-modules. In this section we will introduce different structures on Qi-graded A-modules. More precisely, we will introduce the notions of associative Qi-graded A-algebra and Qi-graded A-Lie algebra. Associated to the notion of an algebra is the notion ofa derivation. It is shown that there exists a natural way to identify E* as a collection of derivations ofthe exterior algebra 1\ E. This identification is the algebraic version of the contraction of a vector field with a k10rm, employed systematically in differential geometry.

6.1 Definition. Let 9 be an Qi-graded A-module and m : 9 x 9 -+ 9 an even bilinear map. The couple (g, m) is called an (associative) Qi-graded A-algebra ifm seen as multiplication is associative, i.e., if (g, +, m) is a ring. It is called an Qi-graded commutative A-algebra ifm is Qi-graded symmetric. The couple (g, m) is called an Qi-graded A-Lie algebra if the map m is Qi-graded skew-symmetric and satisfies the Qi-graded Jacobi identity, i.e., for all homogeneous e, f, g E g : [e, [f,g]] = [[e,f],g] + (_I)(c(e)lc(f)) [f, [e,g]], where we have written [e,f] for m(e,f), as we will always do for Qi-graded A-Lie algebras. This will cause no problems concerning leftlright linearity because m is even. Using the Qi-graded skew-symmetry [e,f] = -(-I)(c(e)lc(f))[f,e], this relation can also be written in the more symmetric form

(-1) (c(g)lc(e)) [e, [f, g ]]

+ (-1) (c(e)lc(f)) [f, [g, e ]] + (-1) (c(f)lc(g)) [g, [ e, f]]

= 0.

For Qi-graded A-Lie algebras one usually calls the element [ e, f] == m( e, f) the bracket of e and f. An Qi-graded submodule F of an Qi-graded A-Lie algebra E is called an Qi-graded sub A-Lie algebra of E (or a subalgebra for short) if F is stable under the bracket operation [_, _], i.e., Ve,f E F: [e,f] E F.

6.2 Examples.• The ring A itself is a (rather trivial) Qi-graded commutative A-algebra. • If E is an Qi-graded A-module, the Qi-graded A-module Ends(E) becomes an Qi-graded A-algebra when we take composition of maps as multiplication. • For any Qi-graded A-module E, the exterior algebra 1\ E is a Z x Qi-graded commutative A-algebra, where we give A a Z x Qi-grading by Ao,a = Aa and An,a = {O} whenever n =1= O. • If (g, m) is an (associative) Qi-graded A-algebra, we can introduce an even bilinear Qi-graded skew-symmetric commutator map [ _, _] : 9 x 9 -+ 9 by its action on homogeneous elements e, f E 9 :

[e,f] =m(e,f) - (-I)(c(e)lc(f))m(f,e).

§6. Algebras and derivations

29

Ifwe identify the bilinear maps m and [ _, -l with the associated linear maps 9 Q9 9 -+ g, the definition of the commutator [ _, -l can be written as [ _, -l = m - m 0 D't. We leave it to the reader to verify that (g, [ _, -l) is an Qt-graded A-Lie algebra. In particular, (g, m) is Qt-graded commutative if and only if the commutator map [ _, -l is identically zero.

6.3 Definition. Applying the construction of the last example in [6.2] to the algebra 9 = (EndR(E), 0) gives us a commutator [_, _lR on EndR(E), turning the set of right linear endomorphisms of an Qt-graded A-module into an Qt-graded A-Lie algebra. In the same way 9 = (End L (E), 0) becomes an Qt-graded A-Lie algebra with bracket [ _, _lL given on homogeneous elements by [ ¢, 'ljJ lL

= ¢ 0 'ljJ -

(-1) (c(¢)lc(1/») 'ljJ

= _(_l)(c(¢)lc(1/»)

0

¢

= 'ljJ

0

¢ - (-1) (c(¢)lc(1/») ¢ 0 'ljJ

(¢o'ljJ _ (_l)(c(¢)lc(1/»)'ljJo¢) .

If no confusion is possible, we will omit the subscripts Land R in these brackets and simply write [ _, -l.

6.4 Lemma. For¢,'ljJ E EndL(E) wehave'.r([¢,'ljJlL)

= -['.r¢,'.r'ljJJR.

6.5 Remark. The above lemma is in agreement with the usual interpretation of the transpose of an endomorphism and the commutator. Moreover, written this way, no additional sign is involved according to our guiding principle [1.21], because the elements ¢ and 'ljJ are not interchanged.

6.6 Definition. Let E be an Qt-graded A-module and m an even bilinear map E x E -+ E. A right derivation of the couple (E, m) is a right linear endomorphism of E whose homogeneous parts ¢ satisfy for all homogeneous elements e, fEE the relations

¢(m(e, f)) = m(¢(e), f)

+ (_l)(c(e)lc(¢))m(e, ¢(I))

.

For a left derivation, ¢ has to be left linear and the relation has to be replaced by ((m(e,f)II¢))

= m(e, ((III¢))) + (-l)(c(¢)lc(f))m(((ell¢)),f)

.

The set of all right derivations of(E, m) is called DerR(E, m) C EndR(E), or DerR(E) if the bilinear map m is understood. Similarly the left derivations are denoted by DerL (E, m) or DerL(E).

Chapter 1. 2t-graded commutative linear algebra

30

6.7 Lemma. The set Ders(E, m) is a subalgebra ofEnds(E) when the latter is equipped with the commutator as bracket; in particular Ders(E, m) is an Qi-graded submodule of Ends(E). Moreover, 'r¢ is a right derivation if and only if ¢ is a left derivation. Proof We leave it to the reader to verify that Ders(E, m) is an Qi-graded submodule of Ends(E). Let us show that if ¢, 'ljJ are two homogeneous right derivations, then [¢, 'ljJ] is a homogeneous right derivation as well. Thus let e, lEE be homogeneous and compute:

[¢,'ljJ]([e,/]) = (¢o'ljJ - (-1)(c(cp)lc(1/»)'ljJo¢)([e,/D

= ¢([ 'ljJ( e), I] + (-1) (c(e)lc(1/») [e, 'ljJ(f) ]) - (_1)(c(cp)lc(1/») 'ljJ([ ¢(e), I] = [¢('ljJ(e)), I]

+ (_1)(c(cp)lc(1/>(e)))

+ (_1)(c(e)lc(1/»)

+ (_1)(c(e)lc(cp)) [e, ¢(f)])

['ljJ(e), ¢(f)]

[¢(e), 'ljJ(f)]

+ (_1)(c(e)lc([cp,1/>]))

[e, ¢('ljJ(f))]

- (_1)(c(cp)lc(1/») ['ljJ(¢(e)), 1]- (_1)(c(e)lc(1/») [¢(e), 'ljJ(f)] - (_1)(c(cp)lc(1/>(e))) ['ljJ(e), ¢(f) 1 - (_1)(c(cp)lc(1/»)+(c(e)lc([cp,1/>])) [e, 'ljJ(¢(f))]

= [[¢,'ljJ](e),/] + (-1) (c(e)lc([cp,1/>J)) [e,[¢,'ljJ](f)].

IqEDI

6.8 Discussion. In ungraded Lie algebras, the auto commutator [ e, e] of an element e E 9 is automatically zero, just because of the skew-symmetry. However, for Qi-graded A-Lie algebras this is no longer true. Indeed, for a homogeneous element e, Qi-graded skewsymmetry gives us [e,e] = _(_1)(c(e)lc(e))[e,e], and thus [e,e] necessarily equals 0 only if (c:(e)Ic:(e)) = O. This phenomenon becomes clearer in the context of derivations. If ¢ E Ends(E) is a derivation, its square ¢2 = ¢ 0 ¢ is, in general, no longer a derivation. But if (c:( ¢) Ic:( ¢)) = 1, it follows from the definition of the commutator that ¢2 = ¢, ¢] is again a derivation (see [V.1.23] for an explicit example).

![

6.9 Definitions . • Let (gl, [_, _ h) and (g2, [_, -lz) be two Qi-graded A-Lie algebras and let ¢ : 91 -+ g2 be linear. The map ¢ is said to be a morphism of Qi-graded A-Lie algebras ifitis even and preserves brackets, i.e., Ve, IE 91 : ¢([ e, Ih) = [¢(e), ¢(f) lz· • A left-representation of an Qi-graded A-Lie algebra (g, [ _, _]) on an Qi-graded A-module E is a morphism of Qi-graded A-Lie algebras ¢ : 9 -+ EudL(E). Rightrepresentations are defined similarly . • For any Qi-graded A-Lie algebra (g, [ _, _ ]) we define maps ads: 9 -+ Ends(g) by

adR(e):

I

I--->

[e,/]

&

adL(e):

I

I--->

[/,e].

The fact that the bracket [ _, _ ] is even and bilinear immediately shows that ads is a well defined even morphism of Qi-graded A-modules. The Qi-graded Jacobi identity tells us

§6. Algebras and derivations

31

that ad R is a right-representation of (g, [_, _ Dand at the same time that all adR(e) are right derivations of (g, [ _, _ D. The Jacobi identity plus the Q(-graded skew-symmetry tell us the same things for ad L : it is a left-representation and im(adL) C DerL(g). These two representations are called the (left- and right-) adjoint representations of the Q(-graded A-Lie algebra (g, [ _, _ D.

6.10 Lemma. For any Q(-graded A-Lie algebra g, the left and right adjoint representations are related by '.r 0 ad L = - ad R. Proof This is a direct consequence of the Q(-graded skew-symmetry and [6.4].

IQEDI

6.11 Remark. For generic Q(-graded A-Lie algebras, the condition that a morphism of Q(-graded A-Lie algebras preserves the brackets "implies" that it must be even, just by counting parities: ¢( [ e, f h) = [¢( e), ¢(f) h "implies" that we must have the equality c(¢) + c(e) + c(J) = c(¢) + c(e) + c(¢) + c(J). However, this argument is not valid whenever the zero element is involved, e.g., ifboth brackets are identically zero.

6.12 Remark. We have restricted our attention to even bilinear maps m : 9 x 9 ---., g, because we will need no others. However, in the literature one also finds non-even homogeneous maps m, especially in the context ofQ(-graded A-Lie algebras. Let us give the precise definition. An Q(-graded A-Lie algebra of parity a is an Q(-graded A-module 9 together with a bi-additive map [ _ Iml -1 : 9 x 9 ---., g, (e, f) I--Y [e Iml f 1satisfying:

D

(i) for homogeneous e, f E 9 : c( [ e Iml f = c( e) + c(J) + a (the bracket has parity a), (ii) for all A, f.L E A and e, f E g: [Ae Iml f f.L 1 = A [ e Iml f 1f.L (left linearity in the

first argument and right linearity in the second), (iii) for homogeneous e,f E g: [elmlfl = -(-l)(c(e)+<>lc(f)+<»[flmlel (Q(graded skew-symmetry of parity a), (iv) for homogeneous e, f, 9 E 9 : [e Iml

[f Iml 9 11 = [[ e Iml f llml 9 1+ (-1) (c(e)+<>lc(f)+<» [f Iml [e Iml 9 11

(the Q(-graded Jacobi identity of parity a). Using (iii), (iv) can be written in the more symmetric looking form (_l)(c(e)+<>lc(g)+<» [[ e Iml fllml 9 1

+( _l)(c(f)+<>lc(e)+<» [[ f Iml 9 llml e 1 +( -1) (c(g)+<>lc(f)+<» [ [g Iml e llml f 1 = 0 .

Chapter 1.

32

2t-graded commutative

linear algebra

If (g, [ _ Iml _ D is an Qt-graded A-Lie algebra of parity a, we can define the left and right adjoint maps ads: 9 -+ Ends(g) by adR(e):

I!---->

[elml/l

and

adL(e):

I

f->

[/lml el·

If we drop the condition that a morphism of Qt-graded A-Lie algebras must be even (but still requiring that it be homogeneous), and if we take the parity of the bracket [ _ Iml -l into account in the definition of a derivation (e.g., a right derivation should satisfy ¢([elmI/D = [¢(e) Iml/l + (-l)(c(C/»lc(e)+a)[elml¢U)]), then all results given in [6.9] and [6.10] remain valid. We note that, given conditions (i) and (ii), the conditions (iii) and (iv) are the only natural ones such that [6.9] and [6.10] remain true. If one does not like conditions (i) and (ii), one can easily transform them into, for instance, a right bilinear map iii : 9 x 9 -+ 9 by iii ( e,!) = (-1) (alc(e)) [e Iml I l. The conditions (iii) and (iv) then transform accordingly. In a similar spirit we can define an associative algebra of parity a as an Qt-graded A-module 9 with a map (_Iml_) : 9 x 9 -+ 9 satisfying (i), (ii), and (iii)' for all e, I, 9 E g: ((elml!)lmlg) = (elmIUlmlg)). And as for the even case, one can construct an Qt-graded A-Lie algebra of parity a out of such an associative algebra by [e Iml I 1 = (elml!) - (_1)(c(e)+a 1c(f)+a) Ulmle). If one tries to generalize even more by dropping the homogeneity condition on the bracket (or the multiplication), then the constructions become rather ugly and lose a lot of their nice properties. For instance, if one maintains that the Qt-graded Jacobi identity should tell us that ~ e I adR )) = [e Iml -l is a derivation, then the symmetric formulation of the Qt-graded Jacobi identity will not hold, and adL(e) will not be a derivation.

6.13 Discussion. The conditions (i)-(iv) of an Qt-graded A-Lie algebra of parity a become natural if we imagine that 9 is just an Qt-graded A-module, that we have an Qt-graded A-Lie algebra (~, [ _, _ ]) with even bracket (as in [6.1]), and that we have a homogeneous invertible left linear map ¢ : 9 -+ ~ of parity a. We then can induce a bracket [ _ Iml -l on 9 by the formula [elml/l =¢-l([¢(e),¢U)])· (N ote that we used right evaluation of ¢ instead of left evaluation as we should have done for left linear maps.) This transported bracket indeed satisfies the conditions (i)-(iv), turning (g, [ _ Iml _]) into an Qt-graded A-Lie algebra of parity a and turning ¢ into a bracket preserving map. Another way to make essentially the same observation is to define the parity shift operation. Given an Qt-graded A-module E and an a E Qt, we define a new Qt-graded A-module EDa as follows: as a left A-module EDa is the same as E, but the Qt-grading is given by (EDa)p = E p- a , i.e., we "lower" the parity by a. With this new grading, EDa will (in general) certainly not be equal to E as right A-module. Now if (g, [ _ Iml _ ]) is an Qt-graded A-Lie algebra of parity a, (gDa, [ _ Iml _ ]) is an Qt-graded A-Lie algebra of parity 0, i.e., an Qt-graded A-Lie algebra in the sense of [6.1]. The link with the previous paragraph is the observation that the identity map ¢ : 9 -+ gDa, e f-> e is an invertible left linear map of parity a.

§6. Algebras and derivations

33

6.14 Lemma. Let E be an Qi-graded A-module and D a (left or right) derivation of its exterior algebra (/\ E, 1\). Then DINE = 0 and D is uniquely determined by its restriction DIN E" Proof From the derivation property and the equation 11\ 1 = 1 . 1 = 1 E A ~ /\0 Ewe deduce that D( 1) = 0, and hence DIN E = O. Since an element of /\ E is a finite sum of terms of the form el 1\ ... 1\ ek, the second assertion follows from the derivation property of D. IqEDI

6.15 Corollary. If D is a derivation of (/\ E, 1\) of parity (n, a) with n zero map. Proof D(N E) C N+n E and

N E = {O} whenever k

<

-1, it is the

< O.

6.16 Proposition. There exists a unique even homomorphism of Qi-graded A-modules ~ : E* -+ DerR(/\ E) such that ~(¢) is the unique right derivation whose restriction to N E ~ E is given by e I--Y ¢(e) E A ~ N E. IN has parity a, ~(¢) has parity (-1, a) in the Z x Qi-graded A-Lie algebra DerR(/\ E). Proof Uniqueness is guaranteed by [6.14], so let us show existence. We construct ~(¢) by its restrictions to /\k E. To that end we define a k-linear Qi-graded skew-symmetric map Ek -+ /\k-l E by (6.17)

(el, ... , ek)

I--Y i-I

("01

k

I)-I)

I:"j)

j=l

e"ll\···l\e"i_ll\¢"O(e".)e"i+ll\···l\e"k'

where ¢ = I:"o cP"o and ei = I:", e"i are the decompositions in homogeneous parts. We leave it to the reader to verify that this map is indeed k-linear and Qi-graded skewsymmetric. We thus have an induced linear map ~(¢) : /\k E -+ /\k-l E. Since this map is obviously of parity a whenever ¢ is, the full map ~(¢) E EndR(/\ E) is of parity ( -1, a) whenever ¢ is of parity a. It thus remains to show that ~(¢) satisfies the property of a right derivation, i.e., for homogeneous ¢ and A: ~(¢)(A 1\ B)

=

(~(¢)A) 1\ B

+ (_I)(c(cp)lc(A)) A 1\ ~(¢)B

.

Since this relation is bilinear in A and B, it is sufficient to verify it for elements of the form A = el 1\ ... 1\ ek and B = ek+l 1\ ... 1\ ek+£. But for these elements it is an easy consequence of the defining formula (6.17) for ~(¢). IqEDI

6.18 Corollary. For any two ¢, 'ljJ E E* the commutator [~( ¢), ~('ljJ) 1 E Der R(/\ E) is zero. Proof [~( ¢), ~('ljJ) 1 maps

N E to N-2 E. The result now follows from

[6.15].

IqEDI

34

Chapter 1. 2t-graded commutative linear algebra

6.19 Nota Bene. If in this corollary both ¢ and 'ljJ are homogeneous, the conclusion translates as ~(¢) 0 ~('ljJ) = -(-1) ("(1»1"(1/») ~('ljJ) 0 ~(¢), the "extra" minus sign coming from theZxQt-grading: (c:(~(¢))Ic:(~('ljJ))) = ((-1,c:(¢))1(-1,c:('ljJ))) = 1+(c:(¢)1c:('ljJ)).

6.20 Discussion. As said in [6.16], ~ is an even morphism of Qt-graded A-modules. But the target module is Z x Qt-graded, not only Qt-graded. And, just as the Qt-graded ring A is Z x Qt-graded (by being {OJ for Z-parities different from 0), we can turn E* into a Z x Qt-graded A-module. However, here we have a choice! One obvious choice is to define (E*)n,n = {O} whenevern -I=- O. In that case ~ becomes a morphism of Z x Qt-parity (-1,0); in particular it is no longer even. Another choice is to define (E*)n,n = {O} whenever n -I=- -1 (Nota Bene. This is a valid definition because elements of E can not be multiplied). With this definition ~ is an even morphism of Z x Qt-graded A-modules. Both approaches have their merits: from a theoretical viewpoint the second approach is nicer because one gets even maps; moreover, the second approach concurs with the idea that elements of En ~ (N E)a have parity (1, a). On the other hand, the first approach is the more practical one: if ~ has parity (-1,0) and ¢ E E* has parity (0, a), then the image ~(¢) hasparity (-l,a) = (-1,0) + (O,a).

6.21 Remark. The choice of the symbol ~ for the map E* -+ DerR(/\ E) is motivated by the applications in differential geometry. There we mainly look at right derivations of the exterior algebra /\ *E, where E stands for the module of vector fields. In [7.26] we will show that there is a canonical identification E -+ (*E)*, and thus the combination with [6.16] gives us a map E -+ (*E)* -+ DerR(/\ *E). The image of an element e E E *E ~ *E evaluates exactly as ~(e)¢ == ((ell¢»,justifying our choice of applied to ¢ the symbol ~ for the map E* -+ DerR(/\ E).

EN

7.

IDENTIFICATIONS

In previous sections we have given constructions of Qt-graded A-modules out of other Qt-graded A-modules, such as homomorphisms, direct sums, tensorproducts, and exterior powers. Some of these constructions "commute" under certain circumstances, i.e., applying these constructions in a different order gives results that are canonically isomorphic. In this and the next section we will provide a number of such identifications, of which a summary can befound in [8. 18J. In this section these identifications are introduced as maps, in the next section the technical proofs that they are isomorphisms (and the conditions neededfor that) are given. All definitions of these identifications are straightforward, although we are naturally led to introduce a non-standard ordering in the definition of the tensor product of maps [7.12J. This new ordering avoids a sign in an identification

§7. Identifications

35

involving the exterior power [7.21 J, a sign that is usually present in the ungraded case [7. 22J. This section and the next one are purely technical, and their contents are not neededfor the understanding of the next chapters. The reader can thus safely skip these two sections at a first reading; looking at the summary [8.18J will give him a fair idea of the contents (s )he skipped.

7.1 Discussion. In what follows we will establish identifications between various spaces. Some of these will always be isomorphisms, some will be isomorphisms only with additional conditions. In establishing an identification, we will denote it by the symbol ). Afterwards we will usually omit this symbol and identify the two spaces without noting the explicit identification.

7.2 Construction (sum of maps). Let F and E i , i E I be S2t-graded A-modules, and let
iEI

iEI

This is well defined because only finitely many ei are different from zero, rendering the sum a finite one. We thus obtain a map) : DiE I MaPs(Ei; F) -+ MaPS(EBiEI E i ; F), (
36

Chapter 1. 2t-graded commutative linear algebra

isomorphism. If a finite set I contains n elements, we denote the map I:~=1 ¢Ji also by

(PI + .. , +
7.3 Corollary. Let E i , i E I be Qi-graded A-modules. The map ( I:iEI
isjinite, these identifications are isomorphisms.

7.4 Nota Bene. The notation I:~=1
7.5 Example. Let F and E i , i E / be Qi-graded A-modules. If Qi is finite, the sets Maps and Horns coincide [2.5]. Since for infinite I the sets DiEI and ffJiEI do not coincide, it is easy to construct examples in which the map J : ffJiEI Homs(Ei; F) ----., Horns (ffJiEI E i ; F) is not surjective. On the other hand, it is not obvious that the image of DiE I Horns(Ei; F) under J is not necessarily contained in Horns (ffJ iEI E i ; F). To show that that can happen, we recall example [2.6]. There we defined the Z-graded commutative algebra A to be the trivially graded A = R and we argued that Z-graded A-modules are just real vector spaces. We now choose an infinite dimensional real vector space X and we define the Z-graded A-module En by (En)o = An X and (En)k = {O} for k -I=- O. We also define the Z-graded A-module F = ffJ:=o An X with grading Fk = {O}for k < 0 and Fk = Ak X for k ::::: O. We finally define the maps
7.6 Construction (direct sum of maps). Let E and Fi , i E I be Qi-graded A-modules. We define the identification J : ffJiEI Homs(E; Fi ) ----., Homs(E; ffJiEI Fi ) by the formula

This is well defined because there are only finitely many maps
§7. Identifications

37

7.7 Construction (product of maps). Let Ei and Fi , i E I be Qt-graded A-modules, and let ¢Ji E MaPs(E i ; Fi ), i E I be linear maps. With these we define a new linear map TIiEI ¢Ji E Maps(EBiEI E i ; EBiEI F i ) by the formula

II ¢i : EBiei

I-->

EBi¢i(ei) .

iEI This is well defined because only finitely many ei are different from zero. We thus obtain a map J : TIiEI MaPs(Ei; Fi ) ---., Maps (EBiEI Ei; EBiEI Fi ), (¢i)iEI I--> TIiEI ¢i. We leave it to the reader to show that this J is injective. In case I is finite, we denote the map TI~=I ¢i also by ¢I x ... X ¢n.

7.8 Lemma. The linear map TIiEI ¢i is injective (surjective, of parity a) ifand only ifall maps ¢i are injective (surjective, of parity a). In particular we obtain an injective map J : TIiEI Aut(Ei) ---., Aut(EBiEI Ei ), (¢i)iEI I--> TIiEI ¢i.

7.9 Remark. If Ei and Fi , i E I are Qt-graded A-modules, we can combine the operations of sum of maps and direct sum of maps to obtain an injective map

II EBMaps(Ei; F

j ) ---.,

Maps(EBEi ; EB Fj

iEI JEI

iEI

) .

JEI

Restriction to the "diagonal" TIiEI MaPs(Ei; Fi) would have given us the product of maps. There is one subtle point in this discussion: one has to be very careful with the order in which one applies the operations sum of maps and direct sum of maps, because the operations EBiEI and TIjEI do not commute in general.

7.10 Construction (homomorphisms on tensor products). Let E, F, and G be Qt-graded A-modules. We want to show that HomR(E QSi F; G) and HomR(E; HomR(F; G)) are isomorphic, as well as HomL(EQSiF; G) and HomL(F; HomL(E; G)). In the right linear casewedefinethemapJ: HomR(EQSiF;G) ---., HomR(E;HomR(F; G)) by the formula J(¢)(e) : f

I-->

¢(e QSi f) .

This is well defined because for each e E E the map J (¢) (e) is indeed right linear. U sing that e.\ QSi f = e QSi At, one can show that J (¢) is right linear. If ¢, e, and f are homogeneous, we obviously have c:(J (¢)( e) (f)) = c:( ¢( e QSi f)) = c:( ¢ ) + c:( e) + c:(J). It follows that J is an even linear map. To show that it is bijective, let us construct its inverse. Given 'ljJ E HomR( E; HomR( F; G)), we define a right bilinear map :($ : E x F ---., G by :($( e, f) = 'ljJ(e) (f). This gives us an induced right linear map ¢ : E QSi F ---., G. We leave it to the reader to verify that the map 'ljJ I--> ¢ is actually the inverse to J. We conclude that J is an isomorphism. In the left linear case, we define J : HomL(E QSi F; G) ---., HomL(Fj HomL(E; G)) by the formula U IIJ( ¢))) : e I--> (( e QSi f II ¢» . In the same way as for the right linear case one can show that this is an isomorphism.

38

Chapter 1. 2t-graded commutative linear algebra

7.11 Construction (homomorphisms as tensor products). Let E, F, and G be S2t-graded A-modules. We will construct identifications J : FQSiHomR(E; G) ----., HomR(E; FQSiG) and J : Hom£(E; G) QSi F ----., Hom£(E; G QSi F). In the right linear case we define J as follows. For f E F and ¢ E HomR(E;G) we define iJ!(f,¢) E HomR(E;F QSi G) by the formula iJ!(f, ¢) : e I--> f QSi ¢(e) . Obviously iJ!(f,¢) is right linear, i.e., iJ!(f,¢) is well defined. Moreover, the map iJ! : F x HomR(E; G) ----., HomR(E; FQSi G) is left bilinear and even. The identification J is the induced even linear map F QSi HomR(E; G) ----., HomR(E; F QSi G). In a similar way we define theidentification J : Hom£( E; G) QSi F ----., Hom£( E; G QSi F) as the even linear map induced by the bilinear map iJ! : Hom£(E; G) x F ----., Hom£(E; G QSi F) given by the formula iJ!( ¢, f) : e I--> (( e II ¢)) QSi f. Taking G = Agives us an identification *E QSiF ----., Hom£(E; F)in which the element ¢ QSi f is identified with the map ((ell¢QSif)) = ~eII¢)·f· In a similar way we obtain an identification F QSi E* ----., HomR(E; F) in which f QSi ¢ is identified with the map e I--> f . ¢( e). Nota Bene: One might be tempted to think that, with these identifications, the transposition map is just the interchanging operator in disguise. But things are not so easy. Applying 9't to HomL (E; F) ~ *E QSi F gives us the space F QSi *E, and we then still have to apply the transpose map 'I': *E ----., E* to get F QSi E* ~ HomR(E; F).

7.12 Construction (tensor product of maps). If we combine [7.10] and [7.11] we obtain in particular the following identification concerning two S2t-graded A-modules E and F "

F* QSi E* ~ HornR(E; F*)

which is given on elements e E E,

f

== HomR(E; HomR(F; A)) ~ HomR(E QSi F; A) == (E QSi F)*

,

E F, 'ljJ E F*, and ¢ E E* by

('ljJQSi¢)(eQSif) ='ljJ(¢(e)f) = (-l)(c(1j»lc(¢(e)))¢(e)'ljJ(f). In order to generalize this identification to arbitrary homomorphisms, we consider the map 8 : HomR(E 2; F 2) x HomR(E l ;Ft) ----., HomR(E l QSi E 2; Ft QSi F 2) defined by 8(¢2, ¢I) : (el QSi e2)

I-->

L

(-l)(aIP+-Y)¢lP(el-y) QSi ¢2a(e2) .

a,p,-yE21 We leave it to the reader to verify that 8( ¢2, ¢I) is indeed a well defined right linear map (the sign will be crucial), and that 8 is an even (right) bilinear map. We thus obtain an

§7. Identifications

39

induced identification J : HomR(E 2 ; F2 )@HomR(E I ; Fd ---., HomR(E I @E2 ; F I @F2 ) such that for homogeneous elements we have

We can easily generalize this by induction to two families (Ei)~=l and (Fi)~=lof Qt-graded A-modules, to obtain an identification

I n n J : Q9HomR(Ei ; Fi ) ---., HomR(Q9E i ; Q9 Fi ) i=1

i=n

i=1

such that for homogeneous elements we have

I: (C:(cpi)IC:(cpj (ej ))) (J(¢n @ ... @¢d)(el @ ... @e n ) = (-l)i>j

¢1(eI) @ ... @¢n(e n ) .

Similarly an identification J : ®7=n Hom£(E i ; Fi ) ---., Hom£( ®~=I E i ; ®~=I Fi ) is constructed such that for homogeneous elements we have

I: (C:(cpi)IC:(cpj(ej ))) ((el @ ... @enIIJ(¢n @"'@¢I)))

=

(-l),<j

((ed¢d @ ... @ ((en I ¢n)) .

Of course there is no compelling reason to use the reverse order in the tensor product

®7=n HomL(E i ; Fi), except that [7.10] and [7.11] suggested it that way. There is also a kind of natural explanation: in the expression (J(¢n @ ... @ ¢I)) (el @ ... @ en) the ¢I is next to el and is ready to be applied. Then we have to shift the result out of the way and we are ready to apply ¢2 to e2, and so on. One could look at it as if two rows of dancers meet head on, forming a new row of pairs. It also is more natural in the proof of sufficient conditions for these identifications to be isomorphisms [8.12]. Moreover, it fits better when discussing duals of exterior powers [7.16] (see also [7.22]). Taking all Fi to be A gives us identifications *En @ '" @ *EI ---., *(EI @ '" @ En) and E~ @ ... @Ei ---., (E I @ ... @En)*.

7.13 Remark. If all ¢i E *Ei are even, and in particular also right linear, then the images JL(¢n @ ... @ ¢I) E *(EI @ .. , @ En) and JR(¢n @ ... @ ¢I) E (E I @ .. , @ En)* coincide (and are even), as can be verified easily.

7.14 Lemma. If'lj; E Homs(E; F) is bijective, then ide @'lj; E Homs(E@G; F@G) is bijective with ide @'lj;-I as inverse. Similarly the map 'lj; @ide is bijective with 'lj;-I @ide as inverse.

Chapter 1.

40

2t-graded

commutative linear algebra

7.15 Construction (maps on exterior powers). LetE andFbe Ql-graded A-modules and let ¢ : E ---. F be right/left linear. We want to investigate when this map induces a linear map I\k E ---. N F. According to [7.12] we have a map ¢ Q9 ••. Q9 ¢ : ®k E ---. ®k F, which we can compose with the canonical projection onto I\k F to yield a linear map cI> : ®k E ---. I\k F. This map induces a linear map I\k E ---. I\k F if and only if the map cI> satisfies therelations cI> 0 D't(ii+l) = -cI> [5.1]. It is fairly easy to see that this is the case if ¢ is even. We conclude that for an even linear map ¢ : E ---. F there exists an induced even linear map I\k ¢ : I\k E ---. I\k F such that for ei E E one has

N

N

E = E [5.3], we quite naturally find that ¢ = ¢. For k = 0 we formally Since defined N E = A; for maps we now formally define N ¢ = id(A). An immediate consequence of these definitions and the definition of the exterior product [5.8] is that for A E I\k E and BEN E we have

It should be obvious that if ¢ is bijective, i.e., an isomorphism, then I\k ¢ also is bijective with inverse I\k (c/J-l). In categorical language one would say that I\k is a functor, but one which we only apply to even morphisms.

7.16 Construction (dual of exterior powers). Let E be an Ql-graded A-module. We will construct identifications J : I\k (E*) ---. (N E)* and J : I\k *E ---. *(I\k E). There are two ways to construct this isomorphism, a fast way and a pedestrian way. We start with the pedestrian way for the right linear case. We denote by J 0 the identification J 0 :~®k(E*) ---. (®k E)* of [7.12], and we introduce the modified permutation operators D'to" = D'tTO"T [4.13], where T E 6k is the fixed permutation T( i) = k + 1 - i (the one which reverses the order of k elements). For homogeneous elements ei this implies

where the sign is determined by the permutation (J and the parities of the ei. We now claim that we have the following equality of maps from ®k E to A : (7.17) Since the neighbor interchanges generate 6k, it suffices to check this for (J of the form (J = (ii + 1). The essential computation to verify is the case k = 2, which is left to the reader. We now define the skew-symmetrization operator A. by the formula

§7. Identifications

41

t

where (-1 denotes the sign of the permutation 0'. This skew-symmetrization operator has the important property that for any permutation 0' we have (7.18)

With these preparations we can construct the identification J. We first define the map'ljJ = J 0 oi;h : ®k E* ---., (®k E)*. Using (7.17) we obtain for any cI> E ®k E* the equality 'ljJ(cI» = J 0 (cI» oA. It then follows from (7.18) that 'ljJ(cI» : ®k E ---., A is Qt-graded skew-symmetric [5.1], and thus i~duces a map w(cI» E (I\k E)*. On the other hand, it also follows from (7.18) that W(D'tO'-l (cI») = (_1)0' . w(cI», i.e., the map W : ®k E* ---., (I\k E)* is Qt-graded skew-symmetric. We thus obtain an induced map J : I\k E* ---., (I\k E) *. This identification is given explicitly by the formula (7.19)

J(¢k /\ ... /\ ¢d(el /\ ... /\ ek)

L

=

(_1)0'. J 0 (¢k @ ... @¢I)(D'tO'(el @.··@ek)) .

O'E6 k

So far the pedestrian way to define J. The fast way uses the concept of derivations. Recall [6.16] that for ¢i E E* we have defined a derivation ~(¢) of bi-degree (-1, c:( ¢)) on the exterior algebra 1\ E. In particular the composite X( ¢I, ... ,¢k) = ~(¢I) 0 • • • 0 ~(¢k) restricts to a map I\k E ---., N E = A. It follows from [6.16] and [6.18] ([6.19]) that X : (E*)k ---., (I\k E)* is k-linear and Qt-graded skew-symmetric. We thus have an induced map J : E* ---., (I\k E)*. In [7.21] we will prove that these two definitions coincide, i.e., that we are allowed to use the same symbol. That we obtain equality of both definitions, and not an equality up to a sign, is due to the reversed order in the tensor product in [7.12]. In the left-linear case the construction is similar. The pedestrian way uses the identification J 0 : ®k(*E) ---., *(®k E) [7.12] and the fast way uses the left-derivations equivalent of [6.16] (which gives a map *E ---., Derdl\ E)).

N'

7.20 Example. Let ¢I, ¢2 E E* and el, e2 E E be homogeneous, then the identification between 1\2 E* and (1\2 E)* gives us:

J(¢2/\ ¢t)(el /\ e2) = J 0 (¢2 @¢I)(el @ e2) - J 0 (¢2 @ ¢dD't(12)h @e2) = (-I)(c(¢2)lc(¢1(e Il ))¢I( e l)¢2(e2)(-I)(c(¢1)lc(e 1))¢2(el)¢I(e2)

.

Apart from the additional signs, this formula can be seen as the ordinary determinant ofthe matrix ( ::

i::i :~i:~U· More generally, the value ofJ(¢I/\· .. /\ ¢k) (ek /\ ... /\ el) can be

seen as a generalization of Det( ¢i (ej)). However, this generalization of the determinant should not be confused with the Berezinian or graded determinant to be defined in [11.5.16]. The latter is related to the group Aut (E), whereas the former bears no natural relation to this group.

Chapter 1. 2t-graded commutative linear algebra

42

7.21 Proposition. Let ¢I, ... ,¢k E E* be arbitrary, then as maps from I\k E to A we have the identity J(¢I/\OO'/\¢k) =~(¢I)o", o~(¢k)'

where the

~(¢i)

are the derivations defined in [6.16J.

Proof Since both expressions are left k-linear in the ¢i, it suffices to check the identity for homogeneous ¢i. Since elements el /\ ... /\ ek with ei E E homogeneous generate I\k E, it again suffices to check this identity on elements of this form. Now using the derivation property of ~(¢), it is easy to check the following identity: k

~(¢)(el/\"'/\ ek) =

I)-I)

(i-I)+ I:(c(ei)lc(ej))

j
¢(ei)el'" /\ ei-l /\ ei+I/\"'/\ ek .

i=1 Using the map idk-I Q9 ¢ : ®k E -+ A Q9 ®k-I E ~ ®k-I E and the canonical projection 7r : ®k E -+ I\k E, where we introduced the abbreviation id k for the identity map on ®k E, we can write this identity as

(i-I)+ I:(c(ei)lc(ej))

k

(7ro(idk-IQ9¢))(I)-I)

j
eiQgeIoo.Qgei-IQgei+lQ9oo.Qgek).

i=l We now introduce for any 1 ::::: i ::::: j ::::: n the permutations O'{ E 6 n defined as the cyclic permutation from i to j, i.e., (p) = pfor p < i or p > j, O'{ (j) = i, and O'f (p) = p + 1 for i ::::: p < j. Using the definition ofthe operators D't" [4.12], [4.13] one can easily show that

0';

k

D't,,; (el Q9 ... Q9 ek) =

I:(c(e,)lc(ej))

I) -1)j
ei Q9 el'" Q9 ei-I Q9 ei+1 Q9 ... Q9 ek .

i=1 U sing that the el Q9 ... Q9 ek generate

®k E

and that (-I)"i

j .

.

= (-1 F-', we find

k

~(¢) O7r = 7rO (id k- I Q9 ¢) 0 (I)-I)"; D't"J i=1 as maps from ®k E to I\k-l E. Composing on the left with ~('ljJ) and using the same formula with k replaced by k - 1 gives us: k-I

k

~( 'ljJ) ~(¢ ) O7r = 7r (id k- 2 Q9 'ljJ) (~) -1) ,,{ D't,,{ ) (id k- I Q9 ¢) (~) -1)"1 D't"i) 0

0

0

0

0

j=1

i=1 k

=

7r 0

(id k- 2 Q9 'ljJ Q9 ¢)

0

k

(L L( -1)"~'''; D't"~'''1) . j=2i=1

§7. Identifications

43

To obtain the second line we used that (id k- 2 @'ljJ) 0 (id k- 1 @¢) = id k- 2@'ljJ@¢ and that 9't"j 0 (id k- 1 @¢) = (id k_ 1 @¢)o9't"j+l becausethemapidk_ 1@¢: ®k E -+ ®k-l E 1 2 eats up the first factor. By induction we get the identity

It now suffices to note that any permutation (J E 6 k has a unique representation as (J = (J~k ••• (J~', where i j is the position of (J-l (j) in the list 1,2, ... , k in which the items (J-l(l), ... , (J-l(j - 1) are (already) put in front. Since there are already j - 1 elements in front which are different from (J-l (j), this position is greater or equal to j, so i· (J/

makes sense. We conclude that

where we have omitted the 7r at the right hand side because there 7r : ®o E -+ 1\0 E is the identity map A -+ A. Applying this to el @ ... @ ek and comparing with (7.19) finishes IqEDI the proof.

7.22 Remark. The fact that we have no sign at all in [7.21] is a direct consequence of the reverse order in our definition of J 0 in [7.12]. In ordinary exterior powers (over R), the identification is usually given as

and then one gets the relation (~(¢l)

0

... 0

~(¢k))(el /\ ... /\ ek)

= (_1)k(k-l)/2 Det(¢i(ej)) = (_1)k(k-l)/2J(¢1 /\ ... /\ ¢k)(el/\'" /\ ek) .

7.23 Corollary. For ¢ E E* we can define a left linear map £ (¢) : I\k E* -+ I\k+l E* by ((AII£(¢D = A /\ ¢. With the right linear map ~(¢) : N+l E -+ N E defined in [6. 16J, we have a commutative diagram £(
-------;

I\k+l E*

1J -------;

L(
(N+l E)* ,

44

Chapter 1. 2t-graded commutative linear algebra

i.e., under the identifications /',t E* ~ (I\k E)* and I\k+l E* ~ (I\k+l E)* we may identify the right dual ~(cP)* of ~(cP) with the map £(cP) of right exterior product with cP. (A similar result holds in the left dual case.)

Proof

For

el, ... ,ek+l E E and cPl, ... ,cPk E E* we compute:

(~(J(cPl/\'" /\ cPk)HcP)*) (el /\ ... /\ ek+d = = J(cPl/\'" /\ cPk) (~(cP)(el/\'" /\ ek+l)) =

(~(cPd

0

... 0

~(cPk)) (~(cP)(el /\ ... /\ ek+l))

= J(cPl/\'" /\ cPk /\ cP)(el /\ ... /\ ek+l) . Since the el /\ ... /\ ek+l generate I\k+l E and the cPl /\ ... /\ cPk generate I\k E*, the result follows. IqEDI

7.24 Discussion. So far we have introduced five constructions to create new morphisms out of a family of morphisms: I:i cPi, tfJicPi, 11 cPi, ~icPi' and I\k cP. We also have constructed identifications which allow us to permute the operation taking the dual with the constructions direct sum [7.3], tensorproduct [7.12], and exteriorproduct [7.16]. We now claim that these identifications also allow us to permute the operation taking the dual map with the five constructions of new morphisms. More precisely, one obtains identifications (I:i cPi)* ~ tfJi(cPi), (tfJicPi)* ~ I:i(cPi), (11 cPi)* ~ 11(cPi), (~icPi)* ~ ~i(cPn, and (I\k 'ljJ)* ~ I\k ('ljJ*). We will give the exact formulation of this statement and its proof only for the case I\k 'ljJ; the other cases are similar and are left to the reader.

7.25 Proposition. If 'ljJ E Hom(E; F)o is an even morphism, we have a commutative diagram

N*E

1J ------4)

*(N E) ,

*(N,p)

i.e., under the identifications N *E ~ *(I\k E) and N *F ~ *(I\k F) we may identifY N *'ljJ with *(N 'ljJ). Similarly we may identifY (N 'ljJ)* with N ('ljJ*).

Proof For el, ... ,ek E E and cPl, ... ,cPk E *F the definitions give

~(el /\ ... /\ ek) (* (N 'ljJ )J( cPl /\ ... cPk)) = ~(~( el /\ ... /\ ek) N 'ljJ) J( cPl /\ ... cPk)

= ~(~(el)'ljJ /\ ... /\ t(ek)'ljJ )J(cPl /\ ... cPk) .

45

§7. Identifications

By definition of J, the last term is a sum (with signs) of products of terms of the form ~(~(ei)'ljJ)¢j = ~(ei)(*'ljJ(¢j)). Since 'ljJ is even, the signs do not change and we find

~(el

1\ ... 1\

ek) (* (N 'ljJ)J( ¢1 1\ ... ¢k))

= ~(el

1\ ... 1\

ek)J (*'ljJ( ¢d 1\ ... 1\ *'ljJ( ¢k))

= ~(el 1\ ... 1\ ek)J( N *'ljJ(¢1

1\ ... 1\

¢k)) .

Since the el 1\ ... 1\ ek generate I\k E and the ¢1 1\ ... 1\ ¢k generate I\k *F, the result IqEDI follows.

7.26 Construction (hi-duals). For any Qt-graded A-module E we construct an identification JLR : E ...., *(E*) by the formula

JLR(e) : ¢

I-->

= ¢(e) , HomR(E; A) == E*.

~(¢)JLR(e)

where ¢ : E ...., A is right linear, i.e., ¢ E define an identification J RL : E ...., (*E)* by the formula

In a similar way we

where now ¢ : E ...., A is left linear, i.e., ¢ E *E. The advantage of identifying E with the mixed bi-duals *(E*) and (*E)* is that we never have to change the order of terms. However, there are four "different" bi-duals. *(E*) ...., E** are isomorphisms [2.15]. The transpositions 'rL : **E ...., (*E)* and Using the transposition 'r : *E ...., E* gives us an isomorphism J : *(E*) ...., (*E)* defined on 'ljJ E *(E*) by

erR :

If'ljJ E *(E*) and ¢ E *E are homogeneous we have J('ljJ)(¢)

=

(_l)(c(,p)lc(¢))~('r¢)'ljJ.

The isomorphism J is compatible with the previously defined identifications J LR and J RL in the sense that J RL = J 0 J LR. The verification of this statement is left to the reader. It follows that we do not need the parentheses in the mixed bi-duals and that we may say that we have an identification E ...., *E*.

7.27 Remark. Let ~ : E ...., HomR(A; E) denote the isomorphism of [2.19] characterized by cI>~(e)(l) = e, then we can interpret JLR(e) as the dual map ~(e)* : E* ...., A. Here is how it works. Let e E E and ¢ E E* be arbitrary, then we have the equality ((¢IIcI>~(e)*)) = ¢o~(e) E A* and

(((¢IIcI>~(e)*»)(l) = ¢(e) = ~¢IIJLR(en = cI>~(~¢IIJLR(e))))(l) and thus ((¢IIcI>~(e)*)) = ~(((¢IIJLR(e)))). By abuse of notation, forgetting about the isomorphisms cI>~ and cI>~, we thus find J LR (e) = e*, where we have made the identificationsE ~ HomR(A; E) and HomL(E*; A*) ~ HomL(E*; A) == *(E*).

46

Chapter 1. 2t-graded commutative linear algebra

7.28 Construction (sums of tensor products). Let E be an Qt-graded A-module and let (Fi)iEI be a family of Qt-graded A-modules. We want to show that E @ (EBiEI Fi ) and EBiEI (E @ Fi ) are isomorphic. First recall that 7ri and Si denote the canonical projections and injections between Fi and EBiEI Fi [3.1]. The map E x (EBiEI Fi ) -+ EBiEI(E@Fi ) given by (e, f) !----> ffii (e @ 7ri(J)) is even and bilinear, and thus gives rise to an even linear map E @ (EBiEI Fi ) -+ EBiEI(E @ Fi). On the other hand we have the even linear map I:iEI id@Si : EBiE1(E@Fi ) -+ E@ (EB iE1 Fi). Let us show by computation that these two maps are inverse to each other.

iEI

iEI

and

iEI

iEI

where we used that trj 0 Si = 0 whenever i =1= j. We conclude that we have constructed an isomorphism between E@ (EB iE1 Fi ) and EBiE1(E@ Fi). In the same way we construct an isomorphism between (EBiEI Fi ) @ E and EBiEI(Fi @ E).

8.

ISOMORPHISMS

In §7 we have defined a number of identifications. In this section some technical proofs are given to show sufficient conditions for these identifications to be isomorphisms. It turns out that this is the case if the Qt-graded A-modules are finitely generated and projective. The condition finitely generated and projective for (Qt-graded) modules is equivalent to the condition finite dimensionalfor vector spaces. At the end of this section a summary of the more interesting identifications can befound.

8.1 Definitions. • A subset G of an Qt-graded A-module E is called a set of generators for E iffor each e E E there exist 91, ... ,9k E G (a finite number!) and a 1 , ... , a k E A such that e = I:i ai9i. The 9i and ai are not supposed to be unique. • An Qt-graded A-module E is called finitely generated, or offinite type, if there exists a finite set of generators G. • A subset B of an Qt-graded A-module E is called a set of independent elements if for all e1, ... ,ek E B (a finite number!) and for all a 1, ... ,a k E A one has the implication: I:i aiei = 0 =} \Ii : a i = O. In words: any (linear) relation between elements of B is necessarily trivial. • A subset B of an Qt-graded A-module E is called a basis for E if the elements of B are at the same time independent and generating. For vector spaces it is well known

47

§S. Isomorphisms

that there always exists a basis. However, for S2t-graded A-modules the existence of a basis is no longer guaranteed. It is immediate that an S2t-graded A-module admits a homogeneous basis ifand only ifitis (isomorphic to) afreeS2t-gradedA-moduleF(G,s) on homogeneous generators G [3.4] . • An S2t-graded A-module E is called projective if there exists an S2t-graded A-module E' such that EEBE' admits a homogeneous basis G, i.e., E EB E' = F(G, s). An S2t-graded A-module that is both finitely generated and projective will be calledfg.p.

8.2 Remark. We have defined the notions of independence of vectors, generating sets and bases with respect to the left module structure, completely ignoring that S2t-graded A-modules also have a (compatible) right module structure. Of course we could have developed the right module case parallel to the left module case. However, this is hardly necessary. If E is an S2t-graded A-module, gEE homogeneous and a E A, we have the relation ag = 9 . I:a:E=21( -l)(c(g)la:)aa:' This shows that a (left) generating set of homogeneous elements is also generating for the right module structure. And if a set of homogeneous elements is (left) independent, it is also independent for the right module structure. Hence for homogeneous sets, there is no difference between the notions of generating and independence for the left or right module structures. Moreover, by splitting into homogeneous components, any generating set can be made homogeneous. The only possible difference between left and right module structures thus lies in non-homogeneous independent sets. Since we will not use these, we will not see any difference between our left module structure definitions and the corresponding right module equivalents.

if and only if there exists an S2t-graded A-module E' such that EEBE' is afree S2t-graded A-module on afinite set of homogeneous generators. 8.3 Lemma. An S2t-graded A-module E is fg.p

Proof If E EB E' = F(G,s) with G a finite set, E is projective and 1fE(G) is a finite set of homogeneous generators for E, proving the if part. To prove the only if part, assume that E is projective and admits a finite number of generators. Since E is projective, there exists E' such that E EB E' = F(G', s') for some set of homogeneous generators G'. Our problem is that we do not know whether we can take G' to be finite. Let G = {gl,"" gn} be a finite set of generators for E. By splitting these generators into their homogeneous parts, we may assume that all gi are homogeneous. We thus obtain an even surjective map ¢ : F( G, s) -+ E defined simply by ¢(gi) = gi' Define the even map 'ljJ : E EB E' = F(G', s') -+ E as 'ljJ = id(E) + Q [7.2], where Q: E' -+ E is identically zero. Define also a map X : F(G', s') -+ F( G, s) as follows. For any g' E G' choose an element f' E F( G, s) such that ¢(f') = 'ljJ(g'). This is possible because ¢ is surjective. Since both ¢ and 'ljJ are even, we may assume that f' and g' have the same parity. Now define X by the formula

x:

L 9'EG'

Ag,g'l-->

L g'EG'

Ag'l'

.

Chapter 1. 2t-graded commutative linear algebra

48

This is a well defined even linear map satisfying the relati on 'ljJ = c/J 0 x. Restriction of X to the Qt-graded submodule E gives us the relation id( E) = c/J 0 (xl E). By using the equality x = (x - (XIE)(c/J(X))) + (XIE)(c/J(X)) it follows easily that the Qt-graded submodules ker(c/J) and im(xIE) are supplements. Hence by [3.9] we obtain E EB ker(c/J) = F(G, c:). IQEDI

8.4 Lemma. If an Qt-graded A-module E isfg.p, then so are E* and *E. Proof If E is a free Qt-graded A-module on a single homogeneous generator eo, it follows immediately that E* is a free Qt-graded A-module on the single homogeneous generator c/Jo defined by c/Jo( eo . A) = A. If E is f.g.p, there exists an Qt-graded Amodule E' such that E EB E' = EB~=1 E i , where all the Ei are free Qt-graded A-modules on a single homogeneous generator [3.4]. Using [7.3] we thus obtain an isomorphism E* EB (E')* ~ EB~=l Ei- Since the last one is a free Qt-graded A-module on n generators, the result follows. The left linear case is analogous. IQEDI

8.5 Proposition. If E is fg.p, the identification J : E

---+

*E* [7.26J is an isomorphism.

Proof Consider the special case of*(E*).If E is a free Qt-graded A-module on a single homogeneous generator, the result follows immediately from the proof of [8.4]. If E 1 , •.• , En are Qt-graded A-modules, we have identifications J i : Ei ---+ * (En and JE& : EB~=1 Ei ---+ *((EB~=1 Ei)*). Applying the isomorphism [7.2] twice gives us the following diagram: n EB .=1 *(E*) EB~=1 Ei •

I1~=1

Ji

II EB~=1 Ei

J(j)

--->

is. *((EB~=1 Ei)*) .

We leave it to the reader to verify that this diagram is actually a commutative diagram. Now let E' be an Qt-graded A-module such that E EB E' = EB~=1 Ei where the Ei are free Qt-graded A-modules on a single homogeneous generator. Applying the commutative diagram to the family of two: E and E', shows that JE& is an isomorphism if and only if both J and J' are isomorphisms [7.8]. Applying the commutative diagram to the finite family Ei proves that JE& is an isomorphism because for free Qt-graded A-modules on a single homogeneous generator the J i are isomorphisms. It follows that J is an isomorphism. IQEDI

8.6 Proposition. If E isfinitely generated or if the index set I isfinite, then the identifications J : EBiEI Homs(E; F i ) ---+ Homs(E; EBiEI F i ) [7.6J are isomorphisms. Proof Since injectivity is automatic, we have to assure surjectivity. For an arbitrary c/J in Homs(E; EBiEI F i ) we define the maps c/Ji E Homs(E; Fi) by c/Ji = 7ri 0 c/J, where

49

§S. Isomorphisms

7ri : EBi Fi ---., Fi denotes the canonical projection. It is immediate that J(EBi¢i) = ¢, except for the fact that we do not know whether (¢i)iEI lies in EBiFI Homs(E; Fi), i.e., whether only finitely many ¢i are non-zero. If the index set I is finite, this is obvious and we may conclude that in that case J is an isomorphism. So suppose that el) ... ) en are generators for E. Since ¢( ej) lies in EBi Fi , it follows that only finitely many ¢i (ej) = 7r i( ¢( ej)) are non-zero. Since there are only finitely many ej, which generate E, it follows that only finitely many ¢i are non-zero. Hence (¢i)iEI E EBiEI Homs(E; Fi), proving that J is an isomorphism. IQEDI

8.7 Proposition. Given three Qi-graded A-modules E, F, and G, we want to investigate the following two identifications: J : F @ HomR(E; G) ---., HomR(E; F @ G) and

J: HomL(E; G)

@

F ---., HomL(E; G@ F).

(i) ifF is projective, then the J's are injective. (ii) ifF isfg.p, then the J's are isomorphisms. (iii) If E is fg.p, then the J's are isomorphisms. Proof We only treat the right linear case, the left linear case being similar. Let us start with the special case in which F is a free Qi-graded A-module on a single homogeneous generator fa. By definition, the map p : A ---., F, a I--> fa . a is a right linear bijection of the same parity as fa. Using [7.14] one can show that C : HomR(E; F @ G) ---., HomR(E; A @ G) defined by C( ¢) = (p-l @ id) 0 ¢ is a right linear bijection. We thus obtain a map Co J 0 (p @ id) : A @ HomR(E; G) ---., HomR(E; A @ G). Identifying A @ H with H for any Qi-graded A-module H, it is an elementary verification that this map is the identity on HomR(E; G). Since C and (p@ id) are bijective, we deduce that J is bijective. We conclude that J is an isomorphism in case F is a free Qi-graded A-module on a single homogeneous generator. To prove (i) and (ii), let us fix E and G, and let us consider a family Fi , i E I of Qi-graded A-modules. We thus have identifications J i : Fi @ HomR(E; G) ---., HomR(E; Fi @ G) andJE&: (EBiEIFi)@HomR(E;G)---.,HomR(E;(EBiEIFi)@G). According to [7.28] there exists an isomorphism (EBiEI Fi ) @ HomR(E; G) ---., EBiEI(Fi @ HomR(E; G)). According to [7.28] we also have an isomorphism (EBiEI Fi)@G ---., EBiE1(Fi@G). This last isomorphism combined with the identification [7.6] gives us an injective identification Jh : EBiEI HomR(E; Fi @ G) ---., HomR(E; (EBiEI Fi) @ G). We thus obtain the following diagram of maps:

"'" 1

[7.28J

(EBiEI Fi ) @ HomR(E; G) We leave it to the reader to check that this diagram is commutative. Now take a family of two: F and F'. Since the index set consists of two elements, the map J h is a bijection [7.6]. Using [7.8] we conclude that JE& is injectivelbijective if and

50

Chapter 1. 2l-graded commutative linear algebra

only ifboth 'J and 'J' are injectiveibijective. Ifwe suppose that F is projective, we can take F' such that F EEl F' is a free SZl-graded A-module on a set of homogeneous generators, i.e., FEEl F' = EBiEI Fi where each Fi is a free SZl-graded A-module on a single homogeneous generator. For these we know that all 'J i are isomorphisms. However, for this family we only know that 'J h is injective. We conclude that 'JEll is injective. This proves (i). If F is also finitely generated, it follows that the family Fi can be taken finite, in which case 'J h becomes bijective, hence 'JEll is bijective, and thus 'J is bijective. This proves (ii). To prove (iii) we proceed in the same way. If E is a free SZl-graded A-module on a single homogeneous generator, one can easily establish the existence of homogeneous linear bijections between Hand HomR(E; H) for any SZl-graded A-module H. Using these bijections one then proves that 'J is an isomorphism if E is a free SZl-graded A-module on a single homogeneous generator. Now let us fix F and G and let us take a finite family (Ei)f=l. We thus have identifications 'J i : F 0 HomR(E i ; G) -+ HomR(E i ; F 0 G) and 'JEll : F 0 HomR(EB~=l E i ; G) -+ HomR(EB~=l E i ; F 0 G). According to [7.2] we have an isomorphism EB~=l HomR(E i ; G) -+ HomR(EB~=l E i ; G). Taking the tensor product of this isomorphism with the identity on F (see [7.14]) and composing it with the isomorphism EB~=l (F 0 HomR(E i ; G)) -+ F 0 (EB~l HomR(E i ; G)) gives us an isomorphism 'J v : EB~l (F 0 HomR(E i ; G)) -+ F 0 HomR(EB~=l E i ; G). We thus obtain a diagram of maps:

~ -------+ J(j)

1

[7.2J

HomR(EB~=l E i ; F 0 G) ,

As before, it is left to the reader to check that this diagram is commutative. If we now suppose that E is f.g.p, then there exists an SZl-graded A-module E' such that E EEl E' = EB~=l E i , where each Ei is afree SZl-graded A-module on a single homogeneous generator. Applying the commutative diagram to the finite family of two: E and E', shows that 'JEll is bijective if and only ifboth 'J and 'J' are bijective [7.8]. Applying it to the finite family Ei shows that 'JEll is bijective if and only if all 'J i are bijective. Since the Ei are free on a single homogeneous generator, the 'J i are isomorphisms, and we conclude that both 'J and 'J' are bijective. IQEDI

8.8 Corollary. Let E and F be SZl-graded A-modules. ifF is projective, the identifications *E 0 F -+ HomdE; F) and F 0 E* -+ HomR(E; F) [7.llJ are injective. Ifeither E or F isjg.p, these identifications are isomorphisms.

8.9 Lemma. If Ei = F(G i , ci), i = 1,2 arefree SZl-graded A-modules on homogeneous generators G i , then El 0 E2 is afree SZl-graded A-module on homogeneous generators G 1 x G 2 with parity mapc(91,92) = Cl(91) --'-c2(92).

51

§S. Isomorphisms

Proof Choosing left coordinates for E1 and right coordinates for E 2, we define a map 'I/J : E1 x E2 -- F(G 1 X G2, c) by 'I/J(2::i Ai . 91i, 2:: j 92j . /-Lj) I----' 2::i,j Ai . (91i, 92j) . /-Lj. We leave it to the reader to verify that 'I/J is even and (right) bilinear, and thus induces an identification E1 &; E2 -- F( G 1 X G2, c). It is easily seen that the inverse is given by the map 2::i,j Vij . (91i, 92j) I----' 2::i,j Vij . 91i &; 92j, proving the lemma. IQEDI

8.10 Corollary. If E and F are projective '21.-graded A-modules, then E If they arefinitely generated, then so is E &; F.

&; F

is projective.

Proof The elements e &; j, e E E, j E F generate E &; F. If E is generated by (ei)iEI and F by (Ii) j E J, then E &; F is generated by (ei &; Ii) iE I,j E J. We conclude that if both E and F are finitely generated, so is E &; F. If E EEl E' and FEEl F' are free '21.-graded A-modules on homogeneous generators, then (E EEl E') &; (F EEl F') is a free '21.-graded A-module on homogeneous generators. But by [7.28] this is isomorphic to (E that E

&;

&;

F) EEl ((E

&;

F') EEl (E'

&;

F) EEl (E'

F is projective.

&;

F I ) ) , showing IQEDI

8.11 Discussion. If E and F are '21.-graded A-modules, we have defined the operation of right dual map, which is an even linear map * : HomR(E; F) __ HomL(F*; E*). We also have identifications F &; E* -- HomR(E; F), *(F*) &; E* -- HomL(F*; E*), and F -- * (F*). We leave it to the reader to verify that these identifications fit together in a commutative diagram

F

&;

E*

1

------?

*(F*)

&;

E*

1

If E and F are f.g.p, we know that the three unlabeled arrows are isomorphisms, and thus thatthe map * : HomR(E; F) -- HomL(F*; E*) is an isomorphism as well. Obviously a similar result is true for left dual maps.

8.12 Proposition. Let Ei and F i , i = 1, ... , n be two families of '21.-graded A-modules and let'J : ®t=n Homs(Ei; F i ) -- Homs(®~=l Ei ; ®~=1 Fi ) be the identification given in [7.12J. (i) If all Ei arefg.p then 'J is an isomorphism. (ii) If all Ei and Fi arefg.p, with the possible exception ofa single pair (Ei, F i ), then 'J is an isomorphism.

52

Chapter 1. Qt-graded commutative linear algebra

Proof We give the proof in the left linear case, the right linear case being similar. We create the following isomorphisms: ~

~ ~ ~

HomL(E2; F2) 0 HomL(E 1; Fd HomL(E2; F2 0 HomL(E 1; Fd) HomL(E 2;HomL(E1;Fd 0F2) HomL(E2; HomL(E 1; F1 0 F2)) HomL(E1 0 E 2; F1 0 F2)

by [8.7] with E2 f.g.p using that 9t is an isomorphism by [8.7] with E1 or F2 f.g.p by [7.10]

We leave it to the reader to trace these isomorphisms and to show that the final result is indeed the identification given in [7.12]. It follows that we have proven the proposition for n = 2, where we used that either E1 and E2 are f.g.p, or E2 and F2 are f.g.p. The general result follows by induction. In case (i) one uses [8.10]. In case (ii) one uses that all Ei and Fi are f.g.p for i = 1, ... , n - 1. Since permuting factors in a tensor product is an isomorphism, we may indeed assume that it is the last couple (En, Fn) that is not IQEDI f.g.p.

8.13 Corollary. Let E 1, ... , En be '21.-graded A-modules. If all but one are fg.p, the identification E~ 0··· 0 Ei -- (E1 0··· 0 En)*, defined on homogeneous elements by

L: (e( 1>i) le(1)j (ej ») (rPn 0···0 rP1)(e1 0··· 0 en) = (-l)i>j

. rP1(ed'" rPn(e n ) ,

is an isomorphism, as is the identification *En 0 ... 0 *E1 -- * (E1 0 ... 0 En), defined on homogeneous elements by ~(e1 0··· 0

en)(rPn 0··· 0

L: (e(1>i)le(1>j(ej))) rPd = (-l)i<j

. ~(e1)rP1'" ~(en)rPn .

8.14 Proposition. If E = F(G, c) is afree '21.-graded A-module on homogeneous generators G, then /\k E is a free '21.-graded A-module. Proof From [8.9] we know that the set B = {91 0 ... 0 9k I 9i E G} is a basis for ®k E. We now choose a total order on G and we define the subsets B g1 ... 9k C B for increasing sequences 91 :::; 92 :::; ... :::; 9k by

B g1 ... 9k = {90-(1) 0··· 090-(k)

I

(J

E

6k } .

B g1 ... 9k form a partition of B in disjunct subsets. We now introduce the modified sets B~1 ... 9k defined as

It should be obvious that the

It is easy to see that the set {91 0 ... 0 k

sub module of ® E as B g1 ... 9k •

9k} U B~1 ... 9k generates the same '21.-graded

53

§S. Isomorphisms

We now recall the construction of /\k E = ®k E / Nk as given in [5.2]. By k-linearity, it is immediate that the szt-graded submodule Nk C ®k E is also generated by the set

T£ = {91 0 ... 09k

+ 91(jj+1) (91 0··· 09k) 11:::; j < k,

9i E G} .

An elementary but slightly tedious verification then shows that Nk is also generated by the union of all sets B~l" .gk (and recall that the index set 91 .. . 9k is an increasing one). Since the B g1 ... gk form a partition of the basis B, it follows that we have found a basis for a supplement of Nk in the form of the set of all 910' .. 09k subject to two conditions (i) that 91 :::; ... :::; 9k and (ii) that it does not belong to the szt-graded submodule generated by B~1 ... 9k' The projection of this basis to /\k E = ®k E / Nk then is a homogeneous basis for /\k E, proving that E is a free szt-graded A-module. IQEDI

N

8.15 Discussion. It can be shown that the condition that 91 0 ... 0 9k belongs to the szt-graded submodule generated by B~1 ... 9k is equivalent to the existence of a permutation (J E 6k such that -(-1)0"910"(91 0··· 09k) = 91 0 .,. 09k. This in turn can be shown to be equivalent to the existence of an index i < k such that 9i = 9i+1 and such that (_I)(o(gi)lo(g,)) = 1. It follows that a basis of /\k E is given by the vectors 91 1\ ... 1\ 9k where the 9i form an increasing sequence such that a 9i is not repeated if (_I)(o(gi)lo(g,)) = 1.

8.16 Proposition. If E is fg.p, the identifications J : N (E*) ~ (/\k E)* and J : N (*E) ~ *(N E) [7.16J are isomorphisms. Proof As usual we treat the right linear case, the left linear case being similar. We denote by n : ®k E ~ /\k E and by n' : ®k E* ~ /\k E* the canonical projections. By [2.21] the dual map n* : (/\k E)* ~ (®k E)* is injective. It then follows from the pedestrian way of constructing J [7.16] that we have the equalities 'I/J = J® 0 A, 'I/J = n* 0 and W = Jon'. We thus obtain the equality n* 0 Jon' = J® 0 A, i.e., the commutative diagram ®k E* (®k E)*

w,

~

J0 ok.

~'

1

NE*

~

J

r~* (N E)* .

The essence of the proof will be "running around this diagram." If E is f.g.p, it follows from [8.13] that J 0 is bijective, i.e., we may write A = J;S/ 0 n* 0 Jon'. Since n' is szt-graded skew-symmetric, we obviously have the equality n' 0 A = (k!) . n'. We now start running:

(k!)· n'

= n' oA = n'

0

(J;s;I on* oJ on') = (n' oJ;S;l on* oJ) on' .

Since n' is sUljective, this implies that we have the equality n' 0 J;S;l 0 n* 0 J = (k!) . id, and thus J must be injective. On the other hand, since each e E (/\k E)* represents an

54

Chapter 1. Qt-graded commutative linear algebra

szt-graded skew-symmetric map on Ek, it follows from (7.17) that we have the equality A 0J,s/ 01l'* = (k!) . J,s/ 01l'*. We now start running again:

(k!) .J,s;101l'*

= AoJ,s;l 01l'* = p,s;101l'*oJ01l")oJ,s;101l'* = J,s;l 01l'* 0 (J01l" oJ,s;l 01l'*) .

Since J,s; 1 01l'* is injective, we deduce the equality J 01l" 0 J,s; 1 01l'* = (k!) . id, and thus J must be surjective. IQEDI

8.17 Corollary. If E is jg.p, there is a natural identification Hornik(Ek, A) ~ /\k *E.

8.18 Summary. The following table summarizes some of the more interesting identifications, as well as where one can find sufficient conditions for the identification to be an isomorphism (a ~ indicates that it is always an isomorphism, and a '----+ indicates that it is always injective). In any case, if all szt-graded A-modules involved are f.g.p, and if the index sets are finite, all identifications are isomorphisms. identification

E&J (EBiEIFi) EBiEI Horns (Ei; F) EBiEI(Ei) EBiEI Horns(E; Fi ) ®i=n Horns (Ei; Fi ) Ei &J ... &J E~

N (E*) E F&JE* *E&JF

definition - isomorphism

'='"

EBiE/(E &J Fi )

[7.28]

'----+

Horns (EBiEI Ei ; F)

[7.2]

- [7.2]

'----+

Ei)* Horns(E; EBiEI Fi )

[7.2]

- [7.3]

[7.6]

- [8.6]

Horns(®~=l Ei ; ®~1 Fi )

[7.12] - [8.12]

(En &J ... &J Ed*

[7.12] - [8.l3]

(N E)*

[7.16] - [8.16]

'----+

-------

(EBiEI

*(E*)

~

(*E)*

[7.28]

[7.26] - [8.5]

HornR(E; F)

[7.11] - [8.8]

HornL(E;F)

[7.11] - [8.8]

Chapter II

Linear algebra of free graded A-modules

In chapter I we have studied general '21.-graded commutative algebra for an arbitrary abelian group '21. and an arbitrary '21.-graded commutative ring A. Starting this chapter we specialize to the case '21. = Z2 and we will abbreviate Z2-graded to simply graded. In order to get "close" to ordinary linear algebra, we also impose two conditions on the graded commutative ring A: it should contain the real numbers R as a sub ring and the nilpotent elements in A should form a supplement to RcA. This allows us to define the notions of basis and graded dimension for free graded A-modules. Except for subspaces and quotients (where one has to be a bit careful), these notions behave exactly as one would expect from ordinary linear algebra. Special attention has to be paid to matrices associated to linear maps. Whereas in usual linear algebra there is a single natural way to associate a matrix to a linear map when a basis has been given, in graded linear algebra there are three natural ways to do so. Each of these three ways has its own advantages and disadvantages. For instance, one of them is particularly adapted to express the graded trace of a linear map. The same representation by matrices is useful to define the graded determinant, also called Berezinian, ofa linear map. However, the proofthat the graded determinant is a group homomorphism is a bit cumbersome. Graded linear algebra is sometimes understood as meaning linear algebra of vector spaces (over R) that split as a direct sum oftwo subspaces: the even and odd parts. In our context that would be the special case in which one takes A to be equal to the real numbers R. However, it only requires a very small step to relate the general case to the special case A = R. This small step is the introduction ofthe notion ofan equivalence class ofbases. This idea allows us to reduce all discussions about free graded A-modules to discussions about bases. And then the only difference between a generic A and the special case A = R isjust the choice ofcoordinates with respect to a basis. 55

Chapter II. Linear algebra of free graded A-modules

56

1.

OUR KIND OF Z2-GRADED ALGEBRA

A

In this section we introduce the additional conditions we will imposefrom now on on the ring A: it should be a special kind of Z2-graded commutative R-algebra. The basic example ofsuch a ring is the exterior algebra ofan infinite dimensional real vector space [/.2J. The most important result of this section is the lemma that says that any finite number of nilpotent elements in A can be "killed" multiplicatively by a non-zero nilpotent element. We also introduce matrices with entries in A and derive some useful properties concerning nilpotent and invertible matrices.

1.1 Definition. Starting this chapter, we will be concerned mostly with Z2-gradings. This being the case, we will abbreviate Z2-graded to simply graded. As usual elements of parity 0 will be called even, but now elements of parity 1 will be called odd. When speaking of graded commutativity, we will always use ordinary multiplication in Z2 for the bilinear symmetric map (_1_) : Z2 x Z2 ~ Z2' For any (Z2-)graded commutative ring A we define the set of nilpotent elements Nby N = {a E A 1 :3k EN: ak = o}.

1.2 Examples. • The real line R is a graded commutative ring if we take AD = R and Al = {o} . • The complex line C is a graded commutative ring if we define AD = C and Al = {a}. Had we defined AD = R and Al = iR, the ring C would have been Z2-graded, but not graded commutative . • Let X be a real vector space (finite or infinite dimensional), and let A be the exterior algebra A = /\ X = EB~=o /\k X. It is a graded commutative ring if we define AD = EB~=o Nk X, Al = EB~=o N k+1 X and the wedge product as multiplication operation. We have 1 E N X ~ Rand N = EB~=I N X.

1.3 Lemma.

N = (N n AD)

EEl Al and

N is an ideal of A.

Proof The proof naturally splits in four steps. (i)If a E Al then the graded commutativity implies a· a = -a· a and hence a 2 = 0, i.e., Al C N. (ii) If n E N then we have 0= (no + nl)k = n~ + kn~-lnl (because of the graded commutativity and (i». Hence ngk = (-kn~-lnd2 = Pn~k-2ni = O. Thus n EN=} no EN. This proves the first assertion. (iii) If n, mEN then (n + m)k = (no + mo)k + k(no + mo)k-l(nl + ml). Since we know that no and mo are also nilpotent, it follows from the binomial formula that (n + m)k = 0 for sufficiently large k. Thus n, mEN =} n + mEN. (iv) If a E A and n EN then an = anD + anI. Since no is even, we have (ano)k = akn~ and thus anD is nilpotent. Since (anl)2 = anI anI = a (aD - al) ni = 0, anI is nilpotent too. Thus a E A, n EN=} an E N. A similar argument applies to na. This proves the second assertion. IQEDI

§1. Our kind of Z2-graded algebra A

57

1.4 Lemma. If nl, ... ,nN is a finite number of nilpotent elements in A, then there exists a non-zero homogeneous nilpotent n such that Vi : nni = O. Proof Without loss of generality we may assume that all ni are non-zero. Moreover, by considering the homogeneous parts of the ni separately, we may assume also that all ni are homogeneous. Define k i E N such that n7 i f- 0 and n7 i + 1 = O. Now define AD = 1 (N ota Bene. This is just formal, A need not have a unit) and Ai = Ai-l n7 i if Ai-l n7 i f- 0, Ai = Ai-l otherwise. Taking n = AN and using the graded commutativity, we find that the product nni can be written as Ainia for some a E A and hence by definition of ki we IQEDI have nni = O.

1.5 Definition. If A is a graded commutative ring, we denote by B the canonical projection B : A ~ A/N and call it the body map. Since N is an ideal in A, A/N is a ring and B a ring homomorphism. If a graded commutative ring A is an R-algebra, then we have in particular that AD, Al,N, and A/N are vector spaces over R and that B : A ~ A/N is a linear map between vector spaces over R. From now on, A will always denote a graded commutative R-algebra with unit such that A/N is isomorphic (as ring) to R. It follows that the body map B : A ~ A/N ~ R has a canonical section R ~ A given by r f--+ r·l (which is well defined because A is a vector space over R). By abuse of notation we will always identify r E R with r·l E A and write a = r + n, when B(a) = r E R, and n EN.

1.6 Examples .• The real line R itself verifies the conditions on A given in [1.5] . • The complex line Cwith Co = C and C 1 = {O} is a graded commutative R-algebra, but it does not verify the conditions of [1.5] . • If X is a vector space over R, its exterior algebra /\ X is a graded commutative R-algebraA verifying the condition of [ 1.5].

1.7 Remark. We restricted the symbol A to denote a graded commutative R-algebra with unit such that A/N ~ R. Although the choice of the field of real numbers is important in the next chapters, the results of this chapter remain valid if one replaces R by any other field of characteristic 0, e.g., C. Moreover, most of the analytic results of the subsequent chapters also remain valid when replacing R by C.

1.8 Discussion (Geometric interpretation of [1.4]). If A is the exterior algebra of a vector space X as in [1.2], [1.6], we can give a geometric interpretation of the element n of [104]. Let ni be a non-zero nilpotent element in A, then by definition of the exterior product as the direct sum of exterior powers of X, ni is a finite sum of monomes, each of which is a finite wedge product of vectors in X. Since there is also a finite number of ni's, we find a finite number of vectors in X that are involved in the definition of the ni's.

58

Chapter II. Linear algebra of free graded A-modules

These vectors span a finite dimensional subspace of X; the wedge product of vectors that form a basis of this finite dimensional subspace is an element n E A which satisfies the requirements of the lemma.

1.9 Definitions. The set M(m x n, A) denotes the set of all matrices of size m x n (m being the row dimension and n the column dimension) with entries in A. The usual matrix multiplication M(m x n, A) x M(n x r, A) __ M(m x r, A) still makes sense on these sets. The body map B extends in a natural way to these matrices: B: M(m x n, A) -- M(m x n, R); it is surjective and preserves matrix multiplication. The set M(n x n, A) equipped with matrix multiplication is a ring with unit In (in terms of matrix elements (In); = S} with S} the Kronecker delta) for which the body map B: M(n x n,A) -- M(n x n,R) is a surjective ring homomorphism. Fora E M(m x n, A) we define the rank ofa, denoted as rank(a), as the rank of its body Ba E M(m x n, R) : rank(a) = rank(Ba), i.e., as the number of independent rows or columns in Ba.

1.10 Lemma. Let a E M (n x n, A) such that B (a)

= 0,

then a is a nilpotent matrix.

Proof The condition B(a) = 0 implies that each entry of the matrix a lies in the kernel of the body map, i.e., is nilpotent. It follows that the homogeneous parts of the entries are also nilpotent. Since there are 2n 2 homogeneous entries (i.e., a finite number), there exists a number N such that aN = 0 for all homogeneous entries a of a. Now the entries of the matrix a k are sums of terms, each of which is a k-fold product of homogeneous IQEDI entries of a. Hence for k > (N - 1)· 2n 2 we have a k = o.

1.11 Lemma. An element a E M(n x n,A) is invertible ifandonlYlfDet(B(a))

=I-

o.

Proof If b is the inverse of a in M(n x n, A), then B(b) is an inverse for B(a) in M(n x n, R) and hence Det(B(a)) =I- O. Conversely, suppose Det(B(a)) =I- 0 and write a = B(a) + nwith B(n) = O. Writing x for the inverse ofB(a) in M(n x n, R), we define b = L:~=o( -xn)kx . This is actually a finite sum because of [1.10]; moreover, one IQEDI easily verifies that b is an inverse to a in M(n x n, A).

2.

FREE GRADED A-MODULES

In this section we define the graded dimension ofafree gradedA-module. It consists of two integers: the number of even respectively odd vectors in a homogeneous basis. We show that this is an invariant ofa (finite dimensional)free graded A-module. More precisely, up to isomorphism there exists only onefree graded A-module of graded dimension plq. These results depend crucially on the conditions we imposed on A in [1.5].

§2. Free graded A-modules

59

2.1 Definition. In this section and the subsequent ones we will be mainly interested in free graded A-modules F( G, c) (on homogeneous generators G). We recall that all free graded A-modules admit a homogeneous basis (e.g., G). We will call a free graded A-module finite dimensional if it admits a finite (homogeneous) basis. A homogeneous basis el, ... , en of a finite dimensional free graded A-module is called ordered if all even vectors come first, i.e., ei even and ej odd implies i < j. A subset F of a free graded A-module E is called a graded subspace if it is a graded submodule of E that in itself is (isomorphic to) a free graded A-module.

2.2 Nota Bene. Any finite dimensional free graded A-module is in particular f.g.p, and thus the identifications given in [1.8.18] apply.

2.3 Remark. Since A is in particular an R-algebra with unit, it follows immediately that any A-module, and thus any free graded A-module, is also a vector space over R. Anticipating the fact that the dimension is an invariant of a free graded A-module, it is immediate that the dimension of a free graded A-module E as vector space over R equals the product of the dimension of E asfree graded A-module and the dimension of A as vector space over R. Since our main example for A is the exterior algebra of an infinite dimensional vector space, any free graded A-module (over this A) is infinite dimensional as vector space over R.

2.4 Example. We know already that A is (trivially) a graded A-module. The unit 1 E A is both generating and independent, hence A admits a basis. Since moreover 1 is even, A is a free graded A-module. Note however that not every non-zero element is a basis for A: if a =I- 0 is nilpotent, then a is not generating (l can not be written as ba with b E A), nor is it independent (if an =I- 0, a n +1 = 0 then an a = 0 is a relation with a non-zero coefficient) . More generally, the graded A-module An is a finite dimensional free graded A-module because the vectors (1,0, ... ,0), (0,1,0, ... ,0), ... , (0, ... ,0,1) are generating, independent and even.

2.5 Lemma. Letel, ... , en, iI, ... , 1m be elementsofagradedA-module E, and suppose that there exist ai j E A such that Ii = 2: j ai j ej. If the iI, ... ,1m are independent, then

rank (Baij){;i: ...... :::. = m. In particular m :::; n. Proof If rank(Baij) < m, then there must exist a relation among the rows of Baij, i.e., there exist ri E R, not all of them zero, such that for all j : 2:i riB(aij) = O. We then define iJ,i = 2:i riaij E A and note that B(iJ,i) = 0, i.e., that the iJ,i are nilpotent. Thus, by [104], there exists a non-zero no E A such that Vj : no f.1j = O. But then 2:i(n O ri)Ii = 2:i,i no(riaij) ej = 2: j no f.1 j ej = 0, contradicting the independence of the k We conclude thatthe rank ofBaij is m and hence m :::; n. IQEDI

60

Chapter II. Linear algebra of free graded A-modules

2.6 Proposition. Let el, ... , en be a basis of a graded A-module E.

(i) If iI, ... , f m is another basis of E, m = n. (ii) All other bases {iI, ... , f n} are classified by invertible matrices a E M (n x n, A) with fi = 2:: j aij ej. (iii) If iI, ... , fn is either generating or independent, it is a basis. Proof • Since the fi and ej form a generating subset, there exist ai j , bj i E A such that fi = 2:: j aij ej and ej = 2::i b/ k Since the fi and ej are independent, it follows from [2.5] that n :s; m and m :s; n . • If iI, ... , f n is a basis, by the preceding argument ::Jai j E A such that fi = 2:: j aij ej and rank(Baij) = n. Thus by [1.11] aij is invertible. On the other hand, if fi = 2:: j aijej with an invertible ai j , then it follows easily that the fi form a basis . • If the fi are independent, there exist aij E A such that fi = 2:: j aijej. By [2.5] we know that rank(BaJ) = n, and thus by [1.11] aij is invertible. By the preceding result the fi form a basis. If the fi are generating, there exist b/ E A such that ej = 2::i b/ k As before b/ must be invertible and thus (by applying the inverse of b/) the fi form a IQEDI basis.

2.7 Counter example. To show that not all graded A-modules are free graded A-modules, consider the free graded A-module A2 and choose a non-zero even nilpotent element /-L E Ao. We then define the subset F c A2 by F = Span( {(I, 0), (0, /-L)}), which is a graded submodule by [1.1.25]. We will show that F does not admit a basis, showing at the same time that not every graded A-module is a free graded A-module, and that not every graded submodule of a free graded A-module is again a free graded A-module. Clearly { (1, 0), (0, /-L) } is a generating set and hence a basis of F (if it exists) comprises at most 2 elements. Now let us first suppose that a single element { (a, b/-L) } is a basis, then there exists K, E A with (1,0) = K,( a, b/-L) = (K,a, K,b/-L) and hence K, must be invertible. It follows from [2.6-ii] that (l,0) is a basis, which is clearly false (it is not generating). Let us suppose next that { (a, b/-L), (c, d/-L) } is a basis, then there exist K" A E A with (1, 0) = K, ( a, b/-L) + A(c, d/-L) = (K,a + Ac, K,b/-L + Ad/-L). Since 1 is not nilpotent, it follows that not both a and c are nilpotent. Hence by applying [2.6-ii] with a suitable choice of an invertible matrix in M (2 x 2, A) we may assume without loss of generality that a = 1 and c = O. But then /-Lk-l (c, d/-L) = (0, d/-Lk) = (0,0), where k is chosen such that /-Lk-l -=f- 0 and /-L k = O. Since this contradicts the independence, we conclude that F does not admit a basis in the sense of a subset that is both generating and independent.

2.8 Lemma. Let E be a graded A-module that admits a basis consisting offinitely many elements. Then E admits a homogeneous basis, i.e., E is a finite dimensional free graded A-module. Proof Let {iI, ... , f n} be a basis of E and fi = fiD + fil the decomposition in homogeneous parts. Since (Ii) is a basis, there exist matrices a, b E M(n x n, A) such that

61

§2. Free graded A-modules

j j iiD = L: j ai Ii and iiI = L: j bi Ii· Since the ii are independent we have a j j j Decomposing ai = ai 0 + ai 1 into homogeneous parts we find iiD

= "2)ai

j

oijo

+ b = In.

+ ai j dj!)

j

(the other two terms necessarily cancel because iiD E Eo). Resubstituting the expressions . k . k . k . for iiD and iil we fmd L:k ai ik = L:jk(a/Oaj + ail1bj )ik. or (smce the ii are independent) in terms of matrices:

If we realize that B (aij) = B (ai j 0) and B (ai j 1) = 0, we deduce from this equation B(a) = B(a)B(a) i.e., B(a) is a projection. Standard linear algebra then shows the existence of a real matrix X such that XB(a)X-1 = (I; ~) with :s; p :s; n. Since X is invertible,. g; = L: j Xij Ii is also a basis of E. Defining the matrix c = X aX-I, j n1 we have giO = L: j Ci gj and B(c) = (I; ~). It follows that c = (Ip:3 ~:), where the ni are matrices of the appropriate size with nilpotent entries. We finally define the j vectors ei by ei = giO if i :s; P and ei = gil if i > p. It follows that ei = L: j di gj with d = (I~+nl I n-p n~ n4 ). Since B(d) = In, d is invertible and (ei) is a basis, which n3 obviously consists of homogeneous elements. IQEDI

°

2.9 Proposition. The number of even vectors in a homogeneous basis ofafinite dimensional free graded A-module E is an invariant of E.

Proof Let { e1, ... , en } and {e1' ... , en } be two ordered homogeneous bases with p and peven elements respectively. These two are related by an invertible matrix a with ei

=

n Laijej j=l

=

p (Laijoej j=l

+

n j L ai 1e j) j=p+1

+

p j (L a i 1e j j=l

+

n L aijOej). j=p+1

Independence of the (ei) and homogeneity of (ei) and (ei) then show that ai j is even if i :s; Pand j :s; p or if i > pand j > p; it is odd otherwise. It follows that B (a) = (~ ~ ) where b is a real matrix of size p x p. Since such a real n x n matrix has necessarily determinant zero if p =I- p, it follows from [1.11] that p = p. IQEDI

2.10 Definition. The graded dimension of a finite dimensional free graded A-module E is a pair of integers (p, q) where p, called the even dimension ofE, is the number of even vectors in a basis for E and q, called the odd dimension ofE, the number of odd vectors in (the same) basis. We usually denote this as dim(E) = plq; in particular dim(An) = niO. If the graded dimension of E is plq, we define the total dimension of E as the sum n = p + q, also denoted as dim(E) = n. In general, when we use the word dimension, we will mean the graded dimension, unless it is clear from the context that we mean the total version (which is notably the case when we talk about finite dimensional spaces).

62

Chapter II. Linear algebra of free graded A-modules

2.11 Proposition. Two finite dimensional free graded A-modules E and F are isomorphic ifand only if they have the same graded dimension.

Proof Let 1> : E -- F be an isomorphism and B a homogeneous basis of E. Then 1>( B) is a homogeneous basis for F. Since the parity is not changed by the even map 1>, it follows that dim(E) = dim(F). On the other hand, suppose dim(E) = dim(F) and let (ei) and (fi) be ordered homogeneous bases of E and F respectively. Then the map 1> defined by (( L:i aiei 111>) = L:i ai Ii is an even bijective (left) linear map, i.e., an isomorphism IQEDI between E and F.

2.12 Remark. If one extends the definition of graded dimension to arbitrary free graded A-modules as yielding a pair of cardinals, then [2.11] remains true; even the proof only needs cosmetic changes. Note that our definition of an isomorphism as an even invertible map is crucial in the invariance of the graded dimension. Had we defined an isomorphism as just any invertible map, the graded dimension would not have been an invariant.

2.13 Definition. On a graded A-module E = Eo tB El we define an involution
2.14 Lemma. The involution
2.15 Remark. We could have defined an involution
3.

CONSTRUCTIONS OF FREE GRADED A-MODULES

In this section we show that most constructions ofgraded A-modules preserve the existence of a basis, i.e., if the original graded A-modules are free, then the newly constructed graded A-module is again free. However, this is certainly not the casefor the most elementary "construction" of a graded submodule. We discuss sufficient conditions on generating subsets to generate a graded subspace, i.e., a graded submodule with a basis. For quotients, closely related to graded subspaces, the situation is slightly better: for finite

63

§3. Constructions offree graded A-modules

dimensional graded subspaces ofafree graded A-module, the quotient is always afree graded A-module.

3.1 Counter example. We have seen that the sum 2::iEI Fi of graded sub modules Fi of a graded A-module E is again a graded submodule. However, if all the Fi are graded subspaces, i.e., admit a basis, there is no reason why their sum should admit a basis. An elementary counter example is given by E = A2 and the two graded subspaces F1 generated by (1,0) and F2 generated by (1, /-l) where /-l is an even nilpotent element. Their sum F = F1 + F2 is the graded submodule encountered in [2.7].

3.2 Proposition. Let E be a graded A-module and let BeE be a set of homogeneous independent elements with Ba = B n Ea. Let furthermore Ha C SpanR(Ba) be arbitrary. Then F = Span(Ho U HI) is afree graded A-module. proof By applying Zorn's Lemma, we know that there exist G a C Ha such that the elements ofG a are independent over R and such that every element in Ha can be written as a finite linear combination with real coefficients of elements in G a (this can be shown along the lines of the proof that every real vector space admits a basis). Since R is a subring of A, it follows that F = Span(G o U G 1 ). We now claim that the elements of G = Go U G 1 are independent over A. To prove this claim, let 2::~=1 /-ligi = 0 be a relation between elements gi E G with coefficients /-li E A. Since G C H, it follows that there are e1, ... , em E B (finitely many!) and Xij E R, i.e., x E M(n x m, R), such that gi = 2::';=1 Xij ej. Since the gi are independent over R, the rank of the matrix x must be n and hence n :s; m. Standard linear algebra then provides the existence of a matrix Y E M (m x n, R) with the property m

.

2:: j =1 XiJYj

k

k

= Ji •

With these preparations we proceed as follows: the equality 2::~1 /-ligi = 0 implies = 0 and hence by the independence of B we have Vj : 2::i /-li Xi j = O. We

2::i,j /-lixijej

j

then multiply by Y to obtain: Vk : 0 = 2::i,j /-li Xi Y/ = 2::i /-liJf = /-lk. Since G is an independent set of homogeneous elements and clearly generates F = Span(G) it only remains to show that F is a graded submodule. But that follows immediately from the observation F n Ea = {2::~=1 aigi In E N, gi E G, a i E A, aigi E Ea}. IQEDI

3.3 Corollary. Let E be a graded A-module and let BeE be a set of homogeneous independent elements. Then F = Span(B) is afree graded A-module admitting B as basis.

3.4 Definition. We will say that a graded subspace F of a free graded A-module E admits a supplement if there exists another graded subspace F, called a supplement to F, such The last condition will be satisfied if and only if F n = {O} and that E = FEB

F.

F

64

Chapter II. Linear algebra of free graded A-modules

F

F + = E. Note that the existence of a supplement to F as graded subspace is a stronger condition than the existence of a supplement to F as graded submodule [1.3.3].

3.5 Corollary. A graded subspace F of a free graded A-module E admits a supplement if and only If there exists a homogeneous basis B of E such that a subset of B forms a basis of F (and then the complementary subset forms a basis for a supplement).

3.6 Proposition. Let F be a graded subspace of a free graded A-module E. Then F admits a supplement ifand only If ElF is a free graded A-module. If that is the case, E is isomorphic to FEEl (E / F).

F,

Proof If F admits a supplement it follows immediately that the canonical projection i.e., 7r : ~ E / F is a bijection. Whence it follows that ElF is a free restricted to graded A-module isomorphic by 7r to and that E is isomorphic to FEEl (E / F). On the other hand, suppose that ElF is a free graded A-module with homogeneous basis (Xj)jEJ. Choose homogeneous Ii E E such that 7r(Ii) = Xj and let (ei)iEI be a homogeneous basis of F. We claim that {ei liE I} U {Ii I j E J} is a homogeneous basis of E. First independence: let L:i )... iei + L: j pJ Ii = 0 be a relation, then applying 7r we get L: j pJXj = 0 and hence pJ = O. Butthen L:i )...iei = 0 and hence)...i = O. Next consider e E E, then 3/-Lj such that 7r( e) = L: j /-LjXj, but then e - L: j /-Lj Ii E F, i.e.,3)...i IQEDI such that e - L: j /-Lj fj = L:i )...i ei . The result now follows from [3.5].

F,

F

F,

3.7 Proposition. ifF is a finite dimensional graded subspace ofafree graded A-module E, then every homogeneous basis C of F can be extended to a homogeneous basis of E.

Proof Let B be a homogeneous basis for E and let C = {h, ... , fn} be a homogeneous basis for F. Since FeE and by definition of a basis, there exist el, ... , em E B (finitely many!) and ai j E A, i.e., a E M (n x m, A), such that fi = L: j ai j ej. By [2.5] and the independence of the fi we conclude that m ~ n. By reshuffling the elements ej we may assume without loss of generality that the matrix b E M (n x n, A) defined by bi j = ai j , 1 :s; i, j :s; n has the property that B (b) is non degenerate, i.e., that b is invertible [1.11]. But then the relation fi = L: j aijej can be rewritten as L:i (b- 1 ) k i fi = L:i,j (b-1)k iaij Ii = ek + L:~l L:j"=n+l (b-1)k iai j Ii- In other words, the el, ... ,en can be expressed in terms of h, ... , f n, e n+ 1, ... , em. It follows that C U (B \ { el, ... ,en}) is a homogeneous basis for E. IQEDI

3.8 Corollary. Let F be a finite dimensional graded subspace ofafree graded A-module E. Then F admits a supplement and ElF is a free graded A-module.

65

§3. Constructions of free graded A-modules

3.9 Counter example. In order to show that [3.7] cannot be extended to infinite dimensional graded subspaces without a more precise knowledge of A, we consider two examples. First we consider A = R. Free graded A-modules are in this case nothing more than real vector spaces split as a direct sum of two subspaces, this splitting being part of the data of the free graded A-module (see also §6). Since the existence of bases is always guaranteed for real vector spaces, [3.7] is true for any graded subspace in the case A = R. In the second example we use the full exterior power 1\ X of an infinite dimensional (real) vector space X as A. Let {Xi liE N} be a set of independent (over R) elements of X, and define ni E 1\ 2 X CAD, i E N by ni = X2i 1\ X2i+l. Obviously we have n: = 0, but any (finite) product ofdifferent n/s is non-zero because of the independence of the Xi. Now let B = {ei liE N} be any countable abstract set and define the free graded A-module E by E = F(B, c), where c is constant 0 on B, i.e., E is the free graded A-module on the even symbols in B. In E we define the even vectors fi, i E N by fi = ei - ni ei+l. Our first claim is that the fi are independent. To see this, suppose "L,i=p Ai fi = O. An elementary calculation shows (3.10)

i=p

i=p+l

and the result follows from the fact that the ei are independent. We conclude that the subset F = Span ( {fi liE N }) is a graded subspace of E. Our second claim is that F is a proper subset of E, and more in particular, that eo rt- F. But this follows as easily from (3.10) as the independence of the j;. Finally suppose that there exists a basis for E of which a proper subset forms a basis for F. By replacing this subset by {fi liE N} it remains a basis for E. This is not as obvious as it might seem, but the proof (of the independence) follows along the same lines as the proof of [3.7]. Without loss of generality we thus may assume that {fi liE N} c E. Now choose e E E \ {fi liE N} arbitrarily. Since B is a basis for E, there exist Ai E A such that e = "L,i=p Aiei. An elementary induction argument shows the equality

E

q

ei

k-l

q-l

k=i e=i

e=i

= L(II ne)fk + (II ne)nqeq+l

,

from which we deduce the following expression for e :

q

e

=L

q

k-l

q

q-l

e=i

i=p

e=i

L Ai (II ne)!k + (2: Ai(II ne))nqeq+l .

i=p k=i

MUltiplying this formula by nq kills the term with eq+l and we are left with a non trivial relation between e and the fi (nq =f:. 0). This contradicts the independence of E. We conclude that there does not exist a basis for E of which a subset forms a basis for the graded subspace F.

66

Chapter II. Linear algebra offree graded A-modules

3.11 Proposition. Let E and F befree graded A-modules and let 1> E Homs(E; F) be homogeneous. If ker ( 1» is a graded subspace of E admitting a supplement, then im( 1» is a graded subspace ofF. If in addition 1> is even, then E 2:! ker( 1» EEl im( 1». proof Let H be a supplement to ker( 1» in E. It follows that 1> restricted to H is injective. If C is a homogeneous basis of H, then one easily shows that 1>( C) is a set of independent homogeneous elements generating im( 1> ). In other words, im (1)) is a graded subspace of F. If in addition 1> is even, 1> is an isomorphism between Hand im( 1». Since H is isomorphic to E/ker(1)), the result follows. IQEDI

3.12 Counter examples. The counter examples given here are intended to prove that the conditions in [1.3.9] and [3.11] cannot be weakened easily. On the other hand, they also show that these conditions, although sufficient, are not necessary. Define 1> : Al -+ A2, 1/J : A2 -+ AI, and X : Al -+ Al by 1>(a) = (a,/la), 1/J(a, b) = b - I_La, and x(a) = /la, where /l E A is (for the moment) arbitrary. These three maps are right linear. Obviously 1> is injective and 1/J is surjective, which implies that ker( 1» and im( 1/J) are graded subspaces and in particular graded submodules. Moreover, im( 1» and ker( 1/J) are both the submodule of A 2 generated by the vector (1, /l). However, if /l is odd, then 1> and 1/J are not homogeneous and neither im( 1» nor ker( 1/J) is a graded submodule. If /l is invertible but not homogeneous, then X is not homogeneous but ker(x) and im(x) are (trivially) graded subspaces. On the other hand, if /l is homogeneous, nilpotent and non-zero, then X is homogeneous but neither im(x) nor ker(x) is a graded subspace (though they are graded submodules). Let E be a (the) A-vector space of dimension 111 with homogeneous basis e, f. If we define 1> : E -+ A by 1>(ae + f3f) = a - f3, then 1> is left linear but not homogeneous. ker( 1» is generated by e + f and is a submodule but not a graded submodule: the even and odd parts of ker (1» do not belong to ker (1».

3.13 Lemma. Suppose E I , ... , En are finite dimensional free graded A-modules with dim(E i ) = Pilqi, then EB~l Ei is afinite dimensional free graded A-module of dimension (2:~=I Pi) I (2:~=I qi).

3.14 Lemma. If E and F are finite dimensional free graded A-modules of dimension plq and r Is respectively, then E &; F is a finite dimensional free graded A -module of dimension

(pr+qs) I (ps

+ qr).

Proof This is an immediate consequence of [1.8.9].

3.15 Lemma. If E is a finite dimensional free graded A-module, then the k-th exterior power /\k E is a finite dimensionalfree graded A-module.

67

§3. Constructions of free graded A-modules

Proof This is an immediate consequence of [1.8.14] and [1.8.15].

3.16 Nota Bene. Let E be a finite dimensional free graded A-module of dimension plq, and let (ei)f~{ be an ordered homogeneous basis of E. It follows from [1.8.15] that the set

B = { ei 1

1\ ... 1\ eik I i l

< ... i j < i H l :s; ... :s;

ik

&

c( eil) = ... = c( ei j ) = 0, c( eiJ+l) = ... = c( eik) = 1 } is a basis of /\k E. An elementary computation then shows that the total dimension of E is given by

N

min(k,p) ( )

'"' L 8=0

p. s

(

q- 1 + k -s ) . k-s

In this sum, the terms with even s contribute to the even dimension of /\k E, terms with odd s contribute to the odd dimension.

3.17 Construction (dual basis). Let E be a finite dimensional free graded A-module and (ei)f=l a homogeneous basis. We define the maps ei E E* by e i (2::i eja j ) = a i . We leave it to the reader to verify that these (ei)f=l are independent, generating and homogeneous of degree c( e i ) = c( ei). We conclude that E* is a finite dimensional free graded A-module of the same graded dimension as E. The basis (ei)f=l for E* will be called the right dual basis to (ei)~l' In a similar way one shows that *E is a finite dimensional free graded A-module of the same graded dimension as E with as basis the maps ie E *E defined by (( 2:: j a j ej I ie)) = a i . The basis (ie )i=l is called the left dual basis to (ei)f=l'

3.18 Proposition. If E l , ... , Eb and F arefree graded A-modules, of which the Ei are finite dimensional, then Homs(E l , ... , E k ; F) is afree graded A-module.

Proof According to [1.4.9] the graded A-module Homs(E l , ... , E k ; F) is isomorphic to Homs(El 0 ... 0 E k ;F). According to [1.8.9] El 0··· 0 Ek is a finite dimensional free graded A-module, and thus by [1.8.8] HomR(E l 0··· 0 E k ; F) is isomorphic to F 0 (El 0· .. 0 Ek)*, which is a free graded A-module by [3.17] and [1.8.9]. A similar argument holds for HomL. IQEDI

3.19 Corollary. Given free graded A-modules E and F, E being finite dimensional, then Hom~k(Ek; F) is afree graded A-module.

Proof According to [1.5.5] there exists an isomorphism between Horns (/\k E; F) and Hom~k(Ek; F). But according to [3.18] the former is a free graded A-module. IQEDI

68

4.

Chapter II. Linear algebra of free graded A-modules

LINEAR MAPS AND MATRICES

In ordinary linear algebra over a field one often identifies linear maps with matrices. Of course this requires bases, but since these are usually understood, there is no real problem. Unfortunately, for free graded A-modules the situation is not so easy. Even after the choice ofa basis, there still are three different natural ways to identify a linear map with a matrix. We discuss the merits of all three ways and we point out some of the dangers involved in this kind ofidentification (see also [VI. 1.20J). The comparison between linear maps and matrices allows us to explain why we called the operator 'I' linking left and right linear maps the transposition operator.

4.1 Construction (linear maps and matrices). Let E and F be free graded A-modules of dimension n = plq and m = rls, and let (ei)f=l and (Ii )';:1 be homogeneous bases of E and F respectively. We denote by (ie)i=l' (e i )i=l' (If)'j=l' and (Ji)'j=l the left and right dual bases. We know that (Ii 0 ei)~j'::l is a basis for F 0 E*. Since finite dimensional free graded A-modules satisfy the conditions of [1.8.8], F 0 E* is isomorphic to HOillR(E; F). We thus are allowed to say that (Ii 0ei)~j'::1 is a basis for HOillR(E; F). Since this basis is labeled in a natural way with two indices, coordinates of a map 1> with respect to this basis can be seen as matrices of size m x n. However, the position of the coordinates is not uniquely determined. We have three types: left coordinates eMR (1))j i, right coordinates MR (1))j i, and middle coordinates NIR(1»j i, determined by

1> = L eMR(1))ji' fj 0e i

L

i,j

Ii 0NIR(1))ji' ei

Lfj0 ei .MR(1))ji. i,j

i,j

The coefficients of these three matrices are related by eMR(1»j i =

M R(1))ji

([E(fj)

(NIR (1))j i) and

= ([E(e d (NI R(1»ji)'

In order to visualize these relations, we suppose that the bases of E and F are ordered, and we write any matrix X E M (m x n, A) in block form X = ( ~ ~ ) with A of size r x p and all others of the appropriate corresponding sizes. In terms of such a decomposition, the relation between the three systems of coordinates is given by

NIR (1)) =

A ( c DB)

===?

eMR (1)) =

(A B) \!:(c) \!:(D)

Me (1)) = R

(A \!:(B)) c \!:(D)

.

We thus see that we have three different maps HOillR(E; F) -+ M(m x n, A), all three "honest" coordinates, and all three obviously additive. However, their behavior under left or right multiplication by A is quite different: for 1> E HOillR(E; F) and>. E A one easily verifies the following results:

eM (>'1» =

(>'A >'B) >.c >'D

NI (>'1» _

(>'A >'B) \!:(>.)C \!:(>.)D

R

R

-

69

§4. Linear maps and matrices

Since we did not (and never will!) give the set M(m x n, A) the structure of a graded A-module, we cannot say that these maps are either left or right linear. Actually, saying that one of them is left (or right) linear gives M(m x n, A) the structure of a free graded A-module, but each choice gives a different structure. For left linear maps we do the same: we represent HomL (E; F) in three different ways by matrices in M(n x m, A). For any 'I/J E HomL(E; F) we introduce left coordinates eML('I/J)i j , right coordinates M1( 'I/J)i j and middle coordinates ML( 'I/J)i j , determined by

'I/J =

2: eML('I/J)i

j

·

2: ie 0iJ .M1('I/J)i

ie 0 iJ i,j

i,j

j

.

i,j

The visualization of the relation between these three types, as well as their behavior under multiplication by elements of A is the same as for right linear morphisms, except that in the left linear case the matrix A has size p x r (instead of r x p) and the others of the appropriate corresponding sizes. We thus obtain the maps eMs, Ms, and Ms which are additive bijections between Horns (E; F) and matrices. But one should never forget that these maps do depend upon the choice of bases for the free graded A-modules E and F. Note also the position of the indices: it is always the first index that is the row index and the second one the column index. On the other hand, the height of an index depends upon the height of the corresponding basis and is more a question of custom than necessity.

4.2 Lemma. Let E, F and H be free graded A-modules of total dimension n, m, and f.

respectively, each equipped with a homogeneous basis, then for 1>s E Homs( E; F) and 'l/Js E Homs(F; H) we have ML(1)L

0

'l/Jd = ML(1)d . ML('l/Jd

and MR('l/JR °1>R)

= MR('l/JR) . M R (1)R)

,

i.e., MR and ML are homomorphisms with values in M(m x n,A) and M(n x m,A) respectively. Proof This is an elementary verification using the definition of these matrices. The fact IQEDI that Ms is in between the tensor product symbols is crucial.

4.3 Remark. The fact that ML is a homomorphism when using the alternative composition o once again confirms the idea that left linear maps should be seen as operating on the right rather than on the left.

4.4 Discussion. According to [4.2], it is Ms that transforms the composition of homomorphisms into matrix multiplication. (We leave it to the reader to verify that the others do not.) In particular when E is the same as F, Ms becomes a ring isomorphism

Chapter II. Linear algebra of free graded A-modules

70

NIs : Ends(E) __ M(n X n, A). These facts make that NIs is by far the most attractive matrix representation of the three types. Since the visualizations allow for an easy translation between these three types, we thus will study only the maps NIs in more detail.

4.5 Lemma. Let E be afinite dimensional free graded A-module, (ei)f=l a homogeneous basis with the associated dual bases. Then't(ie) = (_l)(E(e
4.6 Lemma. Let E and F befree graded A-modules, ofwhich E is finite dimensional. Under the identification *E &;F ~ HomL(E; F) the transposition maps 'tHomL(E;F) and 'I'E are related by 'tHomL(E;F) = 91 0 ('I'E &; id(F)) = (id(F) &; 'I'E) 091. Proof An elementary calculation using homogeneous bases and their duals shows that all three maps send the basis elements ie &; fj E HomL(E; F) to the same element (_l)(E(ei)IE(e,))( _l)(E(e,)IE(fj)) /j &; ei

=

(_1)(E(e i )IE(ie0 fj)) /j &;

ei E HomR(E; F).

IQEDI

4.7 Corollary. Let E and F be finite dimensional free graded A-modules, both equipped with homogeneous bases, and let ¢ : E -- F be left linear. Then:

i,j

i,j

i,j

=

L/j

&; ((_l)(E(ei)IE(ei)+E(fj)). Q:E(ei)+€(fj) (Ci j )) . e i .

IQEDI

i,j

4.8 Discussion. The result of [4.7] is quite unintelligible, except for experts in indices. So let us visualize this relation as we did for other relations in [4.1]. For a left linear map ¢ E HomL (E; F) we write:

and where AL is of size p x r, AR of size r x p, and all other submatrices of the appropriate corresponding sizes. In terms of these submatrices, [4.7] becomes (4.9)

§4. Linear maps and matrices

71

where the superscript t indicates the ordinary transpose of a matrix, i.e., interchanging the roles of row and column indices. If both q and s are zero, i.e., if there are no odd basis vectors, (4.9) reduces to !VJR (,r1» = !VJL( 1> ) t, explaining the name transpose for the operator '3:. The transformation (4.9) obviously represents an internal operation on e e l matrices given formally as MR 0'3: 0 (ML)- . If both q and s are non-zero, this internal operation has order 4. This fact has no interpretation in terms of linear maps, essentially because applying '3: twice does not make sense. In case a left linear 1> is even, it is also right linear and '3:1> = 1>. Moreover, all entries of As and Ds are even and all entries of Bs and Cs are odd. We thus can apply both!VJL and !VJR to 1>, and (4.9) reduces to the equality (4.10)

This operation, which for even 1> relates !VJL(1)) to !VJR (1)), is sometimes called graded transpose or super transpose. However, as above, the reader is warned that this operation is not its own inverse.

4.11 Example. For a free graded A-module E of dimension plq, consider the free graded A-module HomL(A; E), which we have identified with E via f---+ P 111») [1.2.19]. For e E E, we will denote (in this example only) the element
In other words, the map !VJL applied to the map ee identified with the vector e gives us the left coordinates of this vector with respectto the basis (ei )i= l ' The case E ~ Hom R (A; E) is similar: the map !VJR applied to a right linear map R1 ( e) == ee identified with the vector e gives us the right coordinates of this vector with respect to the basis (ei)i=l :

For'I/JR E HomR(E;F) wecanwrite'I/JR(e) = ('l/JRoee)(l), i.e., evaluation corresponds to composition. Similarly for 'l/JL E HomL(E; F) wehave (( e II 'l/Jd = (( llleeo'I/Jd. Since !VJs transforms composition into matrix multiplication [4.2], we see that the sequence of numbers !VJR ( ee) i, i = 1, ... , n should be interpreted as column vector, and that the sequence of numbers !VJL(ee)i, i = 1, ... , n should be interpreted as row vector. Ifwe now recall that '3: relates these two identifications: '3:(ee) = ee [1.2.19], we see that our interpretation of '3: as a (kind of) transpose operator is coherent with the interpretation of !VJs(e) as row/column vectors.

72

Chapter II. Linear algebra of free graded A-modules

But we can learn something more: in this case the visualization (4.9) reduces to the equality

In this equation one can see the danger in identifying a free graded A-module of total dimension n with An by means of the left or right coordinates: in An one does not see that left and right coordinates are quite different (see also [1.3.5]). In fact, in An the right and left coordinates are the same because its dimension is niO.

4.12 Proposition. Let E and F be finite dimensional free graded A-modules equipped with homogeneous bases, and let1>s be in Homs(E; F). Then !VJR (*1>£) = !VJL(1)£) and !VJL(1)'R) = !VJR(1)R).

Proof We consider the case for 1>£. Let (ei)f=l and (fj)'J:=l be homogeneous bases for E and F respectively. Using the dual bases we have 1>£ = L:i,j ie 0 !VJL(1)£)i j . Ii. Under the identification E 3:! (*E)* [1.7.26], [1.7.27] the vectors ei and Ii form the right dual bases to the bases (ie)~l and (if)'J:=1 of *E and *Frespectively. We thus have *1>£

=

ie0 !VJR (*1>R)i j • Ii E HomR(*F; *E). The matrix elements !VJR (*1>R)i j are given by j the equation !VJR (*1> R)i = (ei 0 *1> L) (if) = ei (1)£ <> jf) = (( ed 1> £ <> j 1)) = !VJL( 1> L)i j , where the third equality follows from the identification E 3:! (*E)*. IQEDI

L:i,j

4.13 Discussion. In standard linear algebra over a field, taking a dual map and transposing a matrix are corresponding operations. In the graded case this is no longer the case: the dual of a left linear map becomes right linear, and in general we cannot apply the same matrix representation (!VJ£ or !VJR) to both. In order to be able to apply the same matrix representation, we have to apply the transposition operator't. However, if 1> is even, we can apply the same representation, in which case it follows from (4.10) that the matrices of 1> and its dual map 1>* are related to each other by the graded transpose.

4.14 Definitions. We denote by M (plq, A) the ring M((p + q) x (p + q), A). The reason for the notation plq rather than the obvious notation p + q is because we will usually think of elements X E M(plq, A) as matrices with a block structure: X = (~~), where A is of size p x p and E, C, and D of the corresponding appropriate sizes. We also define Gl(plq, A) as the set of those invertible elements in M(plq, A) for which all entries of the submatrices A and D are even elements in A and all entries of E and C are odd. We stress that we do not turn M (plq, A) into a free graded A-module and that these definitions are purely intended to facilitate discussions (see also [4.11]). If E is a free graded A-module of dimension plq, all three maps -MR , !VJR , MR are bijections from EndR(E) to M(plq, A). Declaring one of these to be an isomorphism of graded

§5. The graded trace and the graded determinant

73

A-modules would equip M(plq, A) with the structure of a graded A-module, but then the remaining two bijections will fail to be isomorphisms.

4.15 Notation. As we noted already, the maps NIs depend upon the choice of bases in the free graded A-modules concerned. If needed we will make this dependence explicit by denoting the matrix NIs (1)) E M ( ~::xx,: A) defined in [4.1] as NI~,j (1)) to indicate that it depends upon the bases (ei) and (iJ). In the special case when E = F has dimension plq and Ji = ei, we will simplify this notation to NIB : Ends(E) ~ M(plq, A), i.e., we will denote NI~,e as NIB'

,

4.16 Lemma. For a given ordered homogeneous basis (ei)f~{ of a free graded Amodule E ofgraded dimension plq, the maps NIH. : (EndR(E), 0) ~ M(plq,A) and NIL : (EnddE), 0) ~ M(plq, A) are isomorphisms of rings. In particular, the maps NIB : Auts(E) ~ Gl(plq, A) are isomorphisms of groups (when equipped with the corresponding composition operator).

4.17 Lemma. Let E and F be A-vector spaces of dimension plq and rls respectively. Let (ei )f!{ and (ei )f!{ be two ordered homogeneous basesfor E and let (gj) j~r and Cih) j~r be two ordered homogeneous bases for F. Let furthermore x, y, and fj be matrices that relate these bases by ek = 2:i Xk i ei = 2:i ei Xi k and!ie = 2: j Ye j gj = 2:i gj ffi e· Then x, x E Gl(plq, A) and y, fj E Gl(rls, A). Moreover, for 1>s E Homs(E; F) we have

x,

and

x

Proof That x, E Gl(plq, A) and y, fj E Gl(rls, A) is an immediate consequence of the invariance of the graded dimension and the fact that the bases are ordered. The relations

between NI;;fJ and

NI~,g follow by a direct computation.

IQEDI

4.18 Definition. For future use we define the rank ofa morphism 1>s E Horns(E; F), denoted as rank( 1>s), by rank( 1>s) = rank( NIs (1)s), where on the right hand side we use [1.9], i.e., rank( 1>s) is the number of independent rows or columns in the matrix with real entries BNIs (1>s). A priori this depends upon the choice of the bases, but from [4.17] we know that BNIi'g( 1>£)

= (Bx) . (BNI1,g) . (By)-l.

Since Bx and By are invertible,

it is a classical result of linear algebra over R that the rank of B NIi,g (1) £) is the same as that of BNI1,g, i.e., the rank of 1> £ is a well defined number (and similarly for 1>R)'

74

5.

Chapter II. Linear algebra of free graded A-modules

THE GRADED TRACE AND THE GRADED DETERMINANT

In this section we define the notion of the graded trace on Ends(E) in abstract terms before we give its expression in terms ofa matrix representation [5.7J. We also prove the graded commutativity of the graded trace. We then go over to the discussion of the graded determinant for elements of Aut(E). We first define this graded determinant on matrices in Gl(plq, A) [5.16J. Some lengthy calculation are then needed to show that the graded determinant of matrices is multiplicative, which in turn is necessary to show that we can define it intrinsically on Aut(E).

5.1 Construction (elementary contractions). Let E, F and G be free graded A-modules of which E is finite dimensional. Consider the map 1'E : F x E* x E x G -- F 0 G given by 1'E(1, 1>, e, g) =

f

0 (1)(e)) 9 .

It is easy to check that 1'E is even and 4-linear. We thus obtain an induced linear map 0 E* 0 E 0 G -- F 0 G. This induced map is usually called the elementary contraction of E* and E; it satisfies the relation

rE : F

r E (1 01> 0 e 0 g) = f

01>( e) 9 .

In the same way one can define the elementary contraction of E with *E (note the order!) as the map Er : F 0 E 0 *E 0 G -- F 0 G satisfying

5.2 Examples. • For finite dimensional free graded A-modules the examples in [1.4.10] can all be interpreted as elementary contractions. For instance, let us consider the composition ~: HomL(E; F) 0 HomL(F; G) -- HomL(E; G). If we make the identifications HomL(E; F) ~ *E 0 F, HomL(F; G) ~ *F 0 G, and HomL(E; G) ~ *E 0 G, which are valid for finite dimensional free graded A-modules, then the map ~ becomes a map *E 0 F 0 *F 0 G -- *E 0 G. In this guise, ~ is exactly the elementary contraction • As a slightly more elaborate version of the previous example, let us now consider the identification HomR(E 1 ; F 1 ) 0HomR(E2; F 2) ~ HomR(E 2 0 E 1 ; F2 0 Fd. Using the identification of morphisms with tensor products as before, evaluation of an element 1>1 01>2 on an element e2 0 el can be interpreted as a map

Fr .

We leave it to the reader to verify that this map decomposes as represents the permutation (1,2,3,4) 1--7 (2,3,1,4).

r E1

091".0

rE

2 ,

where (J

§5. The graded trace and the graded determinant

75

5.3 Construction (graded trace). Let E be a finite dimensional free graded A-module and consider the space *E 0 E. As we know, it is isomorphic to EnddE). To this space we cannot apply directly the contraction of E with *E, but after interchanging the factors, we can. We thus obtain an even linear map gtrL = Er 0 91: EnddE) ~ A- For homogeneous elements e E E and 1> E *E this map is given by

Another way to obtain this map is as the linear map induced by the even bilinear map *E x E ~ A defined on homogeneous elements by (1), e) 1--7 (-1) (e(4» lee e)) (( e 111> )). The map gtr L is usually called the (left) graded trace of a left linear endomorphism. The (right) graded trace gtrR is defined similarly: gtrR = rE 091: EndR(E) ~ AFor homogeneous elements e, 1> it is given as gtrR(e 01» = (_l)(e(e)le(4»)1>(e).

5.4 Proposition. Foranyfinitedimensionalfree graded A-module E we have the identities gtrL = rE 0 ('I'E 0 id(E)) and gtrR = Er o (id(E) 0 't::-J).

Proof Let (ei)f=l be a homogeneous basis of E. The first identity follows from the fact that both maps send the basis element ie 0 ej to (-1) (e(ei)le(ej)) 53 = (-1) (e(ei)le(ei)) 53' The second identity follows from a similar argument.

IQEDI

5.5 Corollary. gtrR o'tEndL(E) = gtrL' 5.6 Lemma. For a finite dimensional free graded A-module E, a homogeneous basis (ei)~l of E and 1>s E Ends(E) we have

Proof We prove the formula for gtrd1>d, the case gtrR(1)R) being analogous. First note that 1> L = L:i,j ie 0 !VJl (1) L)i j . ej = L:i,j ie 0 ej . ([.e(ej) (!VJl (1) L)i j ), and thus:

i,j

= 2)_1)(e Ce)le(e j ))5; ([.e(ej) (!VJl(1>L)i j ) ij

5.7 Discussion. It will be clear from the formula; for the graded trace that in both the left and right case the operation on the corresponding matrix is the same: a sum over diagonal

76

Chapter II. Linear algebra of free graded A-modules

elements, adding a sign and applying the involution
gtrs(1)s) = tr(As) - is even, both gtr s apply. It follows immediately from [5.5] that gtr R (1)) = gtr L (1)). But the same result also follows from the facts that (i) for even 1> the submatrices A and D have only even entries, (ii) the ordinary trace is invariant under ordinary transposition, and (iii) formula (4.10). For even 1> the formula for the graded trace reduces to the formula gtrs(1)) = tr(A) - tr(D).

5.8 Counter example. One might be tempted to think that gtr Land gtr R give the same result when a map is both left and right linear. The following example shows that this need not be the case. Suppose A = /\ X, where X is a finite dimensional vector space over R. Let (xi)f=l be a basis of X and denote 0 -=f- b = Xl 1\ ... 1\ Xn E /\n X. Consider furthermore the free graded A-module E of dimension 011 with odd basis vector el. Finally define the map 1> : E ~ E by

This map is obviously right linear, but is also left linear! To see this, let A E A be odd, then 1>(Aela) = 1>( -eIAa) = -elbAa = 0, because for any odd A E A the product bA = O. Similarly A1>(ela) = O. And thus 1> is also left linear. However, ifn is odd, elb = -bel, and thus we find gtr R (1)) = -
5.9 Lemma. If E and F are finite dimensional free graded A-modules, then we have for homogeneous 1> E Homs(E; F) and 1fJ E Homs(F; E) the equality

IfE

= F this

equality can be written as gtrs([1>, 1fJ]s)

= o.

Proof In order to prove the right linear case (the left linear case being similar), we consider the free graded A-module HomR(E; F)0HomR(F; E) 2'! F0E*0E0F*. Theequality we have to prove amounts to proving that the maps gtr R 0 r E and gtr R 0 r F 091(23) defined on this tensor product with values in A are the same. But this follows from the fact that both map a basis vector fi 0 e j 0 ek 0 fe to the same value

where the last equality follows from the Kronecker 8's.

§5. The graded trace and the graded determinant

77

5.10 Remark. Let E be a free graded A-module of dimension plq. It is an elementary exercise to prove that gtrs : Ends(E) -- A is the unique even linear map such that (i) gtrs(id(E)) = p - q, and (ii) gtrs([1>, 1/J]s) = 0 for all 1>, 1/J E Ends(E).

5.11 Definition. The ordinary determinant, defined for square matrices with real coefficients, is a polynomial function in the matrix entries. It follows that we can extend the determinant to square matrices with coefficients in any commutative ring, yielding a value in this ring. Since Ao is a commutative ring, we thus can extend the determinant function to M(n x n, Ao). We note that the determinant so extended preserves most of its properties, in particular that it is a ring homomorphism and that it can be calculated by expansion according to a row or column.

5.12 Lemma. Let A E M (n x n, Ao) be an invertible matrix, and let V E M (n xl, Ao), WE M(l x n,Ao) anda E Ao = M(l x 1,Ao) be arbitrary. Then

Det

Proof

(it

~) = (a -

WA-IV). Det(A) .

(~~) = (W~-l ~)(~~) andDet(W~_l~) = Det

e; a_W~-lv)·IQEDI

5.13 Lemma. For V E M(n x 1, Ad and WE M(l x n, AI) we have Det(In + VW) = (1 + WV)-I . Proof First note that VW E M(n x n, Ao) and hence that the determinant function can be applied. The actual proof is by induction on n. For n = 1 we have Det(l + VW) = 1 + VW = 1 - WV = (1 + WV)-I (because V 2 = W 2 = 0). For n = + 1 we define V and W to be the column vector consisting of the first entries of V respectively the row vector consisting of the first entries of W With these notations we have

n

n

In

n

+ VW = (I; + VW VnW

VWn ) . 1 + VnWn

We then can apply the previous lemma and the induction hypothesis; we obtain

Det(In + VW) = (1

+ VnWn - VnW(I;+ VW)-IVWn )· Det(I;+ VW) 00

= (1 + VnWn - VnW(2:)-ll(VW)k)VWn)' (1 + WV)-I 00

= (1

+ Vn(2:)-ll(WVl)Wn)·

(1

+ WV)-I

+ WV)-IWnVn ) . (1 + WV)-I = (1 + (1 + WV)-IWnVn)-I. (1 + WV)-I = (1 + Wv + WnVn)-1 = (1 + WV)-I .

=

(1 - (1

78

Chapter II. Linear algebra of free graded A-modules

5.14 Corollary. Let A E M (n x n, AD) be an invertible matrix, and let V E M (n xl, AI) andW E M(l x n, AI) be arbitrary. ThenDet(A+ VW) = Det(A). (1 + W A-1V)-1.

5.15 Lemma. For X

E

M(p x q, AI) and Y Det(Ip

E

M(q x p, Al)we have

+ XY) . Det(Iq + Y X) = 1 .

Proof The proof proceeds by induction on p; for p = 1 it is [5.13]. Ifp = p+ 1 we define X E M(px q,A 1), Y E M(q x p,A 1), V E M(q x 1,A 1) and WE M(l x q,A 1)

such that X = ( ~) and Y = (Y V). With these definitions we have

Ip

+ XY =

( I~+WYXY ~

p

XV) l+WV

Iq

and

+ YX = (Iq + YX) + VW .

Using [5.12] we find

Det(Ip

+ XY) = Det(Ip+ XY) . (1 + WV - WY(Ip+ Xy)-l XV) = Det(I~+ XY) . (1 + W(Iq + YX)-l V) . p

Using [5.14] we find Det(Iq+ Y X) = Det(Iq + YX). (1 + W(Iq + YX)-l V)-l. Since AD is commutative the result now follows. IQEDI

5.16 Definition. We define the graded determinant gDet : Gl(piq, A) -- AD (also called the Berezinian) by the following procedure. For X = (~ ~) E Gl(piq, A) we know that the entries of A and D are even and those of Band C odd. This plus [1.11] imply that both A and D are invertible matrices. gDet(X) is then defined as

gDet(X)

= Det(A - BD-1C) . (Det D)-I.

5.17 Proposition. The map gDet : Gl(piq, A) __ AD is a homomorphism. Proof Given two such matrices X

gDet(XX)

= (~

D) and X = ( ~ B), we compute: B

~

~

~

= Det(AA + BC - (AB + BD)(CB + DD)-l(CA + DC)) x Det(CB + DD)-l

= Det(AA + BC - (ABD- 1 + B)(D-1CBD- 1 + Iq)-1(D-1CA + C)) x Det(D)-l . Det(D)-l . Det(D-1CBD- 1 + Iq)-l .

79

§5. The graded trace and the graded determinant

We introduce the matrices Z

= D-1C and Y = jjD-l

and compute:

(AjjD- 1 + B)(D-1CjjD- 1 + Iq)-l(D-lCA + C)

= (AY + B)(ZY + Iq)-l(ZA + C) = AY(ZY + Iq)-l ZA + B(ZY + Iq)-l ZA + AY(ZY + Iq)-lC + B(ZY + Iq)-lC 00

= A(2) _l)k(YZ)k+l)A + BZ(YZ + Ip)-l A + A(YZ + Ip)-lyC k=O + B(ZY + Iq)-lC and thus:

AA + BC - (AjjD- 1 + B)(D-1CjjD- 1 + Iq)-l(D-lCA + C) 00

= A(Ip - 2)-ll(YZl+1)A - BZ(YZ + Ip)-l A - A(YZ + Ip)-lyC k=O 00

+B(Iq - 2)-ll(zY)k)C k=O = A(YZ + Ip)-l(A - YC) - BZ(YZ + Ip)-l A + BZ(YZ + Ip)-lyC

= (A - BD-1C)(YZ + Ip)-l(A - jjD-1C) We thus finally find

gDet(XX) = Det(A - BD-1C). Det(YZ + Ip)-l. Det(A - jjD-1C) x Det(D)-l . Det(D)-l . Det(Iq + Zy)-l

= gDet(X) . gDet(X) . (Det(YZ + Ip) . Det(Iq + Zy))-l = gDet(X) . gDet(X) because of [5.15]

5.18 Remark. Using the decomposition (~ ~) = ( ~ ~) [5.15], it is elementary to show that we have the equalities

A gDet ( C

1

. (D!1C A-I B),

[5.17], and

B) _ Det(A - BD-1C) Det(A) D = Det(D) = Det(D - CA-IB) .

5.19 Definition. Let E be a finite dimensional free graded A-module of dimension plq, then we extend the definition of graded determinant to Aut(E) by the following procedure. Choose an ordered homogeneous basis (ei) for E and define

gDet(¢) = gDet(Nrs(¢)) .

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Chapter II. Linear algebra of free graded A-modules

5.20 Proposition. The value of gDet( ¢) does not depend upon the choice of NIL or NI'R> nor upon the choice of the basis (e;).

),

Proof Let (ei) be an ordered homogeneous basis of E, then, using the block matrix form NI (¢) = (~= ~= we have gDet(NIL(¢)) = Det(AL - BLDr;ICL). Det(DL)-1 and

s

gDet(NIR(¢))

=

Det(AR - BRDi/CR) . Det(DR)-I. But the equality of these two quantities is immediate if we use (4.9), realize that the ordinary determinant is invariant under transposition, and realize also that C Lt(DL t)-l BL t = -(BLDr;ICL)t because Band C have odd entries that produce a minus sign when interchanging. The fact that gDet (¢) does not depend upon the choice of the basis is an immediate consequence of [5.17], [4.15] and [4.17]. IQEDI

5.21 Corollary. The map gDet : Aut(E) morphism.

~

Ao is a well defined multiplicative homo-

5.22 Remark. The graded determinant gDet is a generalization of the ordinary determinant in the sense that it is the integrated form on the A-Lie group Aut(E) of the graded trace on the A-Lie algebra Ends(E) (see [III.3.14] and [VI.2.1S]), just as the ordinary determinant is the integrated form of the usual trace.

6.

THE BODY OF A FREE GRADED A-MODULE

In this section we discuss in detail the relation between a generic A and the special case A = R. We define, in analogy with the case of the ring A, the set ofnilpotent vectors in a free graded A-module. This allows us to extend the body map B tofree graded A-modules, yielding free graded R-modules (i. e., a direct sum of two ordinary vector spaces over R). We then introduce the notion ofequivalent bases and we define an A-vector space to be a free graded A-module together with an equivalence class ofbases. Restricting the notion ofa subspace slightly, we then can prove that there is "no" difference between a generic A and the special case A = R. All is encoded in terms ofa basis, and the only difference is what kind of coefficients one puts in front of basis vectors.

6.1 Discussion. As has been said before, the real numbers perfectly fit the conditions imposed on A in [1.5], so let us startthis section with a discussion what happens if we use A = R. It will be obvious that a graded R-module is a vector space over R in the usual sense, but ... there is more to a graded R-module than that. A graded R-module splits into an even and an odd part, both of them vector spaces over R. More precisely, if X is a graded R-module, then it defines two subspaces Xo and Xl such that X = Xo EEl Xl. On

§6. The body of a free graded A-module

81

the other hand, the reader can convince himself easily that, given a pair (Xo, Xd of vector spaces over R, the vector space X = X 0 EEl Xl is a graded R-module when we define X a to be the part of parity cx. We conclude that graded R-modules are nothing more nor less than vector spaces over R together with a splitting into a direct sum of two subspaces. In particular, a graded R-module is always a free graded R-module. However, instead of taking A = R, we could have used any A to obtain R, just by taking the body map B : A ~ R. A natural question now is, can we obtain graded R-modules in a similar way from any graded A-module? As we will see, the answer is positive when we restrict our attention to free graded A-modules.

6.2 Definition. For a free graded A-module E we define the set NE of nilpotent vectors by N E = {x EEl :3a E A : a # 0 & ax = 0 }.

6.3 Lemma. NE consists of those elements in E that have nilpotent coefficients with respect to any basis of E. Moreover, NE is a subspace (over R) of E, ANE e NE and

NE eNE . Proof Let GeE be a basis for E and let x E E, then there exists eI, ... , en E G and aI, ... , an E A such that x = 2:~ 1 a i ei. If all the a i are nilpotent, it follows from [1.4] that x ENE. On the other hand suppose x E NE,i.e., there exists an a E A,non-zero such that ax = O. It follows from the properties of a basis that then Vi = 1, ... ,n : aa i = O. If a i is not nilpotent, it is invertible and we deduce that a = 0, in contradiction with the IQEDI hypothesis. This proves the first part of the lemma; the rest follows easily.

6.4 Counter Example. Of course the definition of nilpotent vectors can be given for any graded A-module. And although the results of [6.3] remain valid for any graded submodule of a free graded A-module, they are not valid in the more general setting of arbitrary graded A-modules. To appreciate the problem, consider the graded commutative ring A with Ao = R and Al = X, where X is a vector space over R of dimension at least 2, and where the multiplication in Al is trivial, i.e., AI' Al = {O}. This A verifies the conditions given in [1.5]. In X we choose two independent vectors x and y, and we consider the sub module F ofA2 generated by the two vectors (x, 0) and (0, y). Since these vectors are homogeneous (they are odd), F is a graded sub module of A2, and thus the quotient E = A2 / F is a graded A-module. We claim that for this E the subsetNE is not a subspace (over R) of E. If7r denotes the canonical projection 7r : A2 ~ E, the vectors 7r(l,O) and 7r(0, 1) belong to NE because X· 7r(1, 0) = 7r(x, 0) = 0 and y. 7r(0, 1) = 7r(0, y) = O. However, we claim that 7r(1, 1) does not belong to NE. For suppose it did, then there should exist 0# a E A such that a . 7r(1, 1) = 7r(a, a) = 0, i.e., (a, a) E F. This implies that a must be a multiple of both x and y (remember that multiplication in Al = X is trivial), and thus must be zero. Since this contradicts the assumption a # 0, we conclude that 7r(1, 1) does indeed not belong to NE.

Chapter II. Linear algebra of free graded A-modules

82

6.5 Definition. We extend the notion of the body map B to any free graded A-module E as being the canonical projection (in terms ofreal vector spaces) B : E -- E/NE. It maps the free graded A-module E into a vector space over R, called the body of E. Note that the definition ofB for the free graded A-module A coincides with the original definition of body map just because of our requirement A = REB N, i.e., NA = N.

6.6 Lemma. Let E be a free graded A-module, thenfor all a E A andfor all x E Ewe have B(ax) = (Ba)(Bx). Moreover, ifG is a set of homogeneous elements in E, then G is an independent set in E ifand only if BG is an independent set (over R) in BE.

= (Ba)(Bx) = B((Ba)x) (the last equality because B is linear over R), we have to show that ax - (Ba)x = (a - Ba)x ENE. But a - Ba E A is nilpotent, from which we deduce that indeed ax - (Ba)x ENE. For the second assertion, let us first assume that G is independent and let us suppose that there exist r1, ... , rn E Rand e1, ... , en E G with 2:7=1 ri(Bei) = 0, or equivalently 2:7=1 riei ENE. Combining [3.3] and [3.7], we may assume thate1, ... , en are elements in a homogeneous basis of E. It thus follows from [6.3] that the coefficients ri are nilpotent. Since they are also real by hypothesis, they must be zero. In the other direction, suppose BG is independent over R and suppose there are aI, ... , an E A and e1, ... , en E G with 2:i aiei = O. Applying B and the independence of BG shows a i E N. Now let B be a homogeneous basis for E, then there exist II, ... , f m E Band bi ] E A such that ei = 2:] bi ] Ii. Again applying B and using the independence of BG and BB (just proved!) then shows that m 2 n and that the matrix Bb i ] contains an invertible n x n submatrix. Without loss of generality we may assume that it is the submatrix (Bbi])~]=l' From 0 = 2:i aiei = 2:i,] a i bi ] Ii and the independence of B we deduce that Vj : 2:i a i bi ] = O. But by [1.11] the matrix bi ], j :::; n is invertible, and hence a i = 0, finally proving that G is an independent set. IQEDI Proof To show the relation B(ax)

= EoEBE1 isafreegradedA-module, then BE = (BEo)EB(BE1) is a (free) graded R-module of the same graded dimension; in particular B(Eo.) = (BE)o..

6.7 Corollary. If E

6.8

Lemma. Given any (left or right) linear map 1> : E -- F between two free graded A-modules, there exists a unique linear map of (free) graded R-modules B1> : BE __ BF making the following diagram commutative:

E

~

F

BE

-----4

BF .

B.p

If X : F -- G is a linear map, then B (X 0 1» parity a then B1> is too.

=

(BX)

0

(B1». If 1> is homogeneous of

§6. The body of a free graded A-module

83

Proof If B¢ exists, it must be defined by (B¢)(Bx) = B(¢(x)). From this formula, linearity and uniqueness follow immediately. It thus remains to show that ¢ exists. For that it suffices to show that B(¢(x)) = 0 whenever Bx = O. To see this, choose a basis for E. The vector x has nilpotent coefficients with respect to this basis, and thus by linearity of ¢ its image is a sum of terms, each a product of a vector of F and a nilpotent coefficient. From this it follows that ¢( x) is a nilpotent vector in F. The equality B (X 0 ¢) = (BX) 0 (B¢) follows from the defining equation for B on maps. The last statement follows immediately IQEDI from the fact that (BE)o = B(Eo) (and idem for F).

6.9 Remark. The above result can be rephrased as saying that the map B is a functor from the category of free graded A-modules with its linear maps (left or right) to the category of (free) graded R-modules with its linear maps (for which the distinction left and right does not exist).

6.10 Remark. If E is a finite dimensional free graded A-module, Homs(E; F) is also a free graded A-module. In that case we have two different definitions of B¢ for a linear map ¢ : E ~ F: the one of [6.8], but also the one as given by application of the map B to the free graded A-module Homs(E; F). In the first case B¢ lies in Homs(B(E); B(F)) and in the second case in B(Homs(E; F)). Declaring these two definitions to coincide gives us an isomorphism between these two spaces. Actually, this is a special case of the more general fact that the body map B "preserves" all our constructions offree graded A-modules, e.g., 0 k (BE) is isomorphic to B( 0 k E), where the first tensor product is that of graded R-modules and the second that of free graded A-modules. The proof (using homogeneous bases) of these and similar isomorphisms are left to the reader (see also [6.26] and [6.27]).

6.11 Discussion. Since any free graded A-module E is a vector space over R, it follows from [6.6] thatB : E ~ BE is a linear map between vector spaces over R whose kernel is the linear subspace N E . However, no canonical choice for a supplement (over R) to NE presents itself. The purpose of the next definition is to create such a supplement.

6.12 Definition. Two bases (ei)iEI and (fj )jEJ of a given free graded A-module E are said to be equivalent if they are related to each other by real coefficients, i.e., if there exist aij E R such that ei = L: j aij Ii. Reflexivity and transitivity are obvious. For symmetry, suppose Jj = L:i bj iei with bj i E A. From Ii = L:i k b/a/ Jk it follows that L:i a/ = 5j. Taking bodies, we obtain the equation L:JBbj i)a/ = 5j. Defining

b/

Jj = L:i(Bb/)ei, we find Jj = L:i,k(Bb/)a/ Jk follows that the bj i are necessarily real.

= L:k 51!k = h

From this it

84

Chapter II. Linear algebra of free graded A-modules

6.13 Discussion. Specifying a basis B for a free graded .A-module E automatically gives a supplement RE for NE (over R) defined by RE = SpanR(B). It will be obvious that an equivalent basis defines the same supplement RE. We thus obtain a map from the set of equivalence classes of bases to the set of supplements for N E . This map is injective but in general not surjective [6.15].

6.14 Lemma. If E is a finite dimensional free graded .A-module, then any supplement

RE forJVE defines an equivalence class of bases of which RE is the real span. Proof By definition of a supplement, B : RE __ BE is an isomorphism of vector spaces over R. Let B be a homogeneous basis for the R-vector space BE and denote by jj C RE its image under B- 1 . Imitating the proof of [6.6], we see that jj is independent in E (using that is independent but not needing that jj is homogeneous in E). Combining [6.7], [2.6-iii], and the finite dimensionality, we conclude that jj is a basis whose real span is RE. IQEDI

BB

6.15 Counter example. To show that the condition of finite dimensionality is not void, let us consider the infinite dimensional free graded .A-module E of [3.9] with its basis {ei liE N} and the vectors fi = ei - niei+1. Since the ni are nilpotent, we have Bei = Bfi, Since the Bei form a basis of BE, it follows that the fi generate a supplement (over R) for NE. However, the fi do not form a basis for E as shown in [3.9].

6.16 Corollary. For finite dimensional free graded A-modules equivalence classes of bases can be identified with supplements RE (over R)for N E.

6.17 Discussion. So far in talking about supplements for NE we have interpreted E as an ordinary vector space over R. However, we can interpret E as a graded R-module because Eo andE1 are subspaces of E (over R). Moreover,NE is a graded subspace of the graded R-module E, i.e., NE = (NE n Eo) EEl (NE n Ed. It thus is reasonable to require that a supplement RE to NE is a graded subspace. If that is the case, the map B : RE -- BE is an isomorphism of graded R-modules. A little reflection will show that a supplement RE generated by an equivalence class of bases is a graded subspace (of E, seen as a graded R-module) if and only if the equivalence class contains a homogeneous basis.

6.18 Counter example. To show that an equivalence class need not always contain a homogeneous basis, consider the free graded .A-module E of dimension 111 with homogeneous basis {e, fl. Let MEND be non-zero, then the vectors = e + Mf and = f + Me form a basis for E whose equivalence class does not contain a homogeneous basis.

e

f

§6. The body of a free graded A-module

85

6.19 Definition. By an A-vector space E we will mean afinite dimensional free graded A-module together with an equivalence class of bases whose associated supplement RE to NE is a graded supplement (in other words, the equivalence class should contain a homogeneous basis). Moreover, when using a basis for E, it will always be a (homogeneous) basis within the given equivalence class. If E is an A-vector space, the supplement RE is part of the A-vector space structure. We thus can use the isomorphism B : RE -- BE to identify BE with RE c E and to forego the notation RE. This means in particular that we see the map B as a projection B : E -- E whose kernel is NE and whose image is BE == RE c E. The image BE can be described as those points of E that have real coordinates with respect to a basis in the equivalence class.

6.20 Remark. There is no special reason to require A-vector spaces to be finite dimensional. However, since we do not need infinite dimensional ones, our definition avoids having to say all the time that they are supposed to be finite dimensional.

6.21 Nota Bene. In case A = R we obtain the definition of an R-vector space. The reader should not confuse this with the notion of a vector space over R. Since for a graded R-module there exists only one equivalence class of bases (there are no nilpotent vectors and all linear maps are smooth), an R-vector space is exactly the same as a graded R-module. We conclude that an R-vector space is the direct sum of two vector spaces over R.

6.22 Definition. Given two A-vector spaces E and F, a linear map 1> : E __ F will be called smooth if and only if 1>(BE) c BF c F. Using one of the matrix representations of [4.1], 1> is smooth if and only if all its matrix elements are real. This is most easily seen by noting that basis vector~iin E belong to BE and that the matrix elements of 1> are (up to conjugation) the components of 1>(ei). Using [6.19] this can be stated as 1> E B Horns (E; F) or equivalently as B1> = 1> (see also [6.10]).

6.23 DiscussionlDefinition. With the above definitions, we have created a new category: the category of A-vector spaces, together with the smooth linear maps. In this category, we can still perform our constructions of new A-vector spaces. In fact, for direct sums, tensor products, exterior powers and endomorphism spaces we have given a homogeneous basis of the new free graded A-module in terms of homogeneous bases for the old free graded A-modules. The equivalence class of bases for the new A-vector space then is generated by the basis constructed from bases for the old A-vector spaces that lie within the given equivalence classes. We leave it to the reader to check that changing the original bases wi thin their equivalence classes does not alter the equivalence class of bases of the

86

Chapter II. Linear algebra of free graded A-modules

newly created A-vector space. For the A-vector space F(G, c) generated by the graded symbols G the obvious choice for a homogeneous basis is G itself. For graded subspaces (and thus quotients) the question is more delicate since there is no obvious way to induce a basis on a graded subspace once a basis for the total A-vector space is given, nor is there an obvious way to induce a basis on a quotient. We therefore restrict the notion of a graded subspace in our new category as being a graded subspace in the old sense with the additional restriction that there exists a homogeneous basisfor the total A-vector space within its equivalence class such that a subsetforms a basisfor the graded subspace in question. The equivalence class induced by this subset defines the equivalence class for the graded subspace (again, different bases within the equivalence class on the ambient A-vector space induce the same equivalence class on the graded subspace). With this definition of a graded subspace, a quotient space is well defined in our new category. Ifwe have a homogeneous basis within the equivalence class, a subset of which defines the graded subspace, the complement induces a homogeneous basis on the quotient. And as before, changing the basis on the original space (with all the restrictions specified above) does not change the equivalence class on the quotient.

6.24 Lemma. Let 1> E Homs(E; F) be a smooth homogeneous linear map between A-vector spaces. Then ker( 1» and im( 1» are graded subspaces in the sense of[6.23]. Proof Let P c BE and Q c BF be homogeneous bases in the corresponding equivalence classes. Left and right linear being similar, we consider the right linear case and we start with the even case. Since 1> is smooth, the matrix representation [4.1] of 1> (all are equal) is of the form 1> ~ ( ~ ~) with A and D matrices with real entries. In particular for any e E Po, 1>( e) is a linear combination with real coefficients (the matrix elements of A) of elements of Qo. Elementary linear algebra over R then tells us that we can change the basis vectors in Po and in Qo by real coefficients in such a way that there exist subsets PfJ C Po and Q~ C Qo such that PIJ is a basis for ker( 1>1 BEo), such that Q~ is a basis for im(1)I BEo), and such that 1> is a bijection between Po \ PfJ and Q~. A similar result holds for the odd basis vectors in Hand Ql. We conclude that there exists homogeneous bases P and Q within the corresponding equivalence classes and subsets pi C P and Q' C Q such that P' is a basis for ker(1)IBE), that Q' is a basis for im(1)IBE), and that 1> is a bijection between P \ pi and Q'. It follows easily that Span( Pi) = ker( 1» and that Span( Q') = im( 1». The result then follows from [3.3]. If 1> is odd, the matrix representation [4.1] of 1> is of the form 1> ~ (~ ~) with Band C matrices with real entries. In particular for any e E Po, 1>( e) is a linear combination with real coefficients (the matrix elements of C) of elements of Ql. Elementary linear algebra over R then tells us that we can change the basis vectors in Po and in Ql by real coefficients in such a way that there exist subsets PIJ C Po and Qi C Ql such that PIJ is a basis for ker( 1>IBEo)' such that Qi is a basis for im( 1>I BEo), and such that 1> is a bijection between Po \ PIJ and Qi. A similar result holds for the basis vectors in PI and Qo. The rest of the proof is as for the even case. IQEDI

§6. The body of a free graded A-module

87

6.25 Counter example. In [3.12] we have seen that the condition that 1> be homogeneous is not superfluous. To show that smoothness is also not superfluous, we consider the map 1> : A2 ~ A2 defmed by 1>(a, b) = (a + /-Lb, /-La + /-L2b) for some /-L E Ao, /-L (j. R. This 1> is not smooth because /-L (j. R, ker phi is generated by (- /-L, 1) and im 1> is generated by (1, /-L). Since /-L is not real, ker phi and im 1> are not graded subspaces in the sense of [6.23]; they are however graded subspaces in the sense of [2.1].

6.26 Remark. Let E be afinitedimensional A-vector space with basis (ei)f=l and F an A-vector space with basis (Ii )jEJ (homogeneous, in the appropriate equivalence classes). It follows that the elements Ii 0 e i define the equivalence class of bases for the A-vector space HomR(E; F). A right linear map 1> : E ~ F is smooth if and only if its matrix elements are real numbers, i.e., its coefficients withrespectto the basis (Ii 0ei)f=1,jEJ are real. The analogous result holds for left linear maps. We conclude that 1> E Horns (E; F) is smooth if and only if 1> E B(Homs(E; F)) c Homs(E; F), i.e., the smooth linear maps E ~ F are just the maps belonging to the body of Homs(E; F). Now recall that for amap 1> E Horns (E : F) wehave B1> = 0 if and only if its matrix elements are nilpotent (see the proof of [6.8]). This can be stated intrinsically either as 1>(BE) C NF or as 1> E NHoms(E;F), descriptions that should be compared with the definition of smoothness: 1>(BE) c BF or 1> E B(Homs(E; F)). With these preparations we can describe the isomorphism between Homs(BE; BF) and B(Homs(E; F)) (see [6.10]). The kernel of B : Homs(E; F) ~ Homs(BE; BF) defined in [6.8] is the space NHoms(E;F), which is a supplementto B(Homs(E; F)). It follows that B : B(Homs(E; F)) ~ Homs(BE; BF) must be an isomorphism.

6.27 Discussion. On the category of A-vector spaces and smooth maps we can still apply the body map B as a functor to the category of R-vector spaces. The upshot of what we will do in the remainder of this section is that we will construct a functor G from the category of R-vector spaces to our new category, which is "inverse" to B in the sense that it allows to show that these two categories are equivalent. However, we do not insist on this categorical language, we will merely point out some key constructions and properties that will be useful in the future. Readers versed in the language of categories will find it easy to complete the proof that these two categories are equivalent. The fact that these two categories are equivalent immediately implies that the constructions of direct sums, free modules, and tensor products coincide, being solutions to universal problems. But we have more: our definition of graded subspace in the category of A-vector spaces [6.23] is such that they correspond exactly with the graded subspaces in the category of R-vector spaces.

6.28 Remark. Even with the additional data of an equivalence class of bases and requiring maps to be smooth, the only invariant of a free graded A-module remains its graded

88

Chapter II. Linear algebra of free graded A-modules

dimension. More precisely, if E and F are two A-vector spaces, then they have the same graded dimension if and only if there exists a smooth isomorphism between E and F.

6.29 ConstructionIDiscussion. If E is an A-vector space, one can ask whether E is completely determined by the R-vector space BE. We will show that the answer is positive. For an R-vector space X = Xo EEl Xl we define GX as

If we introduce left multiplication by A as a· (b 0 x) = (ab) 0 x, and the grading (GX)a = (Ao 0R Xa) EEl (AI 0R Xl-a), then GX becomes a free graded A-module. If (ei)iEI is a basis of X, (1 0 ei)iEI is a basis of the free graded A-module GX (use [1.7.28]). Since the equivalence class of this basis does not depend upon the choice of the original basis (ei)iEI, we conclude that we can turn GX into an A-vector space of the same graded dimension as the R-vector space X. Using [6.3] it is elementary to show that B(GX) c GX is just the subset of elements of the form 10x, i.e., we have a canonical identification between X and B(GX). Coming back to our original question, if B is a basis for E (homogeneous, within the specified equivalence class), then BB = B (sic!) is a basis of BE, and thus 1 0 B is a basis of G(BE). We thus can identify canonically G(BE) with E, thus answering the question in the positive. Strictly speaking the spaces B(GX) and X are not the same, nor are G(BE) and E. The correct terminology is that they are canonically isomorphic. However, in the sequel we will forget such subtleties and we will write B(GX) = X and G(BE) = E, pretending that Band G are really inverse to each other.

6.30 Lemma. Let E and F be A-vector spaces and let 1> : BE

-+ BF be a linear map of R-vector spaces. Then there exists a unique smooth (left or right) linear map ofA-vector spaces G1> : E -+ F making the following diagram commutative:

E

~

F

BE

-------+

BF .



Moreover,

if1> is homogeneous of parity Ct, then G1> is too.

Proof G1> is determined by its values on a basis. Since a basis of E is contained in BE (by definition of an A-vector space), proving existence and uniqueness. Smoothness

follows because this G1> obviously satisfies (G1»(BE)

c BF.

IQEDI

§6. The body of a free graded A-module

89

6.31 Corollary. B (G1» = 1> and B is an isomorphismfrom smooth linear maps between A-vector spaces to (smooth) linear maps between R-vector spaces.

Proof The equality B(G1» = 1> follows from the uniqueness in [6.8]; that B is an IQEDI isomorphism follows from the uniqueness in [6.30].

6.32 Discussion. We can summarize the above results by saying that there is no real difference between the category of A-vector spaces (with smooth linear maps) and the category of R-vector spaces. This fact is most easily seen when we think in terms of bases: the operations Band G do not change the basis, only the set of scalars one puts in front of a basis element. From this observation it follows immediately that all constructions that are performed using (homogeneous) bases are preserved by the functors Band G. As said before, for B we have to be careful with graded subspaces and quotients, but for G there are no problems since for vector spaces over R one can always complete an independent set to a basis.

6.33 Nota Bene. Having claimed that there is no real difference between the categories of A-vector spaces and R-vector spaces (more precisely, we claim that G and B are isomorphisms of categories), we immediately have to warn the reader for a pitfall: for this statement to be true, one has to take as morphisms in the category of A-vector spaces either left or right linear smooth maps, but not both at the same time! Whenever it is useful to consider both at the same time (and it often is), Band G are no longer isomorphisms (e.g., for every odd linear map between R-vector spaces, there exist two smooth odd maps between A-vector spaces: a right and a left linear one).

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Chapter III

Smooth functions and A-manifolds

For those readers who skipped or have already forgotten the first two chapters, we suggest that they imagine that the ring A is the exterior algebra A = /\ X of an infinite dimensional vector space X over R, which we usually split into an even and odd part: AD = EB~=o X and Al = EB~=o + I X. We will never need these details about A, but it fixes the ideas. Moreover, an A-vector space E can be thought of as being a direct sum of two ordinary vector spaces over R (the even and odd parts) in which the coefficients with respect to a basis are replaced by elements ofA. In this interpretation the even pan Eo consists of those vectors which have even coefficients with respect to a basis of the even part and odd coefficients with respect to a basis of the odd part, while for the odd part EI the parities are reversed. The only thing one should remember is thatfor the basis vectors of the odd part it makes a difference whether one puts the coefficients on the left or on the right. Finally, we will always assume that our A-vector spaces are finite dimensional, and that a basis is ordered in the sense that we first put the basis vectorsfor the even part and then thosefor the odd part. The main subject of this chapter is the notion of an A-manifold, a generalization of the notion of an ordinary manifold in which R is replaced by a graded commutative ring A and in which Rn is replaced by the even part Eo ofan A-vector space E of dimension plq. Since the standard approach to smooth functions cannot be copied to this more general setting, we provide an alternative definition which works in both cases. We prove that smooth functions on Eo can be identified with ordinary smooth functions on RP times skew-symmetric polynomials on Rq, i.e., cOO(Eo) ~ COO (RP) 0/\ Rq. Once we know what smooth functions are, we define their derivatives, something which is not automatically included in our approach to smooth functions. This requires a condition on A, but this condition is satisfied by the basic example for A given above. It then is elementary to generalize the implicit and inverse function theorems; even partitions of unity pose no problems. With these ingredients we then just copy the definition of ordinary manifolds in terms of charts and transitionfunctions between charts to obtain

Nk

Nk

91

Chapter III. Smooth functions and A-manifolds

92

A-manifolds. Product manifolds are defined easily, but the construction ofa submanifold requires some care. Due to our definition of smooth functions, one cannot always construct submanifolds of lower dimension passing through an arbitrary point of the ambient manifold. We finish this chapter by extending the body map B to A-manifolds and their smooth maps. We prove that the topology ofan A-manifold M is completely determined by the topology of its body B M (which is a manifold in the ordinary sense of the word, or, in our terminology, an R-manifold). This extended body map preserves all constructions, e.g., B(M x N) = BM x BN. In terms of coordinates, the body map is just the projection onto their realpart (i.e., the projection A = /\ X __ R = /\0 X in terms ofour example of A).

1.

TOPOLOGY AND SMOOTH FUNCTIONS

In this section we provide our alternative approach to smooth functions which avoids the use of limits of difference quotients. We show that the set Coo (U; A) of A-valued smooth functions on an open set U c Eo is a graded R-algebra and that the set Coo (U; F) of F-valued smoothfunctions (F an A-vector space) is a free graded Coo (U; A)-module of the same graded dimension as F. Using the parity reversal operator, we show that the restriction to the even part of an A-vector space is in reality not a restriction at all.

1.1 Discussion. Just as smooth functions on open sets in RP are the basic ingredients for ordinary manifolds, we want smooth functions on open sets of ... to be the basic ingredients for A-manifolds. Three questions arise: what are the open sets, what are the smooth functions and, most important for the moment, what kind of spaces to put on the dots? In trying to find reasonable answers to these questions, we will be guided by the idea that R-manifolds, i.e., those obtained by taking A = R, should be the same as the ordinary well known manifolds. The most naive answer to the space question would be to take any A-vector space, but an R-vector space X is a direct sum of two ordinary real vector spaces Xo EEl Xl and for ordinary manifolds there is no subdivision of the (local) coordinates. A less naive answer, and the one we will adopt, is to take the even part Eo of an A-vector space E. If {el' ... ,ep , h, ... ,iq} is a homogeneous basis of E with the ei even and the ii odd, then a point in Eo has even coordinates with respect to the ei and odd coordinates with respect to the Ii. The number of even coordinates gives us the even dimension of E and the number of odd coordinates gives us the odd dimension of E. Moreover, if we take the special case A = R, all odd coordinates are zero, and what remains is essentially RP ,just as we wanted. We note that we thus have a profound distinction between a generic A (with Al #- {O}) and the special case A = R. In the first case we have non trivial odd coordinates and

§1. Topology and smooth functions

93

thus retain information about q (the odd dimension), whereas in the second case we lose all information about q. Another way to see this difference is to consider the body map B : E -> BE, which provides the passage from the generic case to the special case A = R. We have seen that BE contains enough information to reconstruct E as G(BE). On the other hand, BED does not contain enough information to reconstruct Eo (and certainly not the whole of E): the odd dimension of E is missing. It is this difference that gives A-manifolds their extra flavor.

1.2 DefinitionIDiscussion. Let E be an A-vector space of dimension plq. We define a topology on E as the coarsest topology for which the body map B : E -> BE £:! Rp+q is continuous, i.e., U c E is open if and only if U = B- 1 (0) for some open subset 0 of BE. Said differently, U is open if and only if BU is open and U = B- 1 (BU). Yet another way to say the same is to state that the map B : U -> BU induces a bijection between the open sets of E and the open sets of B E ~ Rp+q. This topology is usually called the DeWitt topology. All subsets of E, and in particular Eo, will be equipped with the relative topology. Since BE = E/NE , it follows that open sets U c E are saturated with nilpotent vectors, i.e., U open implies U + NE = U. For R-vector spaces the body map B is the identity, so the DeWitt topology on X is the standard euclidean topology on X. In order to have a closer look at the DeWitt topology in the general case, let us choose a basis (ei)f=l of E. With respect to this basis we consider left coordinates x = 2:i Xi ei E E to identify E with An as sets. This gives us an identification of BE with Rn such that the body map is given by B (xl, ... , xn) = (BXl, ... , Bx n ). Using this identification, a point (Xl, ... , xn) will be in the open set U if and only if (Bxl, ... ,Bxn) lies in the open set 0 = BU c Rn, i.e., the topology is completely determined by the body parts of the coordinates. If we assume furthermore that the homogeneous basis is an ordered one, a point (Xl, ... , xn) lies in Eo if and only if xl, ... ,xP E AD and x p + l , ... ,xn E AI. We deduce that a set U c Eo is open if and only if there exists an open set 0 C RP such that

Ai

c Eo A particular consequence is that the induced topology on a fiber {( Xl, ... , x P )} x is the indiscrete topology (either all or nothing). Another consequence is that for the topology on Eo we have the same characterization as for the topology on E: U C Eo is open if and only if BU is open in BED ~ RP and U = B-l(BU), where we now see B as the restriction B : Eo -> BED. Or again, B induces a bijection between open sets of Eo and open sets of BED ~ RP.

1.3 Corollary. The DeWitt topology on an A-vector space E and on its even part Eo are locally connected (any neighborhood oj a point contains an open connected neighborhood ojthat point, which is equivalent to the connected components ojopen sets being open [Du, theorem V.4.2]).

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Chapter III. Smooth functions and A-manifolds

1.4 Lemma. Let E be an A-vector space, then: (i) the product topology of Eo EEl El coincides with the topology of E; IfF and F are supplementary graded subspaces of E, then the product topology of F EEl F coincides with the topology of E; (ii) the A-vector space operations of addition (E x E ~ E) and multiplication by elements ofA (A x E ~ E and E x A ~ E) are continuous; (iii) left linear maps to A are continuous, and in particular the coordinate projections

je : x

= 2:i Xi ei

f--+

xj

= ~ x II

je )).

1.5 Notation. According to standard abuse of notation, we will often denote the (left) coordinate projections je : (xl, ... ,xPH ) f--+ x j by xj.

1.6 Discussion. So far in discussing the DeWitt topology we have treated the spaces E and BE as different. However, we have identified BE with those points in E that have real coordinates [11.6.19], i.e., we have interpreted B as a projection map B : E ~ E. For an arbitrary subset U c E we thus interpret BU as a subset of E. Moreover, the inclusion BU c U is equivalent to the equality BU = Un BE, but neither condition need be true. However, if U is open in E, and thus in particular U = B-l(BU), we have BU c U, and thus BU = Un BE c U. In other words, for open sets U the body BU consists of those points in U that have real coordinates.

1.7 Discussion. Once we have a topology, we can speak of continuous functions, and in particular we can speak of functions on open sets U of the even part of an A-vector space E with values in an A-vector space F. As usual we will denote these sets by CO(U; F). However, we want to speak of differentiable functions, and there we encounter a problem. The standard way to define a differentiable function is to say that the derivative should exist. Such a derivative is (usually) defined as the limit of a difference quotient. But in our context, we can not always write such a difference quotient because of the existence of nilpotent elements. Even disregarding the nilpotent elements, the DeWitt topology is not Hausdorff, so a limit need not be unique. In order to circumvent these problems, we will use an alternative approach to differentiable functions which does not use limits nor difference quotients. The essential idea of this approach is expressed in [1.8].

1.8 Proposition. Let 0 C RP be open, and let f : 0 ~ RN be afunction. Then f is of classC k+ l ijandonly If there exists an open coverU = {Ua I a E I} of 0 andforall a E I functions 9ai : U; ~ R N , 1 :::; i :::; p of class C k such that for all x, y E Ua : P

(1.9)

f(x) - f(y) =

L 9ai(X, y) . (xi i=l

yi) .

§ 1. Topology and smooth functions

95

Proof If f is of class C k + 1 , we take any cover of 0 consisting of convex open sets (for instance open balls). For each convex Ua in this cover we define the functions 9ai by

Jor

1

(LlO)

9ai(X,y)

=

of oxi(sx+(l-s)y)ds.

The convexity of Ua guarantees that this integral makes sense, and since the of / ox i are of class C k , the 9ai are of class C k as well. Computing L,i 9ai(X, y) . (Xi - yi) immediately gives f(x) - f(y)· If on the other hand the condition is satisfied, we compute on each open Ua the partial derivatives of f in x E Ua by (Lll)

of ux'

.

~(x) = hm

h->O

f(x

+ hei) h

f(x)

.

= hm

h->O

9ai(X + hei, x) = 9ai(X, x) ,

where ei = (0, ... , 0, 1, 0, ... , 0) denotes the i-th basis vector in RP. Since the 9ai are of class C k , this shows that the partial derivatives of f exist everywhere and are of class C k , i.e., f is of class C k + 1 . IQEDI

1.12 Remark. One can prove a stronger version of [1.S] which does not need a cover and which states that f is of class C k + 1 if and only if there exist functions 9i : 0 2 ~ R N of class C k satisfying (1.9) for all x, yEO. The reason we did not give this stronger result in [1.S] is that [1.S] as it is, remains true if we replace the field of real numbers R by the field of complex numbers C (and then it says that f is holomorphic if and only if the 9ai are continuous). On the other hand, if we replace [1.S] by the stronger result without the cover, then we no longer can go over to the complex case. The simple reason is that the stronger result needs a partition of unity argument. In fact, domains in CP for which no cover is needed occur naturally in complex analysis: by a classical corollary ofH. Cartan's theorem B, this is the case for any domain ofholomorphy (== a pseudo convex domain == a Stein domain), see [Ra, corollary 6.26]. Let us now give the proof ofthe stronger result. Since the if part is proved as in [l.S], we attack the only ifpart. We choose a cover V = {Va I a E I} of Ox o having the following property: for V E V we have either (i) :JC convex open set in 0 such that V = C x C, or (ii) :Jj (1 :s; j :s; n) such that V(x, y) E V : x j -=f- yj. The existence of such covers follows easily from the fact that RP is Hausdorff. Given such a cover, we define on each Va E V functions 9ai : Va ~ R N as follows. In case (i) we define 9ai by (1.10); in case (ii) we define 9aj(X, y) = (f(x) - f(y))/(x j - yj) and for i -=f- j we define 9ai = O. In both cases the functions 9ai are of class C k and satisfy (1.9) for all (x, y) E Va. Now let Pa be a partition of unity of class C k subordinated to the cover V (see [5.1S] for an exact definition), anddefinethefunctions 9i : 0 x 0 ~ RN by 9i(X, y) = L,a Pa(x, y) 9ai(X, y). Then for any (x, y) E 0 x 0 wehave Pa(X, y) -=f- 0 =} (x, y) E Va, and thus L,i(X i _yi) 9i(X, y) = L,a Pa(x, y) L,i(X i - yi) 9ai(X, y) = L,a Pa(x, y) (f(x) - f(y)) = f(x) - f(y). This finishes the proof.

96

Chapter III. Smooth functions and A-manifolds

1.13 Discussion. If f : 0 C RP ~ RN is smooth (of class COO), application of [1.8] gives us an open cover {Ua} of 0 and again smooth functions f~~) : U; ~ RN satisfying (1.9). We then can apply [1.8] again to each of the functions f~~), to obtain an open cover

{Uab } of Ua and smooth functions fgl)(bj) : U;b ~ RN satisfying a condition analogous to (1.9). Continuing this procedure, we get an infinite tree of open covers {Ual a2 ... ar } (i.e., U ar Ual a2 ... ar = Ual ... ar-l' and the index family for a r might depend upon the sequence and an infinite tree of smooth functions j 1:S; ij:S; p2 - 1 satisfying for allx,y E Ual ... arb:

al ... ar-l)

f«r)) ( '): alt1 ... artr

U;l ... ar ~ RN,

If on the other hand we have such a tree of covers and functions (without any differentiability assumption) satisfying (1.14), we can argue as follows. If for a fixed value of r all functions fg~ill ... (arq are continuous, then by [1.8] all functions f(~0~j ... (ar_lir_l) are of class C 1 . Continuing this argument we conclude that the initial function f == f(O) is of class Varyingthevalueofrproves [1.15].

cr.

1.15 Proposition. Let f : 0 C RP ~ R N be a function. Then f is smooth if and only if there exists an infinite tree of open covers {Ual ... ar } and continuous functions

fg~ill ... (arq as described in [1.13 J satisfying (1.14).

1.16 Definitions. With the above preparations concerning smooth functions on RP, we now return to the question how to define smooth functions on open subsets U c Eo of the even part of some A-vector space E. We start with some preparatory definitions . • Given a continuous function f : U ~ F with U c Eo open and E and FA-vector spaces, a smooth tree associated to f will be an infinite tree of open covers {Uala2 ... ar} and continuous functions fg~il) ... (arir) : U;l ... ar ~ F satisfying for all x, y E Ual ... arb: 2r dim(E)

(r) f (alill ... (a r ir) (X)

-

f(r)

(alil) ... (arq ( y ) --

(j x - Yj) . f(r+l) (alill ... (a rir)(bj) ( x, ) y .

'"'

L

j=l

• A symbol F is called a smooth system of F-valuedfunctions if for all E and for all U c Eo open, we have a set F(U; F) c CO(U; F) of continuous functions from U to F such that for any f E F(U; F) the following two conditions are satisfied: (Al) there exists afamily Ua C U of open subsets covering U and n = dim E functions 1>a,i E F(U;; F) such that n

\/x,y E Ua

f(x) - f(y) = L(x - y)i ·1>a,i(X, y) , i=l

97

§ 1. Topology and smooth functions

where the Xi are the coordinates of x with respect to some basis of E (homogeneous, in the specified equivalence class); (A2) f(BU) C BF.

1.17 Remark. In case A = R, condition (A2) is completely superfluous. We added it for exactly the same reason we added a similar condition for smooth linear maps in [11.6.22]: to be able to prove that the body map B is a bijection on certain sets of functions (see [2.16]).

1.18 Remark. We have restricted our attention to functions defined on open sets of the even part of an A-vector space. However, one can define as easily differentiable functions on open sets of the entire A-vector space. We also have restricted our attention to infinitely often differentiable functions, but functions of class C k can be defined in a similar way. For more details the reader is referred to [Tu2].

1.19 Example. Let f : U C Eo ~ F be a continuous function, and suppose there exists a smooth tree offunctions associated to f. We then can construct a smooth systemFf (of F-valued functions) as follows.

Ff(U;1 a2··· a; F) r

=

{f«r)) ( .) al1-l ... artr

11 ::; i J· ::; 2j - 1 dim(E)}

,

U;l

and if 0 is not of the form a2 ... a ' then F f (0; F) = ~, with the exception of the first r term of the tree Ff(U; F) = {f}. The definition of a smooth tree immediately proves that Ff is indeed a smooth system.

1.20 Definition. We can order smooth systems by inclusion, i.e., .1'1 ::; .1'2 if and only if for all U : F 1 (U; F) c F2 (U; F). It is also easy to prove that the union Fu of two smooth systems .1'1 and .1'2, defined by Fu (U; F) = .1'1 (U; F) U .1'2 (U; F), is again a smooth system. It thus makes sense to speak about the maximal smooth system. Which brings us to our final definition: Coo is the maximal smooth system of F-valuedfunctions. Elements f E Coo(U; F) will be called smoothfunctions (on U, with values in F). More generally, if D is an arbitrary subset of F, we define the sets Coo(U; D) by

Coo(U; D) = {f

E

Coo(U; F) I f(U) cD} .

The special cases D = Fa, CI'. E Z2 are also denoted as Coo(U; Fa) = Coo(U; F)a. Note that this definition is quite different from the definition of Map s (E; F) "" although both concern sets of maps.

98

Chapter III. Smooth functions and A-manifolds

1.21 Proposition. In the case A = R the above definition of COO (U; F) is equivalent to the usual definition of infinitely often differentiable functions.

Proof If f : U -- F is infinitely often differentiable, we can construct a smooth tree associated to f by [1.15]. As in [1.19] we then can construct a smooth system Ff . By maximality of Coo , F f is included in Coo, and thus f E Coo (U; F). On the other hand, if f E Coo(U; F), then by repeatedly applying property (AI) we can construct a smooth tree associated to f. And then by [1.15] the function f is infinitely often differentiable. IQEDI

1.22 Discussion. We have, intentionally, swept one slightly disturbing aspect of the proof of [1.21] under the rug. It concerns the number of functions to be used in (AI). In case A = R, an R-vector space X of dimension plq is a direct sum of two ordinary vector spaces X = Xo EEl Xl of dimensions p and q respectively. Functions of the even part Xo = Xo EEl {O} are obviously functions ofpreal coordinates, not ofn = p+q coordinates. Now in [1.15] the smooth tree is constructed with p functions at each stage, whereas (AI) requires n = p + q functions. This problem is easily solved by taking the zero functions for the missing ones. Since the additional q coordinates are identically zero, these terms do not contribute to the summation in (AI). On the other hand, when constructing the smooth tree from (AI), we get at each stage n = p + q functions, where we need only p. This is even simpler to solve: we just neglect the functions we do not need before we apply [1.15]. In view offormula (1.11), we will wantto define partial derivatives ad(x) as CPa,i (x, x), i.e., as the diagonal of CPa,ii. Although the functions CPa,i are not unique, we learn from (1.11) that the diagonal of CPa,i is independent of the choice for CPa,i' On the other hand, the above considerations tell us that for A = R, we can choose the functions CPa,i with i > p completely arbitrary, and thus the partial derivatives ad are also arbitrary for i > p. There is no contradiction, because (l.ll) was given only for i :s; p; and no one would try to define partial derivatives ad for i > p if the function f depends only upon p coordinates. But it shows that we have to be careful when we want to define partial derivatives for a more general A. And indeed, we will see that the structure of A is crucial in the definition of partial derivatives by means of (1.11) (see [3.1] and [V. loS]). But before we can address the question of partial derivatives, we need to analyze the consequences of our definition of smooth functions in more detail.

1.23 Lemma. The sets Coo(U; F) enjoy thefollowing properties: (a) All constant maps with real image belong to Coo(U; A), as do the coordinate projections ie. (b) For f E coo(U; A) and ex E Z2 define 9 : U -- A" c A by g(x) = (f(x))",. Then 9 E Coo(U; A)", c Coo(U; A). (c) If>' E Rand f E Coo(U; A) then>.· f E Coo(U; A). (d) Iff, 9 E Coo (U; A), then f + 9 E Coo(U; A). (e) If property (AI) holds for some homogeneous basis, it holds for all homogeneous

bases.

99

§ 1. Topology and smooth functions

(f) For f E Coo(U; A) and E' an A-vector space we define 9 : U x Eb -- A by g(x, x') = f(x). Then 9 E Coo(U x Eb; A). (g) Suppose E = E' EEl E"for two A-vector spaces E' and E" and suppose that

U' C Un Eb is open in Eb. Let x" E BE" n U and f If 9 : U' -- F is defined by g(x') = f(x', x"), then 9

E E

Coo (U; F) be arbitrary. coo(U'; F).

(h) If f, 9 E Coo(U; A) then f· 9 E Coo(U; A). (i) f E Coo (U; F) If and only lffor all j : jf 0 f E Coo(U; A), where jf are the (left-linear) coordinate functions on F. U) Let be an open set of F o, f E Coo(U; F) and 9 E Coo(D; G). If im(f) c then 9 0 f E Coo(U; G). (k) Let Ui CUbe a family of open sets covering U. If f : U -- F is such that Vi: flu; E COO (Ui ; F), then f E Coo(U; F).

D

D,

Proof In this proof we leave it to the reader to verify, whenever necessary, that condition (A2) is satisfied and that the constructed functions are continuous; the conditions in the lemma assure that there will be no problem. The basic line of proof will be the construction of a smooth tree of (continuous) functions satisfying (Al) by means of induction to the leveL As in [1.19] we then can form a smooth system, from which we can conclude that it must be included in Coo because the latter is maximal. We will abbreviate this procedure by saying that we use the recursion argument. • To prove (a), first note that constant functions and the je are continuous [104]. For any constant function f we can choose the functions 1>i = O. Hence by maximality all constant functions belong to Coo(U; F). For the je one can choose 1>j = 1, all others constant 0 (with respect to the same basis as used for the coordinate projections). Note that in both cases the cover {Ua} consists of the single element U itself. • To prove (b), note that the i-th coordinate of x E Eo has parity c(ei). Using the same cover {Ua} as for f, we define the functions Xa,i = (1)a,i)'''+E(ei). It follows that g(y) = g(x) + L,i(y - x)i . Xa,i(y, x). By [104] the Xa,i are continuous; the rest follows by the recursion argument. • For (c), multiply the 1>a,;'s by A and use the recursion argument. • For (d) and (e), let f and 9 be as in (d). First note that, by taking pairwise intersections of elements of the original covers for f and g, we may assume that the same cover {Ua} serves for both f and g. Denote by (ei) and 1>i the basis and functions in (Al) for f and denote by (fi) and Xi the same ingredients fog. Finally denote by zi the (left!) coordinates of the vector x - y with respect to the basis (ei) and by (i its coordinates with respect to the basis (fi). It follows that there exists a matrix Ai j E R such that (j = L,i zi . Ai j and hence we have f(x)

+ g(x) =

f(y)

+L

zi ·1>i(X, y)

+ g(y) + L

(j . Xj(x, y)

j

= f(y) + g(y) + L

zi . (1)i(X,y)

+ LAi j · Xj(x,y))

.

j

We conclude the proof of (d) by noting that Ai j . Xj (x, y) is smooth by (c) and by applying the recursion argument. The proof of (e) follows by taking f = O.

Chapter III. Smooth functions and A-manifolds

100

• To prove (f), note that

g(x, x') - g(y, y')

= f(x)

- f(y)

= 2)x -

y)i ·CPa,i(X, y)

=L..,,(x-y) ·Xa,i(X,x,y,y '""' i "), where the functions Xa,i : (Ua x Eb)2 ~ A are defined by Xa,i(X, x', y, y') = CPa,i(X, y). We then apply the recursion argument after having added zeros for the functions Xa,i for coordinates in Eb. • For (g), let (ei)~l be a basis for E' and let (ei)f=m+1 be a basis for E", making (ei)f=l a basis for E. If we define Xi(X', yl) = CPi( (x', x"), (y', x")), i :::; m on the subset U~ = U' n Ua, then obviously g(x ' ) = g(yl) + 2:::1 (x' - yl)i. Xi(X',yl). The result then follows by the recursion argument. • For (h), let CPa,i and Xa,i be the functions of property (Al) for the functions f and 9 respectively. As before, we may assume that these functions share the same cover {Ua}. By [104] the function f . 9 is continuous. By (b) and (d) we may assume that the images of f andg are homogeneous. Using the functions CPa,i and Xa,i we obtain n

(f. g)(x) - (f. g)(y)

=

2:(x - y)i. (CPa,i(X,y). g(y) i=l

+ (_l)(E(ei)IE(f(x))) . f(x)· Xa,i(X,y)) . The result now follows if we apply (c), (d), (f), and the recursion argument. • To prove (i), suppose first that f is smooth. Using the left-linearity of the jf we obtain (if 0 f)(x) = (jf 0 f)(y) + 2:i(X - y)i . (if 0 CPa,i)(X, y). The recursion argument then shows that the jf 0 f are smooth. For the other implication, suppose that the jf 0 f are smooth. We then have to prove that the function f = 2: j (if 0 f) . fj is smooth, where (fj) denotes the basis of F dual to the coordinate functions jf. By hypothesis each of the functions jf 0 f has locally defined functions CP~,i as in (Al). As before, by taking multiple intersections when necessary, we may assume that the domains of definition of these functions coincide. We then define the functions CPa,i : U; ~ F by CPa,i(X, y) = 2: j CP~,i(X, y) . fj~ By definition of the topology, these functions are continuous. Moreover, they satisfy CP~,i = jf 0 CPa,i. We then conclude by the recursion argument. • To prove 0), let CPa,i and Xb,j be the functions of property (Al) for the functions f and 9 respectively. By taking intersections with the sets g-l(Db) if necessary, we may assume that each f(U a ) is contained in some Db. We then compute:

(g 0 f)(x) = (g 0 f)(y)

+ 2:(f(x) - f(y))j . Xb,j (f(x), f(y)) j

= (gof)(y) + 2:(X-y)i. (2:Cf CPa,i)(X,y). Xb,j(f(x),f(y))) . O

i

j

The result then follows by applying (d), (h), (i), and the recursion argument. • Finally, for (k) let Ui,a be the families of open sets as defined by property (Al) for each Ui. Since the Ui cover U, it follows that Ui,aUi,a = U, proving that f is smooth. IQEDI

§ 1. Topology and smooth functions

101

1.24 Summary. We can reformulate most of{1.23J in the following more manageable

form. (i) Being a smoothfunction is a local property, stable under composition and fixing

ofvariables (to real values). (ii) The set Coo(U; A) = Coo(U; A)o EEl Coo(U; Ah is a graded commutative Ralgebra with unit under pointwise addition and multiplication offunctions. (iii) For any A-vector space F, Coo(U; F) = Coo(U; F)o EEl Coo(U; Fh is afree gradedCoo(U; A)-module of the same graded dimension as F. Proof The graded commutativity of Coo(U; A) follows from the graded commutativity of A and the fact that multiplication in Coo (U; A) is pointwise. That Coo (U; F) is a free module of the same dimension as F is an immediate consequence of the fact that we can write f = 2: j (jE 0 1) . Ej for an element f E Coo (U; F) and that we can interpret Ej as a IQEDI constant function with real image on U.

1.25 Discussion. If f : U ~ F is a smooth function, we get from property (Al) the functions 1>a,i : U; ~ F. We can combine these functions into a single function 1>a : U; ~ Hom£(E; F) defined by 1>a(x, y) = 2:i ie 01>a,i(X, y). It is an immediate consequence of [1.23] that 1> is smooth. It follows that for each function f E Coo(U; F) there exist functions 1>a E Coo (U;; Hom£( E; F)) such that

Vx,y

E

Ua : f(x) - f(y) = ((x - yll1>a(x,y))) .

In this way we obtain a definition of smooth functions which is obviously independent of a basis. It should be noted that we have made a choice here to take left linear endomorphisms, a choice already present in (Al) by using left coordinate functions and writing the coordinates (y - X)i to the left of the cPa,i. In analogy with ordinary real valued Coo functions (see (1.11)) we will (want to) define partial derivatives in our graded setting by the functions 1>a,i (x, x). It follows that we need that these functions are uniquely determined by the function f. We will see in [3.1] that this is not automatically true, but that it requires a condition on A.

1.26 Construction. The reader might have got the impression that only the even part of an A-vector space can be the domain of smooth functions (in the sense of [1.20]). However, any A-vector space E can be interpreted as the even part of an A-vector space E". Before we give the definition of E", let us have a quick look at what we need. A vector x E E is even only if it has homogeneous coordinates Xi of the correct parity with respect to a homogeneous basis (ei)f=l' But a generic vector does not have homogeneous coordinates. On the other hand, if we split the coordinates Xi into their homogeneous components Xi = (Xi)O + (xih, we get homogeneous coordinates, but twice as many and as many even as odd ones. The A-vector space Ea thus should have twice the dimension of E and with the same number of even and odd basis vectors.

102

Chapter III. Smooth functions and A-manifolds

To prepare the definition of E", we recall the parity shift operation introduced in [1.6.13]. For any ex E Z2 we define the A-vector space EDa in the following way: as a left A-module it is the same as E, but the Z2-grading is given by (EDa){3 = E{3-a. Since Z2 has only two elements, we have EDo, which is obviously the same as E, and EDI, which has its parities reversed with respectto E : (EDI)O = EI and (EDI h = Eo. In the literature the operation DI is also called the parity reversal operation. As a right A-module EDI is certainly not the same as E, but the (set theoretical) identity map {d : EDI ---? E is an odd left linear bijection. Using this bijection, we can transport the structure of an A-vector space from E to EDI : if( ei)i=l is a basis in the prescribed equivalence class for E, the vectors ei = {d- 1 (ei) define an equivalence class of bases on EDI, independent of the choice of the basis (ei) in its equivalence class. With these preparations, we define the A-vector space E~ as E~ = E EEl EDI. And then indeed E~ = Eo EEl E5 1 = Eo EEl EI ~ E, but the last identification is not an identification of A-vector spaces. In order to get a better idea of what this space E" is, we define the (left linear) projection 7r : E" ---? E as 7r = id + {d. We also define the map

Since y E E" is even if and only if we have c(yi) = C(ei) and c(lt) = c(ei) = 1- c(ei), it follows easily that

1.27 Discussion. Let'I/J: E ---? F be a (left or right) linear map between A-vector spaces. Identifying Ewith E~ we can interpret this map as a map 'I/J : E~ ---? F. We thus have two notions of smoothness: smooth as a linear map E ---? F in the sense of [11.6.22] or smooth as a map E~ ---? F in the sense of [1.20]. We leave itto the reader to verify thatthese two notions coincide. Instead oflooking at E~, we also could have restricted 'I/J to Eo. It is a direct corollary of [1.23-a] that if'I/J E Homs(E; F) is a smooth linear map in the sense of [11.6.22], then its restriction to Eo is smooth in the sense of [1.20], i.e., 'l/JIEo E COO (Eo; F). Moreover,

§2. The structure of smooth functions

103

if'!/J E Homs(E; F)" is smooth and homogeneous of parity a, then '!/JIEo E Coo(Eo; F)" is also homogeneous of the same parity. On the other hand, for a linear map '!/J E HomL(E; F), knowledge of the restriction '!/JIEo does not necessarily imply knowledge of'!/J itself. For instance, the linear map ¢ defined in [11.5.8] satisfies the condition ¢IEo = 0, which is smooth. However, ¢ itself is neither smooth nor the zero map. However, as we will show in [2.31], if'!/J is a member of a smooth family of linear maps and if A satisfies a condition we will need anyway, then '!/J is completely determined by its restriction to Eo.

2. THE

STRUCTURE OF SMOOTH FUNCTIONS

In this section we analyze in more detail the structure of smooth functions. More precisely, if U is an open set in Eo with E an A-vector space of dimension plq, we show that Coo (U; F) is isomorphic as graded commutative R-algebra with Coo (0; BF) 0 /\ R q, where 0 = BU is open in RP. For this result to be true, the ring A has to be big enough in the sense that there should exist q odd elements whose product is non-zero.

2.1 Lemma. Let f E Coo(U; F), XO E U, and 1 :::; j :::; n be arbitrary. Then there exist a decreasing sequence ofopen sets Xo E Uk C U and functions f(k) E COO (Uk; F) with the following property. If N E N and a E AE(ej) are such that aN = 0, then for all

x E

UN,' 00

f( x 1 , ... , x j-1 , x j

+ a, x j+1 , ... , x n) = '"' ) L ak· f(k)( x. k=O

Proof We introduce the A-vector space if E EEl Span(ej), i.e., adding one more coordinate to E of the same parity as ej. By 1f : E -+ E we denote the canonical projection, by d : E -+ if the diagonal injection d(x) = (x, xj), and by D : if -+ if the map D(y) = (y1, ... , yn, yj). We will say that a subset U C E is diagonal if y E U implies D(y) E U. Now suppose that g(k) is a smooth function on Uk, where Uk is a diagonal open subset of Econtaining d(xo). Then there exist an open subset d(x o) E Uk+1 C Uk and smooth k 1 functions Xi + on Uf+1 such that for y, z E Uk+1 : ~

~

n+l

g(k)(y)

= g(k)(z) + L(y - z)i . X~+1(y, z) . i=1 ~

~

Using [lA-i] and shrinking Uk+1 if necessary, we may assume that Uk+1 is diagonal. We thus can write for all y E Uk + 1 :

104

Chapter III. Smooth functions and A-manifolds

Since the set of2n+2 coordinates of(y, D(y)) only contains n+ 1 different entries, namely those of y, we can apply [1.23-g,i] to deduce thatthefunction g(k+l) (y) = x~ti (y, D(y)) is smooth on fJk+ 1. Starting fromag(O) and an open diagonal set fJo, both to be defined later, we thus create by induction a decreasing se~ence of open diagonal sets fJk and smooth functions g(k) on them such that for all y E UN :

N-l

(2.2)

g(O) (y)

= L (yn+l - yjl. g(k) (D(y)) + (yn+l _ yj)N . gN (y) . k=O

E

To get us started, note that, again by [lA-i], there exists an open diagonal set fJo C such that g(O) (y) = j(yl, ... , yf-l, yn+l, yf+l, ... yn) is a well defined smooth function on it. We finally define Uk = 7r(fJk ) and j(k)(x) = g(k)(d(x)), which is a well defined smooth function because fJk is diagonal. Since open sets are saturated with nilpotent vectors, if x E UN, then not only d(x) E fJN, but also y = (x, x j + a) E fJN . The result IQEDI then follows immediately from (2.2) and the factthat j(k) (x) = g(k) (D(y)).

2.3 Remark. The above result is a kind of Taylor expansion in a single variable but with a nilpotent increment. As such, the functions j(k) look, up to a factor k!, like higher order (partial) derivatives. In [2.27] we will improve this result by showing that there exist functions j(k) on the whole of U, independent of the point XO' In [3.7] we will show that, under suitable conditions on A, the (k!) . j(k) are indeed the higher order (partial) derivatives of f.

2.4 Notation. The result of [2.1] is particularly useful when ej is odd, i.e., when the coordinate x j is itself nilpotent (of order 2). It is often useful to make the distinction between coordinates with respect to even and odd basis vectors visually more explicit. This will be done by introducing the following notational conventions. In the first place, general elements of an A-vector space E will be denoted by lower case roman letters (as we did always). Coordinates with respect to even basis vectors will be denoted by the same lower case roman letter with a superscript, and the coordinates with respect to odd basis vectors will be denoted by the corresponding greek letter with a superscript. More precisely, we will usually assume that our homogeneous basis of E is an ordered one, and then we write p X=

q

Lxi.ei+ L~i.ep+i i=1

Cd

(xl, ... ,xp,e, ... ,~q).

i=1

In particular, x E Eo if and only if the coordinates Xi E AD and ~i E AI. We will call the Xi the even coordinates and the ~j the odd coordinates of the point x E Eo (for which (~i)2

= 0).

§2. The structure of smooth functions

105

2.5Proposition. Foranyf E Coo(U;F) there exist open subsets Ua c U with UaUa = U and functions 9a,i, ... ik E COO (Ua ; F), 0 :::; k :::; q and 1 :::; i 1 < i2 < ... < ik :::; q such that for x E Ua : q

f(Xl, ... ,xp,~l, ... ,~q) = ~

":,ci,

... ":,Cik • 9a,'l,1. ... 'l,k. ( x 1 , ...

P ,X,

0

,'0"

0)

•

Proof A direct consequence of [2.1] using induction on the number of odd coordinates.

IQEDI

2.6 Discussion. Up till now, in our analysis of smooth functions, we only used property (Al), never property (A2). But [2.5] is as far as one can reasonably go using only (Al). Using [1.23-f,g] we can ignore the zero's in the functions 9a,i , ... ik of [2.5], and thus we are led to the study of smooth functions that depend only on even variables. Using (A2) we will show that these are in one to one correspondence with ordinary smooth functions of real variables.

2.7 Definition. If f E Coo(U; F) is a smooth function, then by (A2) it follows that f(BU) C BF. We thus can define a function Bf : BU ~ BF by Bf = fIBU. But then the fact that f is smooth and [1.21] show that Bf is an ordinary smooth function on BU C BEo ~ RP. Using [2.1] it follows that f(x) - f(Bx) is a nilpotent vector, and thus that B(f(x)) = f(Bx). We thus can characterize the function Bf E Coo(BU; BF) as the unique function which makes the diagram

U

f

F

------+

lB BU

j

BF

Bf=fIBU

commutative (uniqueness follows from the surjectivity of B open).

U

~

BU because U is

2.8 Lemma. The map B : Coo(U; A) ~ Coo(BU; R) is a morphism ofgraded commutative R-algebras with unit. For any A-vector space F, B : COO (U; F) ~ COO (BU; BF) is a morphismfromgraded modules overCoo(U; A) to graded modules overCoo(BU; R).

2.9 Definition. Let E be an A-vector space and {el' ... , e p, h, ... ,fq} an ordered homogeneous basis such that the ei are even and the fi odd. Any other ordered homogeneous basis (in the equivalence class) is related to this basis by a matrix with real entries of the

Chapter III. Smooth functions and A-manifolds

106

form ( ~ ~ ). It follows that the graded subspace Ew.o.d = Span( {el' ... ,e p }) is independent of the choice of a homogeneous basis (in its equivalence class !). We thus can associate to an A-vector space E two graded "subspaces" : its body BE, and Ew.o.d, called the part of E without odd dimensions. (Nota Bene. A point in Ew.o.d can have odd coordinates.) The superscripts w.o.d (without odd dimension) will be applied more generally to any subset A of an A-vector space E, and will be defmed as

A w .o . d

= An E w .o .d

.

Note that the definition of Ew.o.d implies that B(Ew.o.d) = BED = (BE)w.o.d. An A-vector space E satisfies E = Ew.o.d if and only if it admits a homogeneous basis without odd elements; such an A-vector space will naturally be called an A-vector space without odd dimensions.

2.10 DiscussionIDefinition. In §11.6 we have seen that the R-vector space BE completely determines the A-vector space E. In [1.1] we have argued that the same is not true for the even part: BED does not determine Eo. If we now insist that we want to reconstruct an open subset U c Eo given the open subset BU c BED in the even part of the R-vector space X = BE, the best we can do is to suppose that Xl = {O}, since we have no data concerning Xl' Let us formalize this idea by means of a map G that applies to open subsets 0 of the even part of an R-vector space X. This G can be seen as an extension of the map G defmed on A-vector spaces in [11.6.29], although it is not a very straightforward extension. We define GO as the unique open subset of the A-vector space without odd dimensions E = G(Xo EEl {Oh) such that B(GO) = O. In other words, GO = B-l(O), with B : E ~ B(G(Xo EEl {Oh)) = Xo. Now suppose that U is an open subset in the even part of an arbitrary A-vector space E, what can we say about G(BU)? The set BU is open in BED and hence G(BU) is open in (G(BEo EEl {Oh))o. But this is just Ew.o.d. We conclude that

G(BU) = Un Ew.o.d == uw.o.d.

2.11 Construction. Let E be an A-vector space without odd dimensions, U an open subset of Eo and f E Coo(BU; BF). By definition of open sets, BU = Un BE, and thus every point x E U decomposes as x = Bx + n with Bx E BU and n ENE. We now define the function G f : U ~ F by the formula (2.12)

(Gf)(x) =

L 00

1

k

k! ((D f)(Bx))(n, ... , n) .

k=O

In this formula Dk f is the k-th order derivative of f (i.e., the "matrix" of all k-th order partial derivatives) seen as amap from BU to the space of k-linear symmetric maps from BE to BF c F. Applying such a map to k vectors nEED makes sense because all the coordinates of n are even (E = Ew.o.d and U c Eo) and thus commute. Finally the infinite sum is actually a finite sum because the coordinates of n are nilpotent.

§2. The structure of smooth functions

107

2.13 Remark. Let E be an A-vector space without odd dimensions, then BED = BE. Hence, if f E COO (BED, BF) is linear, then the map Gf : Eo ~ F constructed in [2.11] is the same as the restriction to Eo of the map Gf : E ~ F constructed in [11.6.30].

2.14 Lemma. For f E Coo(BU; R) and g, h E Coo(BU; BF) we have the equalities + (Gh) and G(f· g) = (Gf) . (Gg).

G(g + h) = (Gg)

Proof The first equality is obvious, the second follows from the formula k

(Dk(f . g)(x))(v, ... , v) = L G>D Cf(x))( v, ... , v) . (Dk-Cg(x))(v, ... , v) , c=o valid for ordinary COO-functions.

2.15 Lemma. If E is an A-vector space without odd dimensions and f E Coo (BU; BF), then Gf E Coo(U; F) and B(Gf) = f. Moreover, iff is homogeneous ofdegree ex, then so is G f. Proof With the usual abuse of notation (writing the same symbol for the coordinate and the coordinate function), we denote by Xi the i-th coordinatefunction on E and by (BX)i the i-th coordinatefunction on BE. It is straightforward to show that B(x i ) = (BX)i as functions and that G(BXi) = Xi. Since f is smooth, there exist local functions CPa,i E Coo((BUa)2; BF) such that

f(By) - f(Bx) = L(Byi - BXi) 'CPa,i(By, Bx) . i

Note that by definition of the DeWitt topology we have Ua = B- 1 (BUa),justifying the definition of the domain of definition of CPa,i in terms of open subsets on E. Applying G to both sides of the defining equality for CPa,i (seeing them as functions on (BUa)2) and using [2.14] gives us

(Gf)(y) - (Gf)(x) = L(yi - Xi) . (Gcpa,i)(y, X) . The recursion argument (see the proof of [1.23]) then proves that Gf E Coo(Uj F). The second assertion is immediate because ((Dk f)(Bx)) (n, ... ,n) is nilpotent for k > O. For the homogeneity statement, note that f is homogeneous of degree ex if and only if f takes values in BFa. Since the n in [2.11] are even, the result ((Dk f)(Bx))(n, ... , n) remains in Fa. IQEDI

2.16 Proposition. Let E be an A-vector space without odd dimensions and U an open subset of Eo. Then B : Coo(Uj F) ~ COO (BU, BF) is an isomorphism. Proof Surjectivity is assured by [2.15]. To prove injectivity, let f E Coo(U; F) be such that Bf == 0, i.e., flBU is identically zero, and let Xo E BU c U. Now look at the proof

108

Chapter III. Smooth functions and A-manifolds

of [2.1] and suppose that g(k) is zero on BUk . It immediately follows that g(k+l) is zero on BUk+l, provided Byn+l =I- Byj. By continuity of g(k+l) and the fact that it takes real values on BUk+l, it follows that g(k+l) is zero on the whole ofBUk+l. This last conclusion would not have beenjustified, had we not known that g(k+l) takes real values on BUk+1 : the non-Hausdorff character allows for non-unique limits! We conclude that I(k) IBUk == 0 for all k. If there were only one even coordinate, the expansion of [2.1] would prove that I == O. For more than one even coordinate we apply induction on their number n. We fix xn = rn + an and apply the expansion of [2.1] with j = n. By the previous argument we have BI(k) == O. Since in the expansion of [2.1] the functions I(k) only appear with the real valued n-th coordinate rn, we may by [1.23-g] interpret the I(k) as smooth functions ofn -1 coordinates. By the (unstated) induction hypothesis we conclude that I(k) == 0, and thus I == o. IQEDI

2.17 Remark. In §I1.6 we showed that B is an isomorphism when applied to smooth linear maps. Here we show that B is an isomorphism when applied to arbitrary smooth functions on an open subset of the even part of an A-vector space without odd dimensions. Said differently, as long as the domain of definition only contains even coordinates, there is no difference between the general case and the special case A = R. The restriction to A-vector spaces without odd dimensions is necessary because we take as domain of definition not (an open subset of) an A-vector space, but only (an open subset of) its even part. The same phenomenon is also reflected in the fact that in reconstructing U from B U by means of G we only recover the part without odd dimensions, i.e., G(BU) = U only if E has no odd dimensions.

2.18 Notation. Up until now we have carefully distinguished an ordinary real valued smooth function of p real variables and its corresponding smooth function G I of p even variables. However, [2.16] has shown that there is no real distinction between these two objects. Therefore we will in the future use the same symbol for both objects. We thus might start with an ordinary real valued smooth function I and use even coordinates as arguments, meaning that we take G f. Or we might start with a smooth function of even variables, plug in real coordinates (a point in BE) and claim that we have an ordinary real valued function, this beingjustified by either taking B I or using (A2).

2.19 Discussion. Let E be an A-vector space of dimension plq and let U be an open subset of Eo. Combining [2.5] and [2.16], we see that we have a nearly complete control over the elements of Coo (U; F). Thefunctions ga,il ... ik E COO (Ua ; F) are only used with all odd coordinates being zero. This means that we only need their restrictions to U;:·o.d, but (by [2.16]) these are uniquely determined by ordinary real valued smooth functions Ia,il ... ik E Coo (BUa ; BF) as ga,iI ... ik (Xl, ... , x P , 0, ... ,0) = Ia,il ... ik (xl, ... ,xP ) (remember, we do not use the symbol G any longer). However, our control is not complete because (i) this description of I is only local on Ua, not on U, and (ii) it is not clear

§2. The structure of smooth functions

109

whether the function f determines the local functions fa,i1 ... i k uniquely. In order to make our control complete, we need to formulate conditions on A, conditions that will also be useful in defining the partial derivatives using the diagonal of the functions CPa,i as in (1.11).

2.20 Definition. Let A be a graded commutative R-algebra and let n E N* be a non-zero natural number. We say that A satisfies (C [nJ) if (C [nJ)

We will say that A satisfies (C [ooJ) if it satisfies (C [nJ) for all n E N*. Obviously (C [001) and (C [n+lJ) imply (C [nJ). In the standard example A = /\ X with X a vector space over R, (C [nJ) is satisfied if and only if dim X ::::: n, a condition valid also for n = 00.

2.21 Proposition. Let E be anA-vector space of dimension plq, let f be in Coo (U; F), and let f i 1... i • E Coo (BU, BF) be such that f(x) = 2:k 2:;. 1 ••• ~ik . f i 1... i k (Xl, ... , x P). J Then the functions fi 1... ik are unique Ifand only If A satisfies (C [kJ).

e

Proof We start by observing thatthe separate additive terms ~i1 .•. ~ik . f i 1... ik (Xl, ... , x P) are always uniquely determined by f: a simple induction argument with respect to k and taking all coordinates ~j = 0 whenever j r:J. {i l , ... , ik} suffices. Now suppose ~i1 ... ~ik . fi 1... ik (Xl, ... , x P ) to be identically zero and take the coordinates Xl, ... , x P to be real. It follows that ~i1 ... ~ik . fi 1... ik (xl, ... , x P ) = 0, which implies that either ~i1 ... ~ik or f i 1""k (Xl, ... , x P ) is zero (because the latter is real, and thus can be inverted if non-zero). If (C [kJ) is satisfied, we can choose ~i1 .•. Eik to be non-zero, implying that fi 1... ik must be zero. On the other hand, if (C [kJ) is not satisfied, the term ~i1 ... ~ik is IQEDI always zero, leaving the function f i 1 ... i k completely undetermined.

2.22 Corollary. Let E be an A-vector space of dimension plq and let U c Eo be open. Then COO(U; F) is in bijection with COO(BU; BF)d, i.e., with dcopies of COO(BU; BF), where d is determined as follows.

(i) If Al = {O}, then d = 1. (ii) If 1 :::; £ < q is such that A satisfies (C [eJ) but not (C [£+lJ), then d (iii) If A satisfies (C [qJ), then d = 2q.

= 2:~=o (3)'

Proof • If Al = {O}, it follows from [2.5] that f(x,O = ga, (x, 0), which proves that f(x,~) = f(x, 0), i.e., f is a function of the even coordinates only. And then the conclusion follows from [2.16] . • Let us now suppose that A satisfies (C [kJ). From [2.5] we obtain (%) local smooth functions ga,i1 ... i k «(k) = the number of increasing sequences 1 :::; i l < ... < ik :::; q). Since fa,i1 ... i k(X l , ... ,xP) = ga,i 1... ik(X l , ... ,xp ,0, ... ,0) depends only upon even coordinates, it follows that fa,i 1... ik E Coo (BU; BF). Butthen we can combine [2.5] and

Chapter III. Smooth functions and A-manifolds

110

[2.21] to conclude that the functions fa,il ... ik E Coo (BU; BF) are uniquely determined by f. Hence they must coincide on overlaps Ua nUb. It thus follows from [1.23-k] that we have global functions fil ... ik E COO(BU; BF) such that fa,il ... ik = fil ... ik Iua . Since (C [kJ) implies (C [k-lJ), the same conclusion holds for all lower order functions. On the other hand, if A does not satisfy (C [kJ), the terms ~il ... ~ik . 9a,il ... ik (x, 0) do not contribute to f. The conclusions of (ii) and (iii) follow. IQEDI

2.23 Discussion. With [2.22] we have complete control over smooth functions. If the A-vector space E has dimension plq and if U c Eo is open, then there is a one to one correspondence between smooth functions f E COO(U, F) and collections of functions fil ... ik E COO (BU, BF) given by (2.24) min(q,C)

L

f(xl, ... ,xp,e, ... ,~q)=

Cil ... ":,Cik .

":,

j.'l.l ... 'I,k. (X 1 , ... ,XP)

= {O}

where £ is the maximal value such that A satisfies (C [cJ) (if Al

,

we take £ = 0).

2.25 Remark. If one takes into account how the functions fil ... ik in (2.24) behave under a change of the odd basis vectors, then it is not hard to show that Coo (U; F) is bijective with min(q,C)

EB

COO(U;F)~COO(BU;BF)0(

N(Rq)).

k=O

It is not surprising that the exterior power /\k (Rq) appears because a k-fold product of the q odd variables ~q is k-linear and skew-symmetric in these variables. If £ ::::: q we have the identification Coo (U; F) ~ Coo (BU; BF) 0 /\ R q, which is sometimes taken as definition of smooth functions of p even variables and q odd variables. Since starting in [3.3] we will assume that A satisfies (C [ooJ) (and thus 00 = £ ::::: q), we obtain this identification as a consequence of a more basic definition of smooth functions.

e, ... ,

2.26 Examples . • According to [II. 1.11] and [1.2.9], the group Aut(E) is an open subset ofEnds(E)o. We claim that taking inverses Inv : Aut(E) -- Ends(E)o is smooth (see also [VI. 1.6]). To prove this, it suffices to show that Inv is of the form (2.24) when using left-coordinates eMs on Aut(E). On the other hand, calculations are much simpler when using the coordinates NIs because these preserve composition (see [11.4.2]), and in particular NIs (1)-1) = (NIs( 1» )-1. Writing tIs (1» = (~ ~ ), A and D are invertible matrices with even entries, and Band C have odd entries. We thus can write:

( AB)-l _ (A-l 0 ) [I CD

-

0

_ (A-

-

0

D- l

•

p

+q

-

(0 _BD- l )]-l _CA- l

00

l

0

D- l

)

•

'"' (

L

k=O

0

_CA- l

-BD0

0

k

l )

'

§2. The structure of smooth functions

111

where the infinite sum is actually finite because Band C are nilpotent (it breaks off after the 2pqth term). We now note that (fortunately) the left coordinates eMs (1)) only differ by a sign from the coordinates NIs (1)) (see [11.4.1]). We thus are allowed to use the entries of A, B, C, and D as coordinates on Aut ( E ). Since taking the inverse of a matrix with real entries is smooth, the maps A 1--7 A-I and D 1--7 D-I are smooth in the even coordinates of A and D. It thus follows that the matrix entries of NI s (1» -1 are smooth functions of the even coordinates and polynomials in the odd coordinates. In other words, Inv: Aut(E) -+ Aut(E) C Ends(E)o is smooth . • The ordinary determinant function is a polynomial in its entries. It thus follows immediately from the definition of the graded determinant and the arguments given in the previous example that gDet : Aut(E) -+ Ao is smooth.

2.27 Proposition. Let f E Coo(U; F) and 1 :::; j :::; n be arbitrary. Then there exist functions f(k) E Coo(U; F), f(O) = f such that Vx E U, Va E N n AE(ej) : 00

(2.28)

f( x 1 , ...

, x j - I ,xj

) + a, x j+I , ... , x n) = '"' L ak · f(k)( x. k=O

by means of (2.24) and [2.16]. If c( ej) = 1, i.e., if the j-th f2?ik = 0 whenever j E {i I , ... , id· If j r:J. {iI"'" id, we define f2.~.ik = ±hO ... jk' where {jo, ... ,jd = {j, iI, ... ,ik} and where the sign is

Proof We will define the

f(k)

coordinate is odd, we define

determined by the position of j within the (increasing) sequence il ... ik : if j is in m-th position, the sign is (_l)m-I. (In particular f~I) = fi,) For k > 1 we define f(k) == O. The result then follows from (2.24). If c( ej) = 0, we first note that by using [2.21] and (2.24) we can reduce the problem to functions of even coordinates only. For a fixed sequence iI, ... ,if we then define fi(,~~.ii by fL~~.ii = k!· (oxj)kfi, ... ii' This makes sense because functions of even coordinates only can be interpreted as ordinary infinitely often differentiable functions. And then a IQEDI simple but tedious computation using definition (2.12) completes the proof.

2.29 Remark. [2.27] is similar to [2.1], but stronger in that the f(k) are defined on the whole of U (and do not depend on the point x o ). We will see in [3.7] that if A satisfies (C [HIJ), then we can interpret the functions k! . f(k) as the k-th order partial derivatives of f with respect to the j-th coordinate. In this way we have a convergent Taylor series expansion for nilpotent increments. The obvious question which comes to mind when seeing this result is: couldn't we prove [2.27] directly, without the intermediate result [2.1]7 The answer is no, and the reason is the presence of a cover in property (Al) of smooth functions. The argument given in [ 1.12] can be used to show that in the real case this cover is not needed, in which case one could prove [2.27] directly. We did not follow this approach because it would not be valid in a complexified setting. In the approach we used here, we are sure that [2.27] is

112

Chapter III. Smooth functions and A-manifolds

valid in a complexified setting as well (i.e., a setting in which we consider commutative C-algebra such that A/N ~ C, see §II.l).

A as a graded

2.30 Discussion. In [1.27] we raised the question whether a linear map is determined by its restriction to the even part of an A-vector space. We now are able to prove that under suitable circumstances the restriction of a smooth family of linear maps to the even part of an A-vector space does indeed determine this family completely.

c Go be open, and let be a smooth map. Suppose furthermore that A satisfies (C [HI]) with q the odd dimension of G. If \/g E U : 'I/J(g)IEo = 0, then \/g E U : 'I/J(g) = 0. In other words, 'I/J(g)IEo uniquely determines 'I/J(g). 2.31 Proposition. Let E, F, and G be A-vector spaces, let U

'I/J: U ~ HOills(E; F)

Proof We treat the right linear case, the left linear case being similar. Let (ei) and (Ii) be bases of E and F respectively (in their equivalence classes !). Since the map HOillR(E;F) x E ~ F, ('I/J,e) f---+ 'I/J(e) is smooth, as are the maps Ji, we conclude that the maps 'l/Jj : U x E ~ A, (g,e) f---+ Ji('I/J(g)(e)) are smooth. Evaluating 'l/Jj on a basis vector ei gives us the middle matrix element 'l/Jj (g) (ei) = !VJR ('I/J(g))j i. Knowledge of these matrix elements completely determines 'I/J. If ei is an even basis vector, our assumption implies that 'l/Jj (g) (ei) = !VJR ('I/J(g))j i = 0, i.e., all the corresponding matrix elements are zero. Now suppose that ei is an odd basis vector. By [1.23-g] the map 1> : U X Al ~ A, (g,ry) f---+ 'l/Jj(g)(eiry) is smooth. By assumption it is identically zero and by linearity it is given by 1>(g, ry) = 'l/Jj (g)( ei) . ry. Since U x Al contains q + 1 odd coordinates and since A satisfies (C [HI]), we deduce from [2.21] that 'l/Jj (g) (ei) is zero. We conclude that all matrix elements of'I/J(g) are zero, i.e., 'I/J(g) is zero. IQEDI

3.

DERIVATIVES AND THE INVERSE FUNCTION THEOREM

The first part of this section is devoted to the definition of the (partial) derivative( s) of a smooth function. This requires a condition on A slightly stronger than the one mentioned in §2. The second part is devoted to the classical theorems ofdifferential calculus: the inverse function theorem, the implicit function theorem, invariance of graded dimension, and the canonical form of a smooth function of maximal rank.

3.1 Proposition. Let E be an A-vector space of dimension plq, let j belong to Coo (U; F), let1>a,i be as in property (Al) and let gi E Coo(U;F) be defined by gi(X) = 1>a,i(X,X). For i :::; p, gi is uniquely determined by j, independent of the index a; for i > p it is uniquely determined if and only if A satisfies (C [HI]).

§3. Derivatives and the inverse function theorem

113

Proof Fix i and let CPa,i and CPb,i be two possibilities. Let x, Y E Ua n Ub be such that x j -=f- yj =} j = i, i.e., x and y differ only in the i-th coordinate. It follows that (yi _ Xi). (CPa,i(y,X) -CPb,i(Y,X)) = O. Denoting by h the difference h = yi - Xi, it follows that the function CPa,i(Y, x) - CPb,i(Y, x) is a function b..(x, h) of the p + q + 1 variables (x, h). We thus are given the equality h· b..(x, h) = 0, while we want to prove that b..(x, 0) = O. Once we have that, smoothness of 9i on U follows from the fact that it is smooth on all Ua and [1.23-k] Consider first the case i :::; p, i.e., h is an even coordinate. According to expansion (2.24) there exist smooth functions (of even variables only) Xil, ... ,ik defined in a neighborhood of(x,O) suchthat

b..(x, h)

'""' i~ i = '""' LL~ k. (Xil ...I ik)(X , ... ,xP,h). 1 .••

k

ij

Taking real values for Xl, ... , x P and h, we deduce (as in [2.21]) from h· b..(x, h) = 0 that ~il ... k . h. Xil .. ,ik (xl, ... , x P, h) must be zero. Since the coordinates (~il, ... , ~ik) and (xl, ... ,xP, h) are independent, it follows that either ~il ... ~ik or h·Xil ... ik (xl, ... ,xP, h) must be identically zero. If the former, the term ~il ... ~ik . Xi1 ... ik (Xl, ... , x P, h) does not contribute to b..(x, h). If the latter, continuity ofXil ... ik (and the fact that it is real valued) implies that h·Xil ... ik (Xl, ... ,xP, h) is identically zero if and only if Xil ... ik (xl, ... ,xP, h) is identically zero. We conclude that b..(x, h) = 0 and in particular b..(x,O) = O. This shows that 9i is uniquely defined by f. Next let i > p, i.e., h is an odd coordinate. Again using expansion (2.24), there exist smooth functions Xil, ... ,ik and '0i 1 , ... ,ik defined in a neighborhood of (x, 0) such that

e

b..(x, h) =

L L e ek . ((Xi1 ... ik )(x\ ... , x P) + h· ('0il ... ik)(XI, ... , x P)) 1

k

.. ,

ij

Sinceh 2 = 0, h·b..(x, h) = 0 implies that eitherh.~il '" ~ik or Xil ... ik(X I , ... ,xP) must be identically zero. !f(C [HIJ) holds, the first is impossible, so Xi1 ... ik is identically zero, proving that b..(x, 0) = 0, i.e., that 9i is uniquely determined by f. On the other hand, if (C [HIJ) does not hold, we can change CPa,i by adding ~k, where k is chosen such that this product is not identically zero, but any k + l-fold product of odd elements is zero (in particular if k = 0, i.e., Al = {O}, we add the constant function 1). By hypothesis such a k :::; q exists. This does not invalidate (Al) but changes 9i also by this non-zero IQEDI amount, showing that 9i is not uniquely determined by f.

e ...

3.2 Remark. As in the proof of [2.16], the fact that we can reduce functions of even variables to real valued functions of real variables is essential. Otherwise, the nonHausdorff character of the DeWitt topology would invalidate the continuity argument.

3.3 Definition/Convention. As suggested already several times, we want to define the partial derivatives of a smooth function f by the diagonal of the functions CPa,i from

114

Chapter III. Smooth functions and A-manifolds

(AI), i.e., by the functions gi from [3.1]. In analogy with ordinary smooth functions and knowing that gi depends only upon f, we will denote these partial derivatives by gi == ad == af / axi, 1 :::; i :::; n = p + q. If we want to make an explicit distinction between even and odd coordinates we will use the notation axi f or af / axi for i :::; p and af.i-p f or af / a~i-p for i > p. According to [3.1] the partial derivatives ad are always defined for i :::; p, i.e., for the even directions. On the other hand, the partial derivatives in the odd directions exist only if A satisfies (C [q+lJ). This immediately raises the question what happens for A = R (or more generally when Al = {O}), because then (C[q+lJ) is certainly not satisfied. Of course this is no problem if there are no odd coordinates present, i.e., when E is without odd dimensions. But in all other cases it poses a problem. Thinking of the case A = R, in which we know that there ought to be only p partial derivatives, not p + q, we can "solve" this dilemma by just ignoring the undefined partial derivatives in the odd directions. That this is reasonable to do is confirmed in [v. loS], where we show that if Al = {O}, then we do not lose anything by ignoring the ad for i > p. For all other cases, i.e., when q > 0, Al =I {O}, and (C [q+IJ) not satisfied, we do have a problem. For that reason we will assume throughout the rest ofthis book that A satisfies (C [=J). Most of the time this is stronger than strictly needed, but it avoids changing (the condition on) A every time we change the odd dimension. It also guarantees that a smooth function f is always represented in (2.24) by the maximal number (only depending upon the odd dimension) of ordinary smooth functions fil ... ik. Of course, with this convention we do not cover the case A = R. However, most of the statements, including their proofs, remain valid for A = R. It is only occasionally that there will be a difference in treatment, but these we will point out in separate remarks.

3.4 Discussion. Let us now for the last time distinguish between an ordinary smooth (vector valued) function f of p real coordinates, and the (smooth) function Gf of p even coordinates. It follows immediately from the proof of [1.21] that aiGf = G( ad), which justifies at the same time our use of the symbol i for this operation, as well as our identification of f with Gf. With this knowledge, the reader should be able to convince himself of the correctness of the following result: if a function f E C= (U; F) is explicitly given by 2q functions fil ... ik E C=(BU, BF) (expansion (2.24)), then the partial derivatives are given by

a

(3.5)

af q ae(x) = L k=O

k

L L

In the last formula, the b~j is the Kronecker delta, always zero except if i j = £ when it

§3. Derivatives and the inverse function theorem

115

is 1. The sign (-l)j-l in the second formula appears when we put the coordinate ~ij in front of the rest.

3.6 Proposition. The maps they commute: [ai, aj ] = O.

ai belong to DerR(CCXl(U; A)), they have parity c(ei), and

Proof We will use the proof of [1.23] to prove the frrst two statements. From part (b) it follows that the parity of i is the same as that of ei. From parts (c,d) it follows that it is linear over R. (Nota Bene. CCXl(U; A) is an R-algebra and as such right and left linear are the same, the difference becomes apparent in the derivation property.) For homogeneous f, part (h) shows that ai(f . g) = (ad) . 9 + (-1) (e(ei)le(f)) . f . (aig), proving that ai is a right-linear derivation of parity c(ei). For the last statement we use (3.5). If Xi and x j are both even coordinates, the equality ai 0 aj = aj 0 ai follows from the same equality for ordinary real valued smooth functions and (3.5). If Xi and x j have different parities, the equality ai 0 aj = aj 0 ai follows immediately from (3.5). Finally, if Xi and x j are both odd coordinates, the equality [ai, aj] == ai 0 aj + aj 0 ai = 0 follows from (3.5) by an elementary calculation. IQEDI

a

3.7 Discussion. In §2 we have given two variants of an expansion that looked like a convergent Taylor series with a nilpotent increment. Let us show that it indeed is a Taylor series (up to factors k!). And remember, our convention that A satisfies (C [CXll) guarantees that the partial derivatives exist for any odd dimension. Looking at the proof of [2.27] and using that for an even coordinate x j we have ajGf = G(ajf), we see immediately that the functions k! . f(k) in [2.27] are exactly the k-th order partial derivatives with respect to an even xj. The same result follows from the proof of [2.27] for odd coordinates, once one realizes that for odd coordinates (a j )2 = 0 [3.6]. We thus are allowed to write (finally) for homogeneous nilpotent a of the correct parity: CXl k CXl k ak a ((aj )kf)( X) -_ ""' a (axj)k f (X) . f( X1 , ... ,x j-l ,Xj +a,x j+l , ... ,xn) -_ ""' L."kf· L."kf· k=O

k=O

In case there are no odd coordinates and if we apply this expansion repeatedly to all (even) coordinates, we obviously recover definition (2.12) of Gf. But also in the case of odd coordinates have we already obtained such an expansion! Using (3.5) we have an alternative way to describe the functions fil ... ik appearing in the expansion (2.24): they are given as partial derivatives of f with respect to odd variables:

It is obviously a circular argument to think that we could define the functions fil ... ik in terms of derivatives. We needed them in order to be able to show that (partial) derivatives

116

Chapter III. Smooth functions and A-manifolds

exist. On the other hand, the expansion (2.24) now can be written in the form of a Taylor expansion: q

I(x,~)

=

L k=O l~i,<···
L (er'· (e)i

2

•••

(~q)iq . ((a~q)iq ... (a~,)i' I) (X,~

= 0) .

3.8 DiscussionIDefinition. Let E and F be A-vector spaces with bases (ei) and (Ej) respectively, and let I : U c Eo -+ F be smooth. Using the left coordinate functions jE on F, we now introduce the functions Ij : U -+ A by Jl = jE 0 I == 10 jE. Obviously we have for x E U the equalities ((f(x) IljE)) = Jl(x) and I(x) = 2:: j Ij(X)Ej. According to [1.23-i] I is smooth if and only if all Ij are. Moreover, from the proof of [1.23-i] we deduce that (a i (iE 0 f)) (x) = (( (ad) (x) I jE ~. This property can also be stated in the following equivalent ways: or

or

af)j (ax'

where Xi now denotes any of the p + qeven or odd coordinates on Eo. If we denote by yj the (even and odd) coordinate functions yj == jE on Fa, then the proof of [1.23-j] shows that we have the chain rule for partial derivatives in the following form (and remember, for this to make sense we have to assume that I(U) c Fa):

or equivalently: (3.10)

a(~x'og) (x) = '" L j

g

aaf1 (x) . aa (f(x») . x'

yJ

The order in these formula; is important and reflects our choice to write the coordinates on the left in (AI). For IE C=(U; F) we define the Jacobian Jac(f) E C=(U; HomL(E; F)) by

It follows from the proof of [1.23-e] that this expression is independent of the choice of a basis for E. With respect to the bases (ei) for E and (Ej) for F, the matrix !Vh (Jac(f)( x) )i j [11.4.1] of this left-linear morphism is (ad j )(x). Since ai has the same parity as ie, it follows immediately that for homogeneous I E C=(U; F)"", the Jacobian

117

§3. Derivatives and the inverse function theorem

has the same parity as f, i.e., Jac(f) E C=(U; HomL(E; F))o:. More explicitly, for a vector v = 2:i Vi ei we have (3.11)

((vllJac(f)(x))) =

L

L

.

af . v' -a . (x) = xt

.. v iap -a (x) Ej xt

2,j

1.

In terms of the Jacobian, the chain rule can be stated as

Jac(g 0 f)(x) = Jac(g) (f(x)) 0 Jac(f) (x) , or equivalently, and preferably because Jac(f) is left linear, as

Jac(f 0 g)(x) = Jac(f)(x)

0

Jac(g)(f(x)) .

3.12 Remark. In the special case A = R, we decided to ignore the (non-existing) partial derivatives in the odd directions. We thus have only the p partial derivatives in the even directions. Hence the Jacobian is an element of HomL(Eo; F) instead of HomdE; F). Moreover, if f is an even function, i.e., taking values in Fo, the Jacobian should be seen as taking values in HomdEo; Fo). We are forced to this decision if we want (3.10) to make sense, because the yj only exist for the even directions. In a certain sense this is the natural thing to do, because for even f all components p (and thus the ad j ) are zero for j > p.

a

3.13 Discussion. As an application of the chain rule, we consider four A-vector spaces E, F, G, and H, open subsets U c Eo, V c Fo, and W c (E x G)o == Eo x Go, -+ Go, and f : W -+ H. With these we define the map Eo x Go byh(x,y) = (x,g(y)). IfVx E UVy E V: (x,g(y)) E W,

and smooth maps 9 : V

h: Eo x V

-+

we can consider the composite function k : U x V -+ G defined by k = f 0 h, i.e., k(x,y) = f(x,g(y)). If we denote by (Xi) the (even and odd) coordinates on Eo, they can be seen either as part of a system of coordinates on Eo x Fo or as part of a system of coordinates on Eo x Go. If we see them as coordinates on Eo x Fo and if we denote by h j one of the coordinates of h in Eo x Go, then by definition of h the partial derivative h j is zero unless h j is the coordinate corresponding to xi, in which case it is 1. The chain rule then tells us that

ax'

k(x,y) = f(x,g(y))

ak af -a. (x, y) = -a. (x, g(y)) . x'

x'

Stated in words, this equation tells us that taking partial derivatives with respect to some coordinates and substituting functions for other coordinates are commuting operations. This applies in particular to the case in which 9 is a constant functions (thus taking values in BG o): taking partial derivatives with respect to some coordinates and fixing some coordinates to real values are commuting operations. This particular case can also be deduced directly from the proof of [l.23-g].

118

Chapter III. Smooth functions and A-manifolds

3.14 Examples. • Associated to a smooth ¢ E Hom L (E; F) we define f : Eo --) F by f ¢IEo E coo(Eo;F). Forx,y E Eo we compute: f(y) - f(x) = {y - xll¢))· It follows that Jac(f) E coo(Eo; HomdE; F)) is the constant map ¢, i.e., Vx E Eo :

=

Jac(f)(x) = ¢ . • If¢ E HomR(E; F) is smooth and f = ¢IEo E COO (Eo; F), a similar computation shows that Jac(f) is the constant map cr- 1 ¢ E HomdE; F) . • Consider the graded determinant gDet E coo(Aut(E)), where we view Aut(E) as an open set in Ends(E)o. In order to compute Jac(gDet)(id(E)) we apply [3.13]. A simple calculation using the matrix expression for gDet shows that the partial derivatives with respect to non-diagonal entries are zero (at id(E) !). The remaining partial derivatives are ±1 in such a way that we have Jac(gDet) (id(E)) = gtrs E HomdEnds(E); A).

3.15 Lemma. Let E and F be A-vector spaces, U c Eo open, V c Fo open and connected, and f : U x V --) A a smooth function. We denote by (yi) the even and odd coordinates on V. There exists a (unique, smooth) function 9 : U --) A such that f = 9 0 7rU if and only if for all i 8 y i f = O.

Proof Let us denote by x even and by ~ the odd coordinates on U, and let us suppose that the dimension of E is plq and that of F rls. If yi is an odd coordinate, it follows from (2.24) that f is of the form f = fOyi it with fo and it function on U x V independent of the coordinate yi. Since 8 yi f = it, it follows from 8 yi f = 0 that it = 0, i.e., f = fo is a function independent ofthe coordinate yi. Repeating the argument for all odd coordinates yi proves that f does not depend upon the odd coordinates, i.e., f is of the form q

f(x,~,yl, ... ,yr+s)

=

~

Now ifyi is an even coordinate (i ::; r), the condition 8 yif = 0 implies, using (3.5) and [2.21], that 8yifi1 ... ik = 0 for all i l , ... , ik. These equations we can see as equations of ordinary smooth R-values functions. But then the connectedness of BV implies that IQEDI the functions fil ... ik are independent of the coordinates yi.

3.16 Corollary. Let U c Eo be an open connected set and f : U --) A a smooth function. Then f is constant if and only if 8d = 0 for all 1 ::; i ::; dim E.

3.17 Discussion. To prepare the inverse function theorem, let E and F be A-vector spaces of dimension plq and r Is respectively, let U c Eo be open and let f : U --) Fo be a smooth function. We denote by (x,~) the even and odd coordinates of a point in U with respect to an ordered basis and by (y, TJ) the coordinates of a point in Fo. It follows from [2.23]

119

§3. Derivatives and the inverse function theorem

that there exist real valued functions

g1 1 ... i k and hL . ik

such that

q

r/=L k=O 1:O;i1 < ...
However, yi is an even coordinate and thus the expansion cannot contain terms with an odd number of~. Likewise, the expansion of1/ cannot contain terms with an even number of ~. It follows that we can write [q/2]

yi=

L

[q/2]

r/= L These expansions are rather awkward, especially when one needs to write them often. We therefore introduce the following notational shorthand. In the first place, we will not write the superscript i (in Xi, yi, ~i, or to denote which coordinate is concerned. In the second place we will abbreviate any product of k factors ~ as ~k (and like wise for the TJ). At the same time we will omit the summation sign over indices i 1, ... , i k as well as these indices in the coefficient functions 9 and h. We thus will write

rn

gt ...

Here we have used the symbol f again for the coefficient functions ,ik and ht, ... ,ik' We also have added a subscript to indicate with which power of ~ they are associated. No confusion between y and TJ is possible, since y only contains even powers of ~ and TJ only odd powers. On the other hand, these simplifications are easily misunderstood, so we will use them only when strictly needed. Taking all ~ to be zero, we obtain that y = f(x, 0) = fo(x) and TJ = O. In other words, fo is the real valued function fo = B f. On the other hand, h (x) represents the matrix valued real function x I--> (x) of size s x q.

hL

3.18 Lemma. Let f : U c Eo -+ Fo be a smooth function as in the above discussion, and let V be an open set in Fo. Then thefollowing two properties are equivalent. (i) f(U) = V and there exists a function 9 E C=(V; E) such that go f = id(U)

and fog = id(V). (ii) fo (BU) = B V and there exist smooth functions go and g1 defined on B V such that go 0 fo = id(BU), fo 0 go = id(BV) and such thatfor all x E BU and all y E BV: g1(fO(X)) . h(x) = Iq and h(go(y)) . g1(y) = Is. Here In denotes the n x n identity matrix. In other words, the function f admits a smooth inverse if and only if fo = Bf admits a smooth inverse and h is everywhere invertible.

120

Chapter III. Smooth functions and A-manifolds

e

ej+l,

Proof Ifwe assume (i), we have y = 2::i2:0 hi(X)' i and ry = 2::r;:~o hj+l(x) . but also x = 2::k2:0 g2k(Y) . ry2k and ~ = 2::£2:0 g2£+1 (y) . ry2£+1. Substituting the first relations in the second ones we obtain

i x = Lg2k(Lhi(X)' e ). (Lhj+l(x), k2:0 i2:0 j2:0 = L gb£) (fo (x)) . (L hi (x) . e i ) £ + £2:0

(3.19)

eH1 fk

i>O

Lg2k(L hi(X) . e i ) . (L hHl(X) . ej+lfk k>O i2:0 j2:0

To obtain the second line, we have applied [2.27] on go and the fact that 2::i>O hi (x) . is nilpotent. In a similar way we obtain for ~ the equation

e

i

(3.20)

Since the coefficients of the separate powers of ~ in an expansion are uniquely determined [2.21], one deduces from (3.19) in order 0 that go 0 fo = id(BU). The order 1 term in (3.20) is given by ~ = gl(fO(X)) . h(x) .~, and thus by the same argument one obtains gl (fo (x)) . h (x) = I q • The rest of property (ii) follows by interchanging the roles of f andg. To prove the implication (ii) =} (i), we define the function 9 by defining the smooth functions gi by induction on i using (3.19) and (3.20). By assumption the functions go and gl exist. Now consider in (3.19) the terms with en. This gives us the equation (3.21)

0= g2n(fO(X))' (h(x) . ~)2n

+ other terms,

where the other terms only contain functions g2k with k < n. To define g2n we now proceed as follows. In (3.21) we replace each occurrence of x by go(y) and each occurrence of ~ by gl(y) . ry. Finally we leave out all the ry's. Since fo(go(y)) = y, we obtain a definition of g2n(y) in terms of the functions gl and g2k' ry2k with k < n, which are defined by the induction assumption. With a similar argument using (3.20) we obtain the functions g2£+1. We thus obtain a function 9 that satisfies by construction (and property (ii)) the condition go f = id(U). Interchanging the roles of f and 9 gives the existence of a smooth function h with ho = fo and hI = h such that hog = id(V). We deduce that 9 is both injective and sUljective and that hence h = f, proving property (i). IQEDI

§3. Derivatives and the inverse function theorem

121

3.22 Remark. In the special case A = R, there are no (non-zero) odd coordinates. Hence an even function f is uniquely represented by the function fo in the shorthand of [3.17]. Part (ii) of [3.18] should thus be replaced by omitting all references to hand g1.

3.23 Theorem (inverse function theorem). Let 0 C Eo be open, f E C= (0; F)o, and if and only if there exists an open neighborhood U ofm such that (i) V = f(U) is open in Fo and (ii) f : U -+ V admits a smooth inverse.

m E 0 arbitrary. Then the Jacobian Jac(f) (m) is invertible

Proof Let us start analyzing the invertibility of the Jacobian: Jac(f)(m) is invertible if and only if its matrix ad j (m) is invertible, which is invertible if and only if its body B( ad j (m)) is invertible [11.1.11]. Using an ordered basis (as we usually do when splitting coordinates in even and odd ones) and applying our knowledge of how partial derivatives work on expanded functions, we obtain m = (x,~) and B( ad j (m)) in block form as

If this matrix is invertible, it follows that both (aftjaxR)(x) and h(x) are invertible matrices. Application of the standard inverse function theorem to the function fo shows that there exists an open neighborhood 0 0 C BEo of x and an open subset Po c BFo such that fo : 0 0 -+ Po admits a smooth inverse. Smoothness of h implies that there exists an open neighborhood 0 1 C BEo of x on which the matrix h is everywhere invertible. We thus conclude that condition (ii) of [3.18] is fulfilled on the open neighborhood U = B- 1 (00 n 0 1 ) and the open set V = B- 1 (fo(00 n 0 1 )) c Fo. It follows that f admits a smooth inverse on U. On the other hand, if f admits a smooth inverse on U, we infer from condition (ii) of [3.18] and the standard inverse function theorem that (aft jaxR)(x) and h(x) are IQEDI invertible on BU, i.e., that the Jacobian matrix is invertible.

3.24 Remark. In the special case A = R, the statement of the inverse function theorem is still valid. However, the Jacobian should be restricted to the even-even part as argued in [3.12]. This is also "obvious" from the proof because h does not exist (is identically zero ).

3.25 Corollary (invariance of dimension). Iff E C=(U; F)o admits a smooth inverse, the graded dimensions ofE :J U and F are the same.

3.26 Remark. In the special case A = R, the invariance of dimension only concerns the even dimension. This follows from the fact that in that case the Jacobian takes its values in HomdEo; Fo) [3.12], in which the odd dimensions have disappeared.

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Chapter III. Smooth functions and A-manifolds

3.27 Theorem (implicit function theorem). Let E and F be A-vector spaces, U open in (E EB F)o and f E C=(U; F)o. Denoting by x the coordinates (even and odd) in E and by y the coordinates in F, we suppose that (x o , Yo) E BU. If the square matrix (8 fi /8yj)( Xo, Yo) of partial derivatives with respect to the coordinates in Fo is invertible, then there exist open sets Xo EVe Eo and Yo EWe Fo with V EB W c U, and a smooth function 9 : V ---+ W with g(x o) = Yo such that

V (x, y) E V EB W : f(x, y)

= f(x o , Yo)

<;=}

y

= g(x)

.

Proof Define the map E C=(U; E EB F)o by (x, y) = (x, f(x, y)). By hypothesis the Jacobian of this map in the point (x o , Yo) is invertible. Hence there exists an oper This inverse must have set of the form V EB W on which admits a smooth inverse the form w(x, y) = (x, 'lj;(x, y)) with f(x, 'lj;(x, y)) = y and 'lj;(x, f(x, y)) = y. We then define the function 9 : V ---+ W by

w.

g(x)

= 'lj;(x, f(x o, Yo))

.

Since f(x o , Yo) has real coordinates, 9 is smooth by [1.23-g]. It obviously satisfies g(xo) = 'lj;(xo,f(xo,yo)) = Yo and f(x,g(x)) = f(x,'lj;(x,f(xo,yo))) = f(xo,yo). Finally, if f(x, y) = f(x o , Yo), then y = 'lj;(x, f(x, y)) = 'lj;(x, f(x o , Yo)) = g(x). IQEDI

3.28 Definition. For f E C=(U; Fo) we define the rank off at m E U, denoted by rank(f)(m), as the rank [11.4.18] of its Jacobian Jac(f)(m), i.e., as the rank of the morphism of R-vector spaces B(Jac(f)(m)).

3.29 Lemma. Let E and F be finite dimensional A-vector spaces and ¢ E Homs( E; F). Then ¢ is injective ifand only ifrank¢ = dimE, which is the case ifand only ifB¢ is injective. Proof Right and left linear being similar, we will treat the right linear case. It is well known from linear algebra over R that a linear map is injective if and only if its rank is equal to the dimension of the source space. This proves the equivalence between rank ¢ = dim E and B¢ injective because rank ¢ is just defined as the (ordinary) rank of B¢ [11.1.9], [11.4.18]. Now assume that ¢ is injective and that v E BE is such that (B¢) (v) = O. Since v = Bv, it follows that B( ¢( v)) = 0, i.e., ¢( v) is a nilpotent vector. Hence there exists A E A, A =I 0 such that (¢(V))A = O. By right linearity and injectivity of ¢ we conclude that VA = O. But v has real coefficients with respect to a basis and A =I 0, and thus all these real coefficients must be zero, i.e., v = O. This proves that B¢ is injective. Finally assume that B¢ is injective. Then n = dim E ::; dim F = m, and thus, with respect to bases of E and F, the matrix t1R ( ¢) has the form ( ~ ) with X a n x n matrix and Y a (m - n) x n matrix. Up to a permutation of the basis vectors for F we may assume that BX is invertible because t1R(B¢) = (:~) is injective. But then X is invertible by

[11.1.11]. We thus can define the right linear map 'lj;: F follows that 'lj; 0 ¢ = idE and thus that ¢ is injective.

---+

Eby t1R ('lj;)

=

(X-l 0). It

IQEDI

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123

3.30 Proposition. Let E and F be A-vector spaces of total dimension nand p respectively andsupposethatf E C=(U;Fo)hasrankkatm E U. Then there exist open sets V C U, m E Vand V' C Eo and a smooth map 9 : V -+ V' with smooth inverse such that f og -1.. (x 1, ... ,xn)

I-->

(1 y, ... ,yP) -_ (x 1, ... ,x k , fk+1( 9 -1()) x,... fP( 9 -1( x ))) ,

with respect to given bases ofE and F (up to a permutation ofthe basis vectors). In case k = n, which is equivalent to Jac(f) (m) being injective [3.29], there exists an open set W C Fo and a smooth map h : W -+ W with smooth inverse such that hofog-1: (x1, ... ,xn)

I-->

(yt, ... ,yP)

=

(xt, ... ,xn,O, ... ,O).

= k implies that there exists an invertible square submatrix of size k x k of the Jacobian Jac(f) (m). By permuting the coordinates we may assume that (a fi / ax]) (m), 1 :::; i, j :::; k is invertible. Now define 9 : U -+ Eo by

Proof The condition rank(f) (m)

=

g(x1, ... , xn)

(f1(x), ... , fk(x), xk+1, ... , xn) .

Our assumption implies that the Jacobian of gat m is invertible, and thus, by the inverse function theorem, there exist open set V C U and V' C Eo such that glv : V -+ V' admits a smooth inverse. This glv satisfies the requirements. If k = n, and thus p :2: n, we define W = { (y1 , ... , yP) E Fo I (y1, ... , yn) E V' } and the smooth map h : W -+ W by .

1

h'(y , ... , yP)

=

Y {

y

i i _

fi( -1( 1 n)) 9 y, ... ,y

3.31 Counter examples. For ordinary real-valued smooth functions the condition k = n can easily be weakened to the condition that the rank of f equals k on the whole of V without affecting the result [Spi, Ch2 Th9]. In the graded case the situation is far more delicate as can be seen in the following counter examples . • Let E be an A-vector space of dimension 012 with coordinates (~, ry). The map f E C=(Eo; Ao), defined by f(~, ry) = ~ry ,

has rank 0 on Eo. However, it is not a constant map . • Let F be an A-vector space of dimension 112 with coordinates (x, ~, ry). The map f E C=(Fo; Fo), defined by f(x,~,ry) = (~ry,~,O),

has rank 1 on Fo. However, an elementary calculation shows that it can not be transformed (not even locally) into the form (z1, z2, z3) I--> (z1, 0, 0).

124

4.

Chapter III. Smooth functions and A-manifolds A-MANIFOLDS

In this section we copy the definition of manifolds in terms of charts and (smooth) transition functions to obtain the definition of A-manifolds. The definition of a smooth function between A-manifolds is standard and the extension ofthe body map to A-manifolds and their smoothfunctions is straightforward. There is only one subtlety involved: since the DeWitt topology is not Hausdorff, we cannot require directly that an A-manifold should be in particular a second countable Hausdorjftopological space. We thus are "obliged" to make a slight detour via proto A-manifolds where we forego these topological conditions.

4.1 Definitions. Let M be a topological space; a chart for M is a triplet 'P : U -+ 0, where 'P is a homeomorphism from an open subset U of M onto an open subset 0 of the even part of some A-vector space E. When no confusion is possible, we will often denote a chart by the open set U c M or by the pair (U, 'P) instead of by the full triplet. Two charts 'Pa : Ua -+ Oa c (Ea)o and 'Pb : Ub -+ Ob C (Eb)O for M are said to be compatible if the two maps 'Pa 0 'P;;l and 'Pb 0 'P;;-l are smooth. More precisely:

'Pa 0 'P;;l E C=('Pb(Ua nUb); Ea)o 'PbO'P;;.l

E

C=('Pa(Ua n Ub);Eb)o.

A smooth A-structure for a topological space M is a collection of charts S

= {'Pa :

Ua -+ Oa I a E I} satisfying three conditions: (SSl) UaE1Ua = M, (SS2) any two charts in S are compatible and, (SS3) if'P: U -+ 0 is a chart for M that is compatible with all elements of S, then this chart belongs to S. A proto A-manifold is a topological space M together with the prescription of a smooth A-structure S; an atlas for a proto A-manifold is a subset U c S satisfying properties (SSl) and (SS2), but not necessarily (SS3).

4.2 DiscussionlDefinition. The attentive reader might have noticed that the above definitions do not really specify the A-vector space in which the open sets Oa lie. This means in particular that the different Oa might lie in A-vector space of different graded dimensions. However, if m E M lies in two different charts Ua and Ub, the even smooth maps 'Pa 0 'P;;l and 'Pb 0 'P;;.l are inverse to each other. It follows from the invariance of the graded dimension [3.25] that Oa and Ob lie in A-vector spaces of the same graded dimension. We thus may conclude that the graded dimension is a locally constant function on M, i.e., the connected components of M have a well defined graded dimension. We will usually assume that all connected components have the same graded dimension. Since the graded dimension is the only invariant of an A-vector space E, we will say that a proto A-manifold M is modeled on the A-vector space E if its graded dimension is the same as that of E, i.e., if all the charts can be thought of as lying in Eo. If M is modeled on an A-vector space of dimension plq, we will say that the (graded) dimension ofM is plq.

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125

4.3 Remark. Of course, for proto R-manifolds the invariance of graded dimension [3.25] should be replaced by the invariance of the even dimension [3.26]. Thus for proto Rmanifolds it is the even dimension of the R-vector space on which a chart is modeled which is the invariant, the odd dimension does not matter (see also [2.10]).

4.4 Lemma. Given a topological space M, a collection of charts U = {Ua I a E I} satisfying (SS 1) and (SS2) defines a unique smooth A-structure for M for which U is an atlas.

Proof To prove the existence, define S to be the collection of all charts for M that are compatible with all charts in U. Since (SS2) is valid for U, it is included in S, and hence (SSl) is satisfied. To prove (SS2), let cp : U ~ 0 and'lj; : V ~ Q be two elements of S. Notice that { 'lj;(U n V n Ua) I a E I} is an open cover of 'lj;(U n V) because U satisfies (SSl). Since on 'lj;(U n V n Ua) we have cpo'lj;-I = cpo cp;;1 0 cpa o'lj;-l, we deduce that cp 0 'lj;-I restricted to 'lj;(U n V n Ua) is smooth as composition of smooth maps. Hence by [1.23-k], cp 0 'lj;-I is smooth. Similarly, 'lj; 0 cp-l is smooth and hence these two charts are compatible. Finally, to prove (SS3), let U be a chart compatible with all elements of S. Since U C S, U is compatible with all elements ofU and hence belongs to S. To prove uniqueness, let S' be another smooth A-structure for M containing U. Any chart in S' is obviously compatible with all elements in U and hence belongs to S, proving S' C S. On the other hand, if U is a chart in S :) S', then it is compatible with all elements in S' and hence belongs to S' because S' satisfies (SS3). IQEDI

4.5 Conventions. From now on we will adopt the following conventions. In the first place, when talking about proto A-manifolds, we will never specify explicitly a smooth A-structure, at most we will specify an atlas for it (which defines the smooth A-structure completely according to [4.4]). In the second place, when we talk about charts, this will always mean a chart in the smooth A-structure, i.e., compatible with all charts in an atlas.

4.6 Corollary. Every set ofcharts covering a proto A-manifold is an atlas.

4.7 Lemma. Let cp : U ~ 0 be a chart ofa proto A-manifold M and let V be an open subset ofU. Then cplv : V ~ cp(V) is a chartfor M.

4.8 Examples. • For any A-vector space E, its even part Eo is a proto A-manifold when equipped with the trivial atlas id : Eo ~ Eo. In particular AD is a proto A-manifold. But Al also is a proto A-manifold. Consider the A-vector space E of dimension 011, i.e., generated by a single odd basis vector e. The standard map Al ~ E, a ~ a . e defines a homeomorphism between Al and Eo, i.e., an atlas for AI.

Chapter III. Smooth functions and A-manifolds

126

• More generally, every A-vector space E can be given the structure of a proto Amanifold by identifying it with with the even part of the A-vector space EU . • If M is a proto A-manifold, any open subset of M is again a proto A-manifold in the obvious way: by restricting an atlas to the open subset in question.

4.9 Construction. Let M be a proto A-manifold and U = { 'Pa : Ua -+ Oa I a E I} an atlas. We then define open subsets Oab C Oa and smooth maps 'Pba : Oab -+ Oba by Oab = 'Pa(Ua nUb) and 'Pba = 'Pb 0 'P;;l. It is easy to verify that the open subsets and maps Oab C Oa, 'Pab have the properties

Va E I

(4.10)

Va, b, eEl

'Paa

= id(Oa) ,

on Oab n Oae : 'Pea = 'Peb 0 'Pba .

What we intend to do is to show that these ingredients completely determine the proto A-manifold M. Let us be more precise. Suppose we are given the following data: (i) a collection {Oa I a E I} where the Oa are open in the even part of some A-vector space, (ii) for all a, bEl an open subset Oab C Oa and (iii) for all a, bEl a smooth map 'Pba : Oab -+ Oba. Suppose furthermore that these data satisfy (4.10). We then can (and will) construct a proto A-manifold M and an atlas U such that these data appear as in the previous paragraph. In order to carry out this construction, first note that implicit in (4.10) are the conditions that (i) Oaa = Oa and (ii) Va, b, eEl: 'Pba(Oab n Oae) C Obe, because else the formula; would not make sense. Next we consider the disjoint union LIaElO a of all the Oa, on which we define an equivalence relation by

x'" y

{=}

:Ja,b E I: x E Oab,y E Oba,y = 'Pba(X) .

This is an equivalence relation because of (4.10). We finally define the topological space M as the quotient M = UaElOai"'. Denoting by Jr the canonical projection Jr : UaElOa -+ M, we define the open subsets Ua of M by Ua = Jr(Oa). Since no two different points in a single Oa are equivalent, Jr : Oa -+ Ua is a homeomorphism. Hence we obtain charts 'Pa : Ua -+ Oa by taking 'Pa = Jr- 1 . It follows easily that 'Pa(Ua nUb) = Oab and 'Pa 0 'Pb 1 = 'Pab, from which one can deduce that the collection of these charts for M satisfies (SS 1) and (SS2), and hence defines a smooth A-structure on M.

4.11 Construction. If E is an A-vector space, we have defined a distinguished subset BE, consisting of those points in E that have real coordinates (with respect to a basis in the equivalence class), and a canonical projection B : E -+ BE. Using these ingredients, we will define similar things for proto A-manifolds. In the first place we define the body B M of a proto A-manifold M by

BM

= {m E

M I :J chart 'P : U

-+

0 : m E U and 'P(m) E BO} ,

127

§4. A-manifolds

i.e., m E BM if and only if it has real coordinates in a chart. This is well defined because ifm has real coordinates in a chart 'Pa : Ua -+ Oa, then its coordinates in a chart 'Pb: Ub -+ Ob are given by ('Pb°'P;;-1)('Pa(m)). Since 'Pb ° 'P;;-1 is smooth, the image ofa point with real coordinates has again real coordinates. We conclude that if a point m has real coordinates in some chart, it has real coordinates in all charts. In a similar spirit we define the canonical projection B : M -+ BM by

'P : U -+ 0 a chart

===}

Blu = 'P- 1 ° B ° 'P ,

where B is the previously defined body projection B : 0 C E -+ BO c BE. Again this definition is independent of the choice of the chart 'P : U -+ 0 because smooth maps preserve points with real coordinates. A quick glance at the examples [4.8] shows that this new definition of B coincides with the previously defined one, i.e., in case the proto A-manifold is an open subset of the even part of an A-vector space.

4.12 Lemma. Let M be a proto A-manifold and U c M an arbitrary subset. Then U is open if and only if BU is open in BM equipped with the induced topology and B-1(BU) = U, in which case we have BU = Un BM c U.

Proof For a chart 'Pa : Ua -+ Oa we have Blua = 'P;;-1 ° B ° 'Pa and hence BUa C Ua. Now let us suppose that for arbitrary m E M we have Bm E Ua' If m lies in the chart 'Pb : Ub -+ 0b, it follows that Bm E Ua nUb, hence B'Pb(m) = 'Pb(Bm) E Oba [4.9]. By the DeWitt topology we then also have 'Pb( m) E Oba and thus m E Ua nUb. Together with BUa c Ua we thus have BUa = Ua n BM and B- 1 (BUa ) = Ua' Since any open U can be covered by charts (use [4.7] to restrict charts to U), the same holds for U and in particular BU = Un BM is open in BM equipped with the induced topology. On the other hand, assume that BU is open in BM and that B- 1(BU) = U. Then in particular there exists an open V C M such that BU = BM n V = BV. But then U = B- 1(BU) = B- 1(BV) = V and thus Uis open. IQEDI

4.13 Corollary. The subset BM ofa proto A-manifold with its induced topology inherits a natural structure ofa proto R-manifold. In particular, ifM has dimension plq, then BM has dimension p.

Proof If 'Pa : Ua -+ Oa and 'Pb : Ub -+ Ob are two (compatible) charts for M, then in the first place 'Pa : BUa -+ BOa and 'Pb : BUb -+ BOb are two charts for BM. Since the restriction of a smooth map to real coordinates is a smooth map in the ordinary sense [2.7], it follows that these two charts are compatible. Hence the smooth A-structure on M induces a smooth R-structure on BM. If M is modeled on an A-vector space E, BM is modeled on BEo. In particular, if M has dimension plq, then BM has dimension p. The fact that the odd dimension disappears is because the charts of M lie in the even part of E, and thus the odd coordinates become zero when taking the body map. IQEDI

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128

4.14 Definition. In the literature, a manifold is usually defined as a second countable (i.e., a countable basis for the topology) Hausdorff topological space equipped with a smooth R-structure. Sometimes the condition second countable is replaced by paracompact. These conditions are added besides the smooth structure condition in order to get rid of some pathological examples and in order to have partitions of unity. Most of the general theory can be developed without these conditions, but not much is gained by doing so. We thus define an A-manifold as a proto A-manifold M for which its body BM is a second countable Hausdorff topological space. It follows immediately that BM is an R-manifold in the usual sense. Although the condition of second countability could have been imposed directly on M, the Hausdorff condition can not because the DeWitt topology is not Hausdorff.

4.15 Definition. A map f : M ~ N between two A-manifolds is smooth if for all charts (U, cp) of M and all charts Cv, 'lj;) of N, one has 'lj; 0

f

0

cp-l is smooth on cp(U n rl(V)) .

Note that this condition is empty whenever U n f-l(V) is the empty set. Since smooth maps are in particular continuous, it follows that smooth maps between A-manifolds are continuous. The set of all smooth maps from M to N is denoted by C=(M; N). A map f : M ~ N is a diffeomorphism if it is a smooth bijection whose inverse is also smooth. If that is the case, the A-manifolds M and N are said to be diffeomorphic.

4.16 Nota Bene. If the map f : M ~ N is smooth and if ¢ : U ~ 0 c Eo and 'lj;: V ~ Pc Fo are charts, then the map 'lj;ofo¢-I: ¢(Unf- 1 (V)) c Eo ~ Fisan even smooth map. In other words, smooth maps between A-manifolds are always even in the sense of [1.20].

4.17 Lemma. Let f : M ~ N be a map, U an atlas for M, and Va set ofcharts on N covering f(M). Iff satisfies the condition given in [4. 15Jfor all charts U E U and V E V, then f is smooth. Proof Denote by (U,cp) and (V,'lj;) arbitrary charts and by (Ua , CPa) and (Vb,'lj;b) charts in U and V respectively. On the set cp(Ua nUn f- 1 (Vb n V)) we have the equality 'lj; 0 f 0 cp-l = ('lj;o 'lj;;;I) 0 ('lj;b 0 f 0 cp;;l) 0 (CPa 0 cp-l), which is smooth by definition of charts and the hypothesis of the lemma. Since the sets Ua nUn f- 1 (Vb n V) cover the set Un f-l(V) (the Ua cover M and the Vb cover f(M)), we conclude by [1.23-k]. IQEDI

4.18 Corollary. Let f : M ~ N be a map between A-manifolds, let U be an atlas for M. Ifwe define the maps fa by fa = flu a , where Ua E U is seen as an A-manifold [4.8J, then the map f is smooth ifand only ifall the maps fa are smooth.

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§4. A-manifolds

4.19 Lemma. Let L, M, and N be A-manifolds, and let J : L smooth maps. Then 9 0 J : L -+ N is smooth.

-+

M and 9 : M

-+

N be

Proof Let (U, 'P), (V, 'l/J), and (W, X) be charts for L, M, and N respectively. What we have to show is that the map xo(gof)0'P- 1 is smooth on 'P(U n (gof)-I(W)). On 'P(Un (go f)-l(W)nJ-I(V)) we have X 0 (go f) 0 'P- I = (X o9 0 X-I) 0 (X 0 J 0 'P- I ), which is smooth since J and 9 are smooth. By varying V, we cover 'P(U n (g 0 f)-I(W)) by sets of the form 'P(U n (go f)-I(W) n J-I(V)). Using either [4.18] or [1.23-k] we conclude that X 0 (g 0 f) 0 'P- 1 is smooth. IQEDI

4.20 Discussion. If M is an A-manifold and F an A-vector space, we could have defined smooth maps J : M -+ F as those maps such that for all charts (U, 'P) of M the map J 0 'P- I is smooth. This makes sense because it is a map from an open subset of the even part of some A-vector space to the A-vector space F. On the other hand, we can interpret F as an A-manifold as in [4.8] and apply [4.15]. Since F has a single chart F -+ F~ it follows from [1.23-b,i] that these two definitions coincide. Now let 0 C Eo and U c M be open. The above discussion gives us the following characterization: a map ¢ : U -+ 0 is a chart [4.1] if and only if ¢ is a diffeomorphism [4.15].

4.21 Lemma. Let M be an A-manifold and F an A-vector space. The set of all A-valued

smooth functions C=(M) == C=(M; A) is a gradedR-algebra andC=(M; F) is afree graded C=(M)-module ofthe same graded dimension as F.

4.22 Proposition. Given a smooth map J : M

-+ N between two A-manifolds, the restriction of J to BM c M is the unique smooth map BJ : BM -+ BN between R-manifolds such that the following diagram is commutative:

M

f

N

---+

B1 BM

1B I

Bf=fIBM

BN.

Proof Uniqueness ofBJ is immediate from the surjectivity of B. Since J is smooth, the image of a point with real coordinates must have real coordinates, hence J(BM) C BN. Smoothness as a map between R-manifolds follows from [2.7]. IQEDI

4.23 Lemma. Let M be an A-manifold and let F be an A-vector space. Then the map B : C=(M;A) -+ C=(BM;R) is a morphism ofgraded R-algebras with unit and

Chapter III. Smooth functions and A-manifolds

130

the map B : C=(M; F) -+ C=(BM; BF) is a morphism from graded modules over C=(M; A) to graded modules over C=(BM; R).

5.

CONSTRUCTIONS OF A-MANIFOLDS

This section treats subjects directly related to the creation of A-manifolds. The definitions ofa direct product and ofa submanifold are straightforward generalizations of the usual definitions. The implicitfunction theorem allows us in the standard way to define submanifolds of Eo by equations (a more general version can befound in [V.2.10J). The only point in which submanifolds of A-manifolds differ from their ordinary counterparts is that it is not possible to find submanifolds of lower dimension through an arbitrary point of the ambient A-manifold. We define andprove the existence of partitions ofunity, and we finish this section by extending the body map B and the "inverse" map G toA-manifolds. As for the even part Eo of an A-vector space E, G can not reconstruct the odd dimension of an A-manifold. More precisely: G(BM) reconstructs the even part of M, omitting the odd coordinates completely.

5.1 Definition. A subset N c M of an A-manifold M modeled on an A-vector space E is a sub manifold if there exists for each point mo E N a chart cp : U -+ 0 c Eo containing mo and a graded subspace F of E such that

cp(U n N) = 0 n Fo == 0 n F .

5.2 Discussion. If F is a graded subspace of the A-vector space E, there exists a homogeneous basis (ei)i=l such that a subset is a basis for F. Without loss of generality we may assume that F is generated by (ei)~=l' It follows that the submanifold N is locally described by the n - k equations

Un N

= {m E U

I If k


:::; n : ie(cp(m))

= O} .

5.3 Nota Bene. The definition of a submanifold looks quite innocuous and completely similar to the ordinary definition for R-manifolds. However, one should realize that, contrary to the ordinary case, one can not guarantee that through every point of an Amanifold passes a submanifold of lower dimension. This is due to the fact that we require that F is a graded subspace of E, and not just any graded submodule. To show the problem, let E be the A-vector space of dimension 012 with basis eb e2. We take M = Eo ~ AI the A-manifold of dimension 012. Any graded subspace

§S. Constructions of A-manifolds

131

of dimension 011 of E must be generated by a vector of the form f = ae1 + be2 with a, bE R. Now let (6,6) be a point of M and suppose that it belongs to a submanifold N of M of dimension OIl. Since there is just one single chart possible, this means that (6, 6) = 6 e 1 + 6 e2 must belong to the subspace generated by ae 1 + be2, i.e., there exists A E A1 such that 6 = Aa and ~2 = Ab. If neither 6 nor ~2 is zero, this implies that neither a nor b can be zero. Since they are real, we deduce that 6 = 6bja. And if A1 is not too small, it is easy to find 6 and 6 that are not a real multiple of each other. Hence there does not exists a submanifold of dimension 011 through such a point.

5.4 Proposition. Let N be a submanifold ofan A-manifold M equipped with the induced topology. Then there exists a (canonically defined) smooth A-structure on N making it an A-manifold for which the canonical injection i : N -+ M is smooth. Moreover, BN is a submanifold of BM in the sense of R-manifolds.

Proof Let 'P : U -+ 0 c Eo be a chart as specified in the definition of submanifold and let F be the graded subspace of E such that 'P(U n N) = 0 n Fo. By definition of induced topology, the set V = Un N is open in N, P = 0 n Fo is open in Fo and the map'l/J = 'Plv : V -+ P is a homeomorphism. In other words, 'l/J : V -+ P is a chart for N. If we denote by V the collection of all such charts for N, it automatically satisfies condition (SSl) of a smooth A-structure. To show that this V also satisfies condition (SS2), let 'l/Ja : Va -+ Pa C (Fa)o and 'l/Jb : Vb -+ Pb C (Fb)o be two charts of N associated to charts 'Pa : Ua -+ Oa C (Ea)o and 'Pb : Ub -+ Ob C (Eb)O of M. The map 'Pb 0 'P;;1 : 'Pa(Ua nUb) -+ (Eb)O is smooth by definition of charts, so by [1.23-g] its restriction to 'Pa(Ua nUb) n (Fa)o is smooth. But 'Pa(Ua nUb) n (Fa)o = 'Pa(Ua n Ub n N) = 'l/Ja(Va n Vb), hence 'P;;l('l/Ja(Va n Vb)) c N and thus 'Pb 0 'P;; 1 ('l/Ja(Va n Vb)) c (Fb)o. It follows from [1.23-i] that 'Pb 0 'P;;1 : 'l/Ja(Va n Vb) -+ (Fb)o is smooth. But with these restrictions, the map 'Pb 0 'P;; 1 is exactly the map 'l/Jb 0 'l/J;; 1, showing that V satisfies also condition (SS2), i.e., V is an atlas for a smooth A-structure on N by [4.4]. Finally BN is a second countable Hausdorff space because it is a subset of BM which has these properties. In terms of the given atlas for N it is immediate from [4.17] that the canonical injection is smooth. To prove the statement about BN, we apply the body map to the definition of a submanifold. Since for open sets we have B-1 (B U) = U and B- 1(BO) = 0 we have B(U n N) = BU n BN and B(O n Fo) = BO n BFo. Since the 'P : BU -+ BO are the charts for BM, it follows immediately that BN is a submanifold of BM in the sense of R-manifolds. IQEDI

5.5 Remark. We will show in [Y.2.17] that the smooth A-structure on a submanifold

N C M given in [5.4] is the unique smooth A-structure such that the canonical injection i : N -+ M is an embedding.

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Chapter III. Smooth functions and A-manifolds

5.6 Lemma. Let M be an A-manifold, N C M a submanifold, and LeN a submanifold of the A-manifold N. Then L C M is a submanifold of M. Proof Let M be modeled on E, FeE a graded subspace on which N is modeled, and H c F a graded subspace on which L is modeled. Let 'lj; : V -+ P be a chart for N such that 'lj;(V n L) = P n Ho. By taking smaller charts we may assume that 'lj; is a chart induced from a chart ¢ : U -+ 0 for M, i.e., V = Un N, P = 0 n Fa, and 'lj; = ¢Iv. It follows that ¢(U n L) = 0 n Ho. IQEDI

5.7 Lemma. Let N be a submanifold of M and fiN: N -+ L is smooth.

f :M

-+

L a smooth map.

Then

Proof Let ¢ : U -+ 0 be a chart for M such that U n N = V, P = 0 n Fa and 'lj; = ¢Iv : V -+ P is a chart for N. Let furthermore X : W -+ Q be a chart for L. According to [4.17] it suffices to prove that X 0 fiN 0 'lj;-1 : 'lj;(V n (fIN )-I(W)) -+ Q is smooth. But this is the map X 0 f 0 ¢-1 restricted to ¢(U n f-l(W)) n Fa, which is smooth by [1.23-g]. IQEDI

5.8 Lemma. Let M and L be A-manifolds, let N C M be a submanifold, and let f : L -+ M be a smooth map. Iff(L) eN, then the map f : L -+ N is smooth when N is equipped with the smooth A-structure of [5.4J Proof Let us denote the map from L to the A-manifold N by fo in order to distinguish it from the initial map to M. (Nota Bene: If fo is smooth, automatically f = i 0 fo is smooth.) Let ¢ : U -+ 0 be a chart for M such that Un N = V, P = 0 n Fa and 'lj; = ¢Iv : V -+ P is a chart for N. Since F is a graded subspace of the finite dimensional A-vector space E, there exists a supplement [11.3.7], [11.6.23], and thus a smooth projection 7r : E -+ F. Since f(L) c N, obviously the restriction of fo to f- 1 (U) is given as fo = 7r 0 f. Since the f- 1 (U) cover L, fo is smooth. IQEDI

5.9 Example. For any A-manifold M modeled on an A-vector space E there exists a (canonically defined) submanifold Mw.o.d modeled on Ew.o.d. Let {Ua I a E I} be an atlas for M and define the set Mw.o.d by Mw.o.d = UaElrp~I(Oa n Ew.o.d). To prove that it is a submanifold, we claim that the charts Ua satisfy the conditions of [5.1]. We first compute rpb(rp~I(Oa n Ew.o.d) nUb) = (rpb 0 rp~I)(rpa(Ub n Ua) n Ew.o.d). Since rpb 0 rp~ 1 is a smooth map, it follows from [3.17] that (rpb 0 rp~ 1 ) (rpa (Ub n Ua ) n Ew.o.d) is contained in rpb(Ub n Ua ) n Ew.o.d. Hence rpb(Mw.o. d nUb) = Ob n Ew.o.d. The submanifold Mw.o.d can also be described as consisting of those points in M for which the odd coordinates in some chart (and hence in all charts) are zero. It follows that we have the inclusions BM C Mw.o.d c M. Moreover, B(Mw.o.d) = BM c M. Given that we have a (canonical) projection B : M -+ BM, one might be tempted to think that there also exists a natural projection M -+ Mw.o.d. That this is a false belief (at least when the odd dimension of M is greater than 1) is shown by the next example.

133

§S. Constructions of A-manifolds

Consider the A-manifold M = Eo with coordinates (x,~, 1]), where E is an A-vector space of dimension 112. The map f : (x,~, 1]) f---4 (x + ~1],~, 1]) is a diffeomorphism. However, projection onto the even coordinate is not invariant under this diffeomorphism (but projection onto the body is). Let f : M -+ N be a smooth map between A-manifolds. If the odd coordinates of m E M are zero, i.e., m E Mw.o. d, the odd coordinates of f (m) are also zero (if all odd coordinates are zero, the parity of the image of a smooth function is necessarily even). We thus have an induced map fIMw.o.d : Mw.o.d -+ Nw.o.d. Since BM C Mw.o.d and BN C Nw.o.d, we can combine this into a commutative diagram

BM fIBM=Bf

C

1

1

BN

1Yfw.o.d

c

M

flMwo.d

C

Nw.o.d

c

5.10 Example (sub manifolds defined by equations). Let f : Eo -+ F be a smooth map and M = rl(O) C Eo. Suppose that for all m EM: rank(f)(m) = dim F = k (implying that k = dim F :s; dim E = n). Let 9 : V -+ V' be as in [3.30], i.e., such that f 0 g-l(x l , ... , xn) = (xl, ... , xk). Since 9 and g-l are smooth, 9 : V -+ V'is a chart for Eo. Moreover, g(V n M) = V' n Span( ek+l, ... , en). We conclude that Mis a submanifold of Eo, and thus in particular that M is an A-manifold.

5.11 Construction. Given two A-manifolds M and N, we equip the cartesian product M x N with smooth A-structure by the following prescription. Let U be an atlas for M and V one for N. Now define an atlas W for M x N by

W = { 'Pa x 'l/Jb : Ua X Vb

-+

Oa

X

Pb I a

E

I, b E J} .

In this construction, the map 'Pa x 'l/Jb is defined by ('Pa x 'l/Jb) (x, y) = ('Pa (x), 'l/Jb (y)); that it is smooth is a direct consequence of [1.23-f,i]. We leave it to the reader to verify that W indeed satisfies (SSI) and (SS2), and thus defines a smooth A-structure for M x N. Since the product of two second countable Hausdorff spaces is again second countable and Hausdorff, it follows that M x N is an A-manifold, called the direct product of the Amanifolds M and N. If M is modeled on E and N on F, then M x N is modeled on E EB F. Associated with this direct product, we have two canonical projections 7r M : M x N -+ M and 7rN : M x N -+ N; it is easy to show that these two maps are smooth maps between A-manifolds. Applying the body map B to the direct product M x N, it is easy to see that we have B(M x N) = (BM) x (BN). Strictly speaking this is not an equality but a canonical diffeomorphism. However, as usual we will forego such subtleties and pretend that it actually is an equality.

Chapter III. Smooth functions and A-manifolds

134

5.12 Lemma. Let L, M, and N be A-manifolds, and let f : L

-+

maps. Then the map (f,g) : L -+ M x N defined by (f,g)(x) ifand only iff = 7rM 0 (f, g) and 9 = 7rN 0 (f, g) are smooth.

= (f(x),g(x)) is smooth

M and 9 : L

-+

N be

Proof If (f, g) is smooth, f =

7rM 0 (f, g) and 9 = 7rN 0 (f, g) are smooth by [4.19]. So suppose f and 9 are smooth. Denote by X : W -+ Q a chart of L and use the notation of [5.11]. We have to show that (CPa X 'lj;b) 0 (f,g) 0 X-I: Q -+ aa X H is smooth. But ((CPa X 'lj;b) 0 (f,g) oX-1)(x) = ((CPaofoX-I)(X),('lj;aogoX-I)(X)), which is smooth by the hypothesis and [1.23-i]. IQEDI

5.13 Lemma. Iff: M

N is a smooth map between A-manifolds, then its graph Gr(f) = { (m, n) E M X N I n = f(m) } is a submanifold ofM x N. Moreover, the projection 7rM : Gr(f) -+ M is a diffeomorphism. -+

Proof Let (m, f( m)) E Gr(f) be arbitrary, let 'lj; : V -+ P c F be a chart of N containing f (m) and let ¢ : U -+ acE be a chart of M containing m such that f(U) c V. Such charts exist by continuity of f. We thus have obtained a chart ¢ x 'lj; : U x V -+ a x P of M x N. Since f is smooth, the map a x F -+ a x F, (x, y) I--> (x, y - 'lj;(f(¢-l(x)))) is a diffeomorphism. Denote by Q the image of 0 x P under this map. It follows that : U x V -+ Q, (m, n) I--> (¢(m), 'lj;(n) - 'lj;(f(m))) is a chart for M x N. Obviously (M x N n Gr(f) = Q n (E EB {O}), proving that Gr(f) is a submanifold of M x N. For the second part, note that M -+ M x N defined by J(m) = (m, f(m)) is smooth by [5.12]; by [5.8] we can see as a smooth map M -+ Gr(f). Obviously 7rM id(M) and (7rMICr(f)) id(Gr(f)). IQEDI

1:

01=

10

=

1

5.14 Lemma. Iff: M -+ Nand 9 : K -+ L are smooth maps, then so is the map f x 9 : M x K -+ N x L defined by (f x g)(m, k) = (f(m), g(k)).

Proof By [5.12] it suffices to show that 7rN 0 (f X g) and 7rL 0 (f X g) are smooth. But (7rN 0 (fxg))(m, k) = f(m), which is smooth essentially by [1.23-f]. A similar argument for the other component finishes the proof.

IQEDI

5.15 Definitions. If f : M -+ A is a continuous map, the set supp(f) eM, called the support off, is defined as the closure of the set of those points where f(m) is non-zero. In other words, m 1- supp(f) if and only ifthere exists an open set U 3 m such that flu is identically zero. A collection of subsets U eM, a E J is called locally finite if for each point m E M there exists a neighborhood of m that meets only a finite number of U 's; in formula: Q

Q

§5. Constructions of A-manifolds

135

5.16 Lemma. Let Uo c M be open and fo : Uo -+ A smooth. If supp(fo) C Uo is closed in M, then f : M -+ A defined by flu o = fo, f == 0 outside Uo, is a smooth function on M.

Proof Let U = {Ua I a E I} be an atlas of M containing Uo . Define a new atlas I a E I} by a o = Uo and aa = Ua n (M\ supp(fo)) if a =f. o. This is indeed an atlas because a subset of a chart is a chart; they cover M because supp(fo) C Uo . It follows that for a =f. 0 we have aa n supp(fo) = ¢ and hence flu == O. Smoothness of a f now follows from [4.18]. IQEDI

i1 = {aa

5.17 Lemma. Let fa : M -+ A, a E J be a collection of smooth maps such that the collection supp(fa), a E J is locally finite. Then f = EaEJ fa is a well defined smooth function on M.

Proof Let U be an atlas of M such that each element U E U meets only finitely many supp(fa)'S. It follows that flu = E~=l faJu, where the ai are the only indices with U n supp(fa,) =f. ¢. As a finite sum of smooth functions, flu is smooth. The result now follows from [4.18]. IQEDI

5.18 Definition. Let V = {Vc, I a E J} be a collection of open sets on M that cover M. A partition ofunity subordinated to the open cover V is a collection of smooth functions Pa : M -+ Ao, a E J satisfying the following conditions. (i) (ii) (iii) (iv)

sUPP(Pa) C Va> the collection {suPP(Pa)} is locally finite, I::!m EM: B(Pa(m)) E [0,1]' and EaEJ Pa = 1.

5.19 Corollary. Let V = {Va I a E J} be an open cover of M, fa E C=(Va), and {Pa I a E J} a partition of unity subordinated to V. Then f = EaEJ Pafa is a well defined smooth function on M.

Proof An elementary combination of [5.16] and [5.17].

5.20 Proposition. Let V be an open cover of an A-manifold M. partition ofunity subordinated to V.

Then there exists a

Proof We proceed in three steps. The first step is to admit the existence of partitions of unity for ordinary manifolds. The reader is referred to [Spi, Ch 2] for proofs. The second step is to prove the existence in case V is an atlas, and the third step is to prove the general case. We thus start with the second step and assume thatU = { 'Pa : Ua -+ Oa I a E I} is an atlas for M. It follows that {BUa } is an open cover of BM. By the first step there exists

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Chapter III. Smooth functions and A-manifolds

a partition of unity (J a : BM --+ [0, 1] subordinated to this cover. Let Uo be a fixed (but otherwise arbitrary) chart in the atlas. It follows thatthe map (J 00 (Brpo) -1 : BOo --+ [0, 1] is a smooth map. Applying G we obtain a smooth map G((J 0 0 (Brpo)-1) : O;:,·o.d --+ Ao. We can interpret this function as the first term in the expansion (2.24) of a smooth function on Oa, the other terms being identically zero. In other words, we obtain a smooth function Po = G((JOO (Brpo)-1) orpo : Uo --+ Ao, a function independent of the odd coordinates in 0 0 • It follows directly from the construction that we have B 0 Po = (J 0 0 B. Moreover, the definition of G by means of derivatives shows that Bm 1- supp((Jo) implies m 1- supp(Po), i.e., supp(Po) c B- 1(supp((Jo)). A priori the set supp(Po) is closed in Uo but not necessarily in M. However, since supp((Jo) is closed in BM, the above inclusion shows that supp(Po) c Uo is indeed closed in M. By [5.16] we may extend Po to the whole of M by zero outside of Uo without losing smoothness. We thus have created smooth functions Pa : M --+ Ao associated to the cover (atlas) U which manifestly satisfy conditions (i) and (iii) of a partition of unity. It also satisfies condition (ii) because sUPP(Pa) C B- 1(suPP((Ja)) and because the (Ja are a partition of unity. We deduce by [5.17] that P = EaEI Pa : M --+ Ao is a smooth function, which obviously satisfies Bop == 1. (Nota Bene. p itself need not be identically 1.) Since inversion is smooth, the functions pa = Pal P are smooth. The collection of these maps is a partition of unity subordinated to the cover (atlas) U. Note that, just as the Pa, the Pa satisfy the relation B 0 Pa = (Ja 0 B, i.e., BPa = (Ja. To prove the third step, let V = {Va I a E J} be an arbitrary open cover of M and let U = {Ua I a E I} be an atlas. It follows that { Uaa = Ua n Va I a E I, a E J} is an atlas. Hence by the previous construction there exists a partition of unity Paa subordinated to this atlas. We thus can define the smooth functions Pa = EaEI Paa. Since the sUPP(Paa) are locally finite, the inclusion SUPP(Pa) C UaEI sUPP(Paa) follows. From this we deduce that the sUPP(Pa) are locally finite, thus proving property (ii) of a partition of unity. The other three properties being obvious, we have shown the existence of a partition of unity subordinated to the (arbitrary) open cover V. IQEDI

5.21 Lemma. Let m be a point in an open set U of an A-manifold M. Then there exists a P E C= (M)o and an open neighborhood V C U of m such that supp(p) C U and such that plv == 1. We will call such ajunction a plateau function around m in U. Proof Let rpo : Uo --+ 0 0 be a chart of M such that m E Uo C U. Since the topology on M is induced by the topology ofBM [4.12], and since any RP is normal, there exists an open subset V of M such that m EVe V c Uo c U, where overlining here means the closure. Finally let {p, T} be a partition of unity subordinated to the open cover

{U, M \ V}.

IQEDI

5.22 Definition. In §2 we have seen that, under suitable circumstances, the body map B admits an "inverse" G. We will now show that under similar conditions the same is true for A-manifolds. To that end we need the notion of an A-manifold without odd dimensions,

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§S. Constructions of A-manifolds

which is an A-manifold modeled on an A-vector space without odd dimensions. It should be obvious that M is an A-manifold without odd dimensions if and only if M = Mw.o.d.

5.23 Construction. Let M be an R-manifold modeled on an R-vector space X. Since the odd part X I does not play any role in the construction of M, we may assume that X is an R-vector space without odd dimensions, i.e., Xl = {O}. Denoting by E the A-vector space without odd dimensions E = GX = G(Xo EB {Oh), we will construct an A-manifold without odd dimensions GM modeled on E such that B(GM) = M; moreover, for any A-manifold without odd dimensions N we will have G(BN) = N. Let U = {'Pa : Ua -+ Oa I a E I} an atlas for M. We then consider the subsets Oab = 'Pa(Ua nUb) C Oa and the maps 'Pba = 'Pb 0 'P;;-l, to which we apply the map G (see §2) to obtain open sets GOa in Eo, open subsets GO ab C GOa, and smooili maps G'Pba. These new data satisfy exactly the conditions of [4.9] and thus define a proto A-manifold GM. Ifwe apply the body map B to the data GO ab C GOa and G'Pba, we recover exactly the data needed to reconstruct M, showing that B(GM) = M. Since M is supposed to be second countable and Hausdorff, GM is an A-manifold. By the same token we see that for an A-manifold without odd dimensions N we have G(BN) = N. However, if N is not "without odd dimensions," we still can construct G(BN), which is an A-manifold without odd dimensions. In the A-manifold G(BN) we recover the even coordinates of N, but not the odd coordinates. We leave it to the reader to convince himself that G(BN) "is" the submanifold Nw.o.d defined in [5.9].

5.24 Lemma. Let M and N be A-manifolds without odd dimensions and f : BM -+ BN a smooth map. Then there exists a smooth map Gf : IVf -+ N such that the following diagram is commutative:

M

Gf

~

Bl BM

N

lB ~

BN.

f

Proof Let U = {'Pa : Ua -+ Oa I a E I} and V = {'l/Jb : Vb -+ Pb I bE J} be atlases for M and N. By taking intersections with f-I(BVb) if necessary, we may assume that for each Ua E U there exists a Vb E V such that f(BUa ) C BVb. We then define maps

Gfa: Ua

-+

Nby

where the G on the right hand side is defined in [2.11]. In order to apply G, it is essential that M is without odd dimensions. By definition of smooth maps, the G fa are smooth maps on Ua .

Chapter III. Smooth functions and A-manifolds

138

Let us show that the Gfa coincide on intersections. For m E Ua n Ua, we have by definition ofB on A-manifolds [4.11] that B(Gfa(m)) = f(Bm) = B(Gfa,(m)). Hence, if fa(BUa) C BVb it follows that Gfa(m) E Vb, but also Gfa' (m) E B- 1(BVb) = Vb' We thus calculate on Ua n Ua , :

Gfa' = 'l/J;;1 0 'l/Jb 0 'l/Ji/ 0 G((B'l/Jb') 0 f 0 (B'Pa' )-1) 0 'Pa' 0 'P~l 0 'Pa = 'l/J;;l 0 G(B'l/Jb 0 B'l/Ji/ 1) 0 G(B'l/Jb' 0 f 0 B'P;:}) 0 G(B'Pa' 0 B'P~l) 0 'Pa

= 'l/J;;l G(B'l/Jb f 0

0

0

B'P~l) 0 'Pa

=Gfa. In going from the first line to the second we have used the definition of the maps B'P and B'l/J and that Nand M are without odd dimensions. We conclude that the Gfa define a global function Gf : M -+ N. [4.18] then tells us that Gf is a smooth map. It follows immediately from the equality B(Gfa(m)) = f(Bm) that B 0 Gfa = flBU a 0 B, proving commutativity of our diagram. IQEDI

5.25 Proposition. Let M be an A-manifold without odd dimensions and f : BM -+ BN a smooth map. Then there exists a unique smooth map Gf : M -+ N such that the following diagram is commutative:

M~

N

BM -----. BN . f

Proof Consider the canonical injection i : GBN ~ Nw.o.d -+ N. Applying the body map gives us the identity on BN, i.e., Bi = id(BN). According to [5.24] there exists a smooth map Gtf : M -+ GBN such that B 0 Gtf = foB. It follows immediately that Gf = ioGtf: M -+ N satisfies our requirements. To prove uniqueness, notefirstthatuniqueness in [4.22] forces B(Gf) = f. So suppose g, h : M -+ N are two solutions, i.e., Bg = Bh = f, and let m E M be arbitrary. Let 'P : U -+ 0 be a chart for M containing m, and let 'l/J : V -+ P be a chart for N containing g(m). Since B(g(m)) = (Bg)(Bm) = f(Bm) = (Bh)(Bm) = B(h(m)) E BV, it follows that h(m) E V = B-l(BV). Using the definition of B on A-manifolds and smooth maps, we then compute BB( 'l/J 0 9 0 'P- 1) = B'l/J 0 f 0 B'P- 1 = B( 'l/J 0 h 0 'P- 1) as maps from BO to BP. Since 0 is an open subset in the even part of an A-vector space without odd dimensions, it follows from [2.16] that 'l/J 0 9 0 'P- 1 = 'l/J 0 h 0 'P- 1, and in particular that g( m) = h( m). Since m was arbitrary, we conclude 9 = h. IQEDI

5.26 Corollary. If M is an A-manifold without odd dimensions, then the body map B : C=(M; N) -+ C=(BM; BN) is bijective.

§S. Constructions of A-manifolds

139

5.27 Remark. Looking at the commutative diagram at the end of [5.9], we now can identify flMwod as the mapGBf = G(fIBM)'

5.28 Discussion. In previous sections we have defined the body map B on subsets of A-vector spaces and on smooth maps. The image was the corresponding R-object. In this section we have extended the notion of the body map B to A-manifolds and smooth maps between them. Obviously the extended definition of B coincides with the old one in case the old and new apply, i.e., when an A-manifold isjust an open subset of the even part of an A-vector space. Moreover, the body map on A-manifolds preserves the notions of product and submanifold as defined in §4. More precisely, if N is a submanifold of M, then BN is a sub manifold of the R-manifold BM; if M and N are A-manifolds, then B(M x N) = (BM) x (BN) as R-manifold.

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Chapter IV

Bundles

In the next chapter we will, among others, introduce the tangent bundle ofan A-manifold M as well as differential forms and their exterior derivatives. These definitions require some knowledge about vector bundles and about the relation between operations on vector bundles and the corresponding operations on the modules oftheir sections. The purpose ofthis chapter is to provide these notions and to put them in a wider context. Let us be more precise (with some liberties). A vector bundle is a natural generalization of a vector space in the sense that it is literally a bunch (bundle) of vector spaces. The set which indexes the various vector spaces (the fibers) is called the base manifold. The various operations one can perform to create new vector spaces out of old ones, such as the direct sum, tensor product, etcetera, can also be performed on vector bundles by doing it in each fiber separately. On the other hand, one canform the set of sections of a vector bundle, meaning the set of functions which assign a point in each of the vector spaces of the bundle. This set is in a natural way a module over the set of functions on the base manifold. It follows that we can perform the operations of direct sum, tensor product, etcetera, also on these modules. In this context there is one important question for differential geometry: how to interpret a k-form. On the one hand a k-form is defined as an element of sections of Homtk( (tangent bundle)k, A) , while on the other hand, especially for the definition of the exterior derivative, it is seen as an element of

Hom tk ( (sections of the tangent bundle) k , (functions on the base manifold) ) . In the first point of view we first apply the construction E ---+ * (I\k E) to the tangent bundle and then we take sections, whereas in the secondpoint ofview we first take sections and then apply this construction to the resulting module. The obvious question is whether 141

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these two points ofview coincide; the answer is positive. Once this question is stated this way, it is natural to ask whether a similar result holdsfor the other constructions. It turns out that for direct sums and quotients the results are almost trivially true. For spaces of homomorphisms it requires some work to prove that a homomorphism of sections depends only pointwise on the values ofthe initial sections. For tensor products and exterior powers though, the proof is much harder. For tensor products and exterior powers we need to know a part ofthe Serre-Swann theorem which states that the space of sections ofa vector bundle is a finitely generated and projective module over the ring of functions on the base manifold. In order to prove this, we need the notion ofa metric on an A-manifold in order to define an orthogonal supplement to a subbundle. For A-vector spaces the notion of a metric is subtly different from what one would expect, and the proof of the existence of an orthogonal complement uses the specific properties of graded smooth functions. The last section of this chapter treats Batchelor's theorem. This theorem could have been stated and proved already at the end ofchapter III. The reason it is included in this chapter is that the interpretation of this theorem requires the notion of a vector bundle, albeit in the classical sense of R-manifolds.

1.

FIBER BUNDLES

In this section we introduce locally trivial fiber bundles with structure group G and typical fiber F over an A-manifold M as well as their morphisms and sections. We show that such a bundle is characterized by its transition functions with respect to a trivializing atlas.

1.1 Definitions . • An A-Lie group is an A-manifold G that admits at the same time the structure of a group for which the multiplication is smooth. More precisely, denoting by m : G x G ---+ G the multiplication in G, m should be a smooth map from the A-manifold G x G to G. As is customary, we will usually denote the multiplication by a dot or just nothing, i.e., m(g, h) = 9 . h = gh. It will be shown in [VI.1.6] that in an A-Lie group the map Inv : G ---+ G, 9 I--> g-1 is automatically smooth . • Let G and H be A-Lie groups. A map p : G ---+ H is called a morphism of A-Lie groups if it is smooth and a homomorphism of (abstract) groups . • Let M be an A-manifold and G an A-Lie group. A smooth (left) action ofG on M is a smooth map : G x M ---+ M satisfying (i) Vm EM: (ide,m) = m(where ide denotes the identity element in G), (ii) Vg, hE G Vm EM: (g . h, m) = (g, (h, m)). For a fixed 9 E G we will denote the map m I--> (g, m) by g. If no confusion is possible, we will denote the image (g, m) in the following shortened ways: as g(m), as g(m), orjust as gm.

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• In a similar way we define a smooth right action by switching the arguments of . More precisely, it is a smooth map : M x G -+ M such that (m, ide) = m and (m, 9 . h) = ( (m, g), h). As for left actions, we denote the map m I--> (m, g) by 9' but evaluation will be denoted on the left, i.e., (m, g) = {m I 9». Again as forleft actions, if no confusion is possible, we will use the shortened form (m, g) = mg. • An action of G on M is said to be effective if 9 = id( M) implies that 9 = ide E G. • An action of G on M is pseudo effective if for all A-manifolds N and all smooth maps 'lj;: N -+ G we have: [Vn EN: ,p(n) = id(M)] ===} [Vn EN: 'lj;(n) = id(G)].

1.2 Nota Bene. If is a smooth (left or right) action of an A-Lie group G on an Amanifold M, it is in particular continuous. It follows that all maps 9 : M -+ M are homeomorphisms. If 9 E G has real coordinates, 9 is a diffeomorphism, but if 9 does not have real coordinates, there is no reason to think that 9 is smooth. See [VI.1.3] for more details.

1.3 DiscussionlExample. Let G = Aut(E) be the group of automorphisms of a finite dimensional A-vector space E. On this group we have defined two different group structures: the usual composition 0 and the opposite composition o. The usual composition 0 corresponds to the point of view that elements of A ut(E) are right linear, i.e., Aut(E) C EndR(E). Moreover, with the composition 0 , the action of Aut(E) on E is a left action. On the other hand, the opposite composition 0 corresponds to the point of view that elements of Aut(E) are (also) left linear, i.e., Aut(E) C EndL(E) (for left linear morphisms we have argued repeatedly that its natural composition law is the opposite 0 defined by ¢o'lj; = 'lj; 0 ¢). And when equipped with 0, the action of Aut(E) on E becomes a right action. Indeed, seen as left linear maps we have {v I ¢ 0 'lj;)) = (( (( v I ¢ )) II 'lj;». It follows that the choice for one of these group structures on Aut(E) depends upon whether we want to see it as having a right or left action on E. Since traditionally one prefers left actions over right actions, we will adhere to this choice. This means that we will use the standard composition law 0 and that we will view Aut (E) as being a subset of the right linear endomorphisms EndR(E). But we can say more about Aut(E). It has a canonical structure of an A-manifold, being an open subset of the even part of the A-vector space EndR(E) [111.2.26]. Taking a basis for E it is immediate from the matrix representation of Aut(E) that the multiplication in Aut(E) is smooth, proving that it is an A-Lie group. With a similar argument one can show that the evaluation : Aut(E) x E -+ E, (¢, e) I--> ¢(e) is an effective smooth (left) action of Aut(E) on E = Eg.

1.4 Remark. It is immediate that an effective action is pseudo effective. Moreover, in the context of R-manifolds the converse is also true: if 9 = id( M), we can take the constant map 'lj;( n) = 9 to obtain that 9 = id( G). However, in the context of A-manifolds this argument no longer holds: not any constant map is smooth!

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1.5 Example. Let E be a finite dimensional A-vector space and Ea one of its homogeneous parts. Since Aut(E) consists of even morphisms, its action on E preserves Ea. We claim that the left action of Aut(E) on Ea is pseudo effective. The case of Eo is a direct consequence of [111.2.31] and the fact that we required A to satisfy (C [(Xll) (in order to be assured that smooth functions have partial derivatives). If'l/J : N -+ Aut(E) is a smooth map such that all 'l/J( n) act as the identity on Eo, all 'l/J( n) - id( E) act as the zero map on Eo. Hence by [111.2.31] they are the zero maps, i.e., all 'l/J(n) = id(E). For E 1 , it suffices to transform this into the previous case by noting that E1 = Egl (see [III. 1.26]) and that the map id: Aut(E) -+ Aut(E0 1) which sends a map to the same map (E and Eol are the same as sets) is smooth. Thus if'l/J : N -+ Aut(E) is a smooth map such that all 'l/J(n) act as the identity on E 1, the smooth map tjj = ido 'l/J : N -+ Aut(E01) is such that all 7jj( n) act as the identity on Eg1. Hence they are all the identity.

1.6 Discussion. The reason we introduce the notion of pseudo effective actions is that it is difficult to prove (if possible at all) that a given action is effective. The problem is that in order to prove effectiveness, one has to disentangle the coordinates of the A-Lie group from those ofthe A-manifold it is acting on. Since A usually contains zero divisors, separating these coordinates is hard. On the other hand, for pseudo effective actions we do not have separate elements that act as the identity, but smooth families of elements acting as the identity. Separating the coordinates of the A-Lie group from those of the A-manifold then is provided by [III.2.21].t

1.7 Definition. Let E, M, and F be A-manifolds, let G be an A-Lie group with a smooth and pseudo effective left action on F, and let p : B -+ M be a smooth surjective map. A local trivializing F-chart (often shortened to a trivializing chart) for p is a chart U for the A-manifold M together with a diffeomorphism 'l/J : p-1 (U) -+ U x F such that ll'U o'l/J = p, where ll'U is the canonical projection on U. To any pair of local trivializing F-charts (Ua, 'l/Ja) and (Ub, 'l/Jb) is associated a map -J;ba : Ua n Ub x F -+ F, defined by the equation ('l/Jb 0 'l/J;; 1 ) (m, f) = (m, -J;ba (m, f)). For a fixed m E Ua nUb, the map -J;ba (m, _) is a map from F to F admitting an inverse -J;ab(m, J, i.e., the -J;ba(m, J are bijections. Two local trivializing F-charts (Ua, 'l/Ja) and (Ub, 'l/Jb) are said to be G-compatible (or compatible if no confusion is possible) if there exist smooth maps 'l/Jba, 'l/Jab : Ua n Ub -+ G such that mE

Ua n Ub

:::}

-J;ba(m, f) = 'l/Jba(m)(f)

&

-J;ab(m, f) = 'l/Jab(m)(f) .

Pseudo effectiveness of the G-action on F implies that the maps 'l/Jba and 'l/Jab, if they exist, are unique. These maps are called the transition junctions between the trivializing tIf A is the exterior algebra of a vector space X over R, the fact that we assume that A satisfies (C (001) implies that X has to be infinite dimensional. In that case it is not hard to show that the action of Aut(E) on E", [1.5] is actually effective (which it is not if X isfinite dimensional; see [II.5.S] for the essential idea for a counter example). I conjecture that this is general: if A is the exterior algebra of an infinite dimensional vector space X over R then any pseudo effective action is actually effective.

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charts Ua and Ub ofthe bundle B, or simply transition functions if the rest is clear from the context. Afiber bundle structure on B with typical fiber F and structure group G is a collection oflocal trivializing F-charts S = { (Ua, 'l/Ja) I a E I} satisfying three conditions: (FBI) UaE1Ua = M, (FB2) any two charts in S are G-compatible, and (FB3) if (U, 'l/J) is a local trivializing F-chart that is G-compatible with all elements of S, then (U, 'l/J) E S. A locally trivial fiber bundle over M with typical fiber F and structure group G (or in short, a fiber bundle over M) is a smooth surjective map p : B ---+ M together with a fiber bundle structure on B with typical fiber F and structure group G. A trivializing atlas for a fiber bundle over M is a subset U c S that satisfies properties (FBI) and (FB2), but not necessarily (FB3). For any point m E M the set p-l (m) C B is called the fiber over m; it is usually denoted by p-l(m) == Bm == Blm- Since every point m lies in a local trivializing F-chart (U, 'l/J), we have a homeomorphism'l/J : Bm ---+ F, explaining the name typical fiber. If m has real coordinates, this homeomorphism is a diffeomorphism.

1.8 Remarks . • For two local trivializing F-charts to be compatible, we required the existence of smooth transition functions 'l/Jba and 'l/Jab. However, existence of one of them implies the existence of the other. Note first that for a fixed m E Ua n Ub the map -J;ba(m, J is a bijection on F with inverse -J;ab(m, J. Existence of'l/Jba implies that -J;ba(m, J = ('l/Jba(m), J. Since ('l/Jba(m)-l, J is its inverse, it follows that the smooth map 'l/Jab : Uan Ub ---+ G defined by 'l/Jab (m) = 'l/Jba (m) -1 satisfies the requirement (remember that inversion in an A-Lie group is smooth) . • In the definition of a trivializing chart we required U c M to be a chart for M, i.e., a coordinate chart for the smooth structure on M. One usually only requires U to be open. The restriction to charts is not really a restriction and avoids awkward phrasing later on . • We have defined fiber bundles using structure groups. In this we have followed [Hi] and [Ste]. Other authors, such as [OHV] and [Hu], do not use structure groups.

1.9 Nota Bene. Note that, due to (FBI), an atlas U = {(Ua, 'l/Ja) I a E I} of trivializing charts for a bundle p : B ---+ M is at the same time an atlas for the smooth structure for M. This justifies to a certain extent the abuse of notation for the word atlas.

1.10 Lemma. Given a smooth surjective map p : B ---+ M, any collection oftrivia Iizing F-charts U = {(Ua, 'l/Ja) I a E I} satisfying (FBI) and (FB2) defines a unique fiber bundle structure on B with typical fiber F and structure group G such that U is a trivializing atlas.

Proof Define S as the collection of all local trivializing F-charts that are G-compatible with all elements in U. The only not completely trivial part of the proof is to show property

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146

(FB2). So let (Ui , 'l/Ji), i

mEU1nU2nUa

= 1,2 be two charts in S, for which we compute

===}

{

(m, ~21(m, f)) = ('l/J2 0 'l/Jl1)(m, f) = ('l/J2 0 'l/J;;lo'l/Ja o'l/Jl 1)(m,f)

= (m, 'l/J2a(m)('l/Jal(m)(f))) .

From this we deduce that 'l/J21 (m) = 'l/J2a(m) ''l/Jal(m) exists and is smooth on U1 nU2nUa. Pseudo effectiveness of the action shows that it is the unique solution on U1 n U2 n Ua . Since the Ua cover (by (FBI)) the manifold M, we can glue them together to a solution on U1 n U2. It then follows from [111.4.18] that (U1, 'l/Jl) and (U2, 'l/J2) are G-compatible.

IQEDI

1.11 Remark. The fact that the structure group G acts pseudo effectively on the typical fiber F is used in the proof of [1.10]. In §V1.6 we will show that if we have a smooth left action of an A-Lie group G on an A-manifold F, then there exists a (proper) normal A-Lie subgroup G1j;o c G acting as the identity on F such that G /G1j;o is an A-Lie group and such that the induced action on F is pseudo effective. This means that the condition that G acts pseudo effectively on F is not restrictive at all: any action can be transformed in a pseudo effective action.

1.12 Terminology. Just as for A-manifolds, we will never mention a fiber bundle structure explicitly, at the most we will specify a trivializing atlas for it. Andjust as for manifolds, a trivializing chart will always mean an element in the fiber bundle structure.

1.13 DiscussionlDefinition. Let p : B ---+ M be a fiber bundle with typical fiber F and let N be a submanifold of M. We then can consider the subset BIN = p-l(N) c B and the restriction PN : BIN ---+ N of the map P to this subset. We claim that BIN is a submanifold of E and that PN : BIN ---+ N is a fiber bundle with typical fiber F (and the same structure group as E). This bundle is called the restriction ofE to the submanifold N. An important class of examples of this kind are sub manifolds N that are open subsets inM. To prove our claim, let ¢ : U ---+ 0 be a chart as in [111.5.1] with ¢(U n N) = 0 n E, where E is a graded subspace of E. By taking intersections with trivializing charts for E, we may assume that there exists a diffeomorphism 'l/J : p- 1(U) ---+ U x F in the fiber bundle structure. Now let X : V ---+ P be any chart for F, then (¢ x X) o'l/J : 'l/J-l (U x V) ---+ 0 x P is a chart for E with ((¢ x X) 0 'l/J) ('l/J-l (U x V) n BIN) = (0 n E) x P. This proves that BIN is a submanifold of B. Finally, the charts Un N for N together with the maps 'l/Jlp-l (UnN) form a trivializing atlas for the fiber bundle PN : BIN ---+ N; the details are left to the reader.

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1.14 Definition. Let p : B -+ M and q : C -+ N be two fiber bundles. A smooth map h : B -+ C is a fiber bundle map if there exists a smooth map 9 : M -+ N such that the following diagram is commutative:

M---+N. 9

Since p is sUljective, such a map g, if it exists, is unique. If the base spaces are the same, we will as a rule require that 9 is the identity on M.

1.15 Remarks. • If a smooth map h : B -+ C induces a set theoretical map 9 : M -+ N making the diagram commutative, then 9 must be smooth. This can be seen by the following argument. Let (U, 'lj;) be a trivializing chart for p : B -+ 111 and choose a point f E BF c F. Commutativity of the diagram implies that glu is given by m I--> q(h('lj;-l(m, I))). Since f has real coordinates, this is smooth. • Some authors use the name strong bundle map for those fiber bundle maps between fiber bundles with the same base space that induce the identity on the base space. Since we will hardly ever use fiber bundle maps over the same base space that do not induce the identity on the base space, we do not introduce this distinction, but will specify it explicitly when needed.

1.16 Definitions .• An isomorphism offiber bundles is diffeomorphism h : B -+ C of A-manifolds between two fiber bundles p : B -+ M and q : C -+ M over the same base space that is at the same time a fiber bundle map (which by our convention induces the identity on M). Two fiber bundles Band C over the same base space are said to be isomorphic if there exists a fiber bundle isomorphism h : B -+ C. • Let M and F be A-manifolds, let B = M x F be the direct product and consider the projection p = 7r M : B = M x F -+ M. For any chart U of M we define the local trivializing F-chart (U, 'lj;) for p by 'lj; = id : p-l (U) -+ U x F. Taking G = {idp}, the set of all such local trivializing F-charts (U, 'lj;) are G-compatible and thus form a trivializing atlas for B. This turns p : B -+ M into a fiber bundle over M with typical fiber F and structure group G = {idp}, called the trivial bundle over M with fiber F. • A fiber bundle p : B -+ M with typical fiber F is said to be trivial if B is isomorphic to the trivial bundle M x F.

1.17 Nota Bene. Let p : B -+ M be a fiber bundle with typical fiber F and structure group G. Suppose furthermore that h : B -+ M x F is an isomorphism of fiber bundles showing that B is trivial. IfU eM is any chart of M, then hlp-l(U) : p-l(U) -+ U x F is a trivializing F-chart for B. However, it is by no means guaranteed that this trivializing

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chart is G-compatible with the trivializing charts of B that define the bundle structure of B. A rather trivial (counter) example is the following. Let B = M x F be the trivial bundle and let H, H =I {id}, be an A-Lie group acting pseudo effectively on F. Let furthermore cp : M ---+ H be a smooth map such that Vm EM: cp( m) =I id. Then the map h : B ---+ M x F, h( m, f) = (m, cp( m) . f) is a fiber bundle isomorphism. But no restriction hlp-l (U) : p-l (U) ---+ U x F is G-compatible with the fiber bundle structure on B = M x F, simply because G = {id} is too small. In [Y.3.l7] a less trivial counter example to this phenomenon can be found. There G ~ Z2 but it is still too small for the actual global trivialization to be G-compatible.

1.18 Discussion. We now can easily explain the name locally trivial fiber bundle. A local trivializing chart (U, 'l/J) for the fiber bundle B is a chart U of M such that the restriction Blu of B to U is trivial, the fiber bundle isomorphism being given by 'l/J. It follows that a locally trivial fiber bundle is just that: every point admits a neighborhood (chart) U such that the restriction of the bundle to that chart is trivial.

1.19 Definition. Let p : B ---+ M be a fiber bundle and N a submanifold of M. A local (smooth) section ofB over N is a (smooth) map S : N ---+ B such that po S = id(N). We will denote the set of all smooth sections of B over N by r N (B); in case N = M we rM(B). From the definition of the restriction of B omit the subscript and write r(B) to N it follows immediately that we can write rN(B) = r(BIN). The reader should be aware that there is no guarantee that the set r N (B) is not empty.

=

1.20 Discussion. Let p : B ---+ M be a fiber bundle with typical fiber F and suppose that U c M is an open set such that Blu is trivial. This implies the existence of a diffeomorphism'l/J: Blu = p-l(U) ---+ U x F such that pu = 1fU 0 'l/J. Associated to any local section S : U ---+ B we then have a map s1j; : U ---+ F defined by s1j; = 1fF 0 'l/J 0 s, i.e.,

('l/J 0 s) (m)

= (m, s1j; (m))

.

Since 'l/J is a diffeomorphism, the map S f---4 s1j; is a bijection between local sections ---+ B and functions s1j; : U ---+ F. Moreover, S is smooth if and only if s1j; is smooth, i.e., if B is trivial over U, then ru(B) ~ C=(U; F). Changing the context slightly, let S : M ---+ B be a section and U = {(Ua, 'l/Ja)} a s1j;a : Ua ---+ F as above. It trivializing atlas for B. We thus obtain local functions Sa is an immediate consequence of [111.4.18] that S is smooth if and only if all functions Sa are smooth. Moreover, the local functions Sa satisfy the equations

S: U

=

(1.21) On the other hand, if we have a set of local functions Sa : Ua ---+ F satisfying these equations, they define a global section S : M ---+ B by the equation s(m) = 'l/J;;l(m, sa(m));

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(1.21) guarantees that s is well defined on intersections Ua n Ub • We conclude that sections s : M -+ B are in 1-1 correspondence with sets oflocal functions Sa : Ua -+ F associated to the trivializing atlas U and satisfying (1.21). Moreover, s is smooth (i.e., s E r( B)) if and only if all Sa are smooth.

1.22 Corollary. Let p : B -+ M be a fiber bundle with typical fiber F. If B is trivial, then any fiber bundle isomorphism h : B -+ M x F induces a bijection f( B) -+ Coo (M; F) given by S I--> 1l"F 0 h 0 s.

1.23 Lemma. Let p : B -+ M be a fiber bundle, U an open set in M and S E fu(B) a local smooth section. Then the image s(U) is a submanifold ofB and p : s(U) -+ U is a diffeomorphism

Proof Since being a submanifold is a local property, we may assume without loss of generality that U is a trivializing chart. If'l/J: 1l"-l(U) -+ U x F is a trivializing diffeomorphism, it transforms the image s(U) into the graph Gr(s1j;). The result then follows from [111.5.13]. IQEDI

1.24 Construction. Let p : B -+ M be a bundle with typical fiber F and structure group G, and let U be a trivializing atlas. It is easy to show, using the pseudo effectiveness of the action, that the collection of maps 'l/Jba has the following properties:

(1.25)

'l/Jaa (m) = ide the identity element in G , Ua nUb n Uc :::} 'l/Jcb(m)· 'l/Jba(m) = 'l/Jca(m) .

Vm E Ua mE

What we will do is show that the collection {Ua, 'l/Jba} completely determines the fiber bundle B. To make this more precise, let M and F be A-manifolds and let G be an A-Lie group with a smooth action on F. Suppose also that we have an atlas U = { 'Pa : Ua -+ Ga } of M and smooth maps 'l/Jba : Ua n Ub -+ G satisfying (1.25). With these data, we claim, we can (re)construct a bundle p : B -+ M with typical fiber F and structure group G such that the maps 'l/Jba appear as described above. To define the A-manifold B and the surjection p : B -+ M we will use [111.4.9]. The ingredients for this construction are the open sets {ja = Ga xF, the subsets {jab = Gab xF and the transition functions rpba defined by rpba(X,f) = ('Pba(X),'l/Jba('P;;l(x))(f)). To realize that these ingredients satisfy (111.4.10), remember first that Gab = 'Pa(Ua nUb) and that 'Pba = 'Pb 0 'P;;l. It follows immediately that the maps rpba are smooth; that they also satisfy (111.4.10) follows from (1.25). According to [111.4.9] we thus may conclude that B is a well defined A-manifold. The projection p : B -+ M we define in local charts by p : (x, f) E {ja = Ga x F I--> X E Ga. It should be obvious from the definition of the maps rpba that this is a well defined, smooth sUljective map as required.

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It thus remains to exhibit the existence of an atlas oftrivializing charts satisfying (FB 1) and (FB2). We claim that the atlas {(Ua, 'l/Ja) I a E I} with 'l/Ja = (cp;;-l X id(F)) 0 i{5a will do. But this should be obvious from the commutative diagram

p-l(Ua) = Ua

pi Ua

-----'Pa

Oa = Oa

X

F

'1';;1 xid(F) )

1

1

KU a

KOa

-----'Pa

Oa

Ua x F

-1 'Pa

------

Ua

and the definition of the transition functions i{5ba for the A-manifold B (and recall that the charts CPa and i{5a are recovered/defined as in [111.4.9]). We thus have shown that the data {Ua, 'l/Jba} determine a bundle p : B -+ M with typical fiber F and structure group G. To show that this bundle is "unique," we proceed as follows. Suppose p : B -+ M and p: -+ M are two bundles with typical fiber F and structure group G. Suppose furthermore that {(Ua, 'l/Ja)} is a trivializing atlas for B and that {(Ua, -J;a)} is one for i.e., the same charts on M but different trivializing maps (by taking intersections we can always accomplish this). Ifwe suppose that the transition functions'l/Jba and -J;ba are the same (i.e., that 'l/Jb 0 'l/J;;l = -J;b 0 -J;;;l as diffeomorphisms of (Ua nUb) x F), then we can define an isomorphism of fiber bundles h : B -+ by hl p -1(Ua ) = -J;;;l 0 'l/Ja. The fact that the transit~on functions are the same guarantees that

ii

ii,

ii

this is well defined. We conclude that Band B are isomorphic fiber bundles. And thus we have proven that the data {Ua, 'l/Jba} determine the bundle p : B -+ M uniquely up to isomorphisms.

1.26 Remark. Attentive readers will object that the above construction is faulty because we cannot apply [111.4.9]. They are right because F is not an open set in the even part of some A-vector space. However, taking an atlas for the A-manifold F and replacing the open sets Oa by direct products of a chart Oa and a chart for F will make the construction a valid one. The details are left to the reader.

1.27 DiscussionlDefinition. What we have shown in the above construction is that giving a bundle B over an A-manifold with typical fiber F and structure group G is completely equivalent to giving an atlas U and transition functions 'l/Jba : Ua n Ub -+ G satisfying (1.25). It thus follows that such a set oftransition functions defines the bundle p : B -+ M. It is this way that we will define and/or construct most of our bundles.

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§2. Constructions offiber bundles

2. CONSTRUCTIONS OF FIBER BUNDLES In this section we discuss various constructions of new fiber bundles out of old ones: associated bundles, pull-back bundles, and product bundles. We also define the notion of a principal fiber bundle and we show that all fiber bundles can be seen as associated to a principal fiber bundle.

2.1 Construction (associated bundles). Let p : B -+ M be a bundle with typical fiber F and structure group G. Let furthermore H be another A-Lie group with a pseudo effective smooth action on an A-manifold E. Now suppose we have an A-Lie group morphism p : G -+ H. Associated to p and B we will construct a new bundle over M with typical fiber E and structure group H, which we will denote as pP : BP,E -+ M, and which is called an associated bundle, associated to B by the "representation" p. To that end, let U = {(Ua,'l/Ja) I a E J} be a trivializing atlas for B with the associated set of transition functions'l/Jba' It is easy to verify that the functions po 'l/Jba satisfy the requirements of [1.24]. We thus obtain a bundle pP : BP,E -+ M with typical fiber E and structure group H defined by these po 'l/Jba. Moreover, the construction is such that we automatically have a trivializing atlas U' = {(Ua , 'l/J~) I a E I} for the new bundle BP,E with the same trivializing charts Ua as U and whose associated transition functions are the po 'l/Jba. (Note that this implies that for each trivializing chart (U, 'l/J) for B we have a corresponding trivializing chart (U, 'l/J') for BP,E, just by adding (U, 'l/J) to U.) There are two ways to assure that the result does not depend upon the choice of a trivializing atlas. The first is to take the whole fiber bundle structure as trivializing atlas (the biggest possible). The second is to show that a different choice oftrivializing atlas leads to an isomorphic fiber bundle. The details of this are left to the reader.

2.2 Construction (pull-back bundle). Let q : 0 -+ N be a fiber bundle with typical fiber -+ N be a smooth map. We will construct a fiber bundle p == g*q : B == g*O -+ M with typical fiber F and structure group G, as well as a fiber bundle map !J : g* 0 -+ 0 inducing g. We thus will have a commutative diagram

F and structure group G and let 9 : M

B == g*O

9

------+

p=g'q 1 M

0

lq ------+

N.

9

Moreover, the construction will be such that the restriction!J : p-l(m) -+ q-l(g(m)) is a diffeomorphism. This fiber bundle will be called the pull-back of Cover g. Let us start with the abstract set-theoretic definition. Consider the direct product M x C and the subset g*O = {(m, c) EM x 0 I g(m) = q(c)}. We claim that the surjection p == g* q = 7r M : g* 0 c M x 0 -+ M is the fiber bundle we are looking for and that the map iJ = 7rG : g*O c M x 0 -+ 0 is the bundle map. What remains to be shown is that

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Chapter IV. Bundles

g*C is a fiber bundle over M in the sense of A-manifolds and that all maps involved are smooth. Let V = {(Va, Xa) I a E J} be a trivializing atlas for C and letU = {Ua I a E J} be an atlas for M such that for each Ua E U there exists Vg(a) E V such that g(Ua ) C Vg(a) (we use the same symbol 9 to denote the map on indices). We now define the map 'l/Ja : p-l(Ua ) C M x C ~ Ua x F by 'l/Ja: (m,c)

f---4

(m,7rp(X g(a)(c))) ,

where Xg(a) : q-l(Vg(a)) ~ Vg(a) x Fis the local trivialization. It is elementary to check that 'l/Ja is a bijection. Moreover, one also can check that we have

where Xa{3 denotes the transition function with respect to the atlas V. In other words, B == g*C is a fiber bundle with transition functions 'l/Jba = Xg(b)g(a) 0 9 (use [1.24]). Finally, there are two ways to show that g is smooth. The first is to note that g*C is a submanifold of M x C and then applying [111.5.7]. The second is to note that on the open set p-l(Ua ) the map g is defined by

g = X;(~) 0 (g x id(F)) o'l/Ja

.

In other words, in the trivializations (u a , 'l/Ja) and (Vg(a), Xg(a)) the map g is given as (m, f) f---4 (g(m), f). The result then follows from [111.4.18]. This also shows that g is a diffeomorphism when restricted to a fiber: in the given local trivializations it is the identity onF.

2.3 Example. Let p : B ~ M be a fiber bundle and N a submanifold of M with canonical injection i : N ~ M. It is immediate from the definition of a pull-back bundle that i* B is isomorphic to the restricted bundle BIN = p-l(N). and that { : i* B ~ B corresponds to the canonical injection of BIN in B. In other words, restriction ofa bundle to a submanifold is a particular case of a pull-back bundle.

2.4 Proposition. Let h : B ~ C be a fiber bundle map between the fiber bundles p : B ~ M and q : C ~ N, inducing the map 9 : M ~ N. Then there exists a unique fiber bundle map hind: B ~ g*C such that h = hind 0 g, i.e., we have a commutative diagram h

B

------>

g*C

------>

pI M

Ig*q ------> id(M)

C

9

hind

M

Iq

------> 9

N.

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§2. Constructions offiber bundles

Proof Let us start with uniqueness. If b E B then, by our definition of fiber bundle map, hind(b) must lie in (g*q)-l(p(b)). Since gis a bijection from (g*q)-l(p(b))to q-l (g(p(b))), the condition h = hind 0 !J implies the sought for uniqueness. To show the existence 0 f hind, we define the map hind : B -+ g*O by the formula hind(b) = (p(b), h(b)) E g*O c M x O. The image lies within g*O because h is a fiber bundle map. To verify that hind is smooth one uses the local charts defined in the IQEDI construction of the smooth structure of g*O; this is left to the reader.

2.5 Corollary. The map h I--> hind is a 1-1 correspondence between fiber bundle maps h : B -+ 0 that induce a given map 9 : M -+ N andfiber bundle maps hind: B -+ g*O (which induce by definition the identity on M).

2.6 Discussion. The above corollary shows that we can always transform a question concerning a fiber bundle map between two fiber bundles over different spaces into a question concerning fiber bundle maps over the same base space. This is another justification not to consider very extensively fiber bundle maps between fiber bundles with the same base space that do not induce the identity.

2.7 Construction (product bundles). Let p : B

-+ M and q : 0 -+ M be two bundles over M with typical fiber F and structure group G respectively E and H. Let { (Ua, 'l/Jba) } respectively { (Ua, Xba) } be the atlas with the transition functions defining the bundle B respectively C. Indeed, by taking, if necessary, pairwise intersections, we may assume that both atlases have the same set of charts Ua • We now define the smooth maps 'l9 ba : Ua n Ub -+ G x H by 'l9 ba (m) = ('l/Jba(m), Xba(m)); they obviously satisfy (1.25). Moreover, there exists a natural smooth action W of G x H on F x E defined by W(g, h)(f, e) = (
2.8 Discussion. In the construction of product bundles we did not prove that the result does not depend upon the choice of the trivializing atlases. One way to prove the independence is to show that a different choice leads to an isomorphic bundle. But there is a nicer way, using pull-back bundles. Consider q : 0 -+ M as a fiber bundle with typical fiber E and consider p : B -+ Mas a smooth map. We thus can form the pull-back bundle p* q : p* 0 -+ B (also with typical fiber E) and the smooth map fJ : p*O -+ O. A glance at the definition of p*O and the maps p* q and fJ shows that it is rather symmetric in p and q; in particular fJ = q* p and

p*O = q* B = { (b, c) E B x 0 I p(b) = q(c) }

c

B x 0 .

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Chapter IV. Bundles

We leave it to the reader to convince himself that the product bundle B x MC is (isomorphic to) the pull-back bundle p*C = q* B with projection po p*q = q 0 q*p, i.e., we have a commutative diagram

B

-------. M . p

We conclude that not only is B x M C a fiber bundle over M (with typical fiber F x E), but also a fiber bundle over B with typical fiber E and a fiber bundle over C with typical fiber F.

2.9 Discussion. So far we did not pay much attention to the structure group G of a fiber bundle p : B -+ M. In fact, the only role it plays is in restricting the possible compatible local trivializing charts. Two reactions are possible: take the smallest one so as to restrict as far as possible the number of compatible charts. And at the other extreme: take the biggest one in order to have the maximum number of compatible charts. To describe this idea in more detail, we suppose that i : G -+ H is an injective A-Lie group morphism and that

ISS

§2. Constructions offiber bundles

2.10 Remarks. • The process of enlarging the structure group can be interpreted as a special case of an associated bundle. The map i : G -+ H is a smooth group homomorphism, and G and H have pseudo effective smooth actions on F. We thus can apply the construction of associated bundles. Of course, in this case the result is the same bundle, seen as an A-manifold . • As an application of reduction of the structure group, one could say that a fiber bundle is trivial if and only if its structure group can be reduced to the identity.

2.11 Definition. A principal fiber bundle over M with structure group G is a fiber bundle p : P -+ M with typical fiber G and structure group G whose pseudo effective smooth

left action on the typical fiber is the multiplication m : G x G

2.12 Lemma. Let p : P

-+

G (which is effective).

M be a principal fiber bundle with structure group G. There exists a canonically defined effective smooth right action of G on P with the property that each local trivialization (U, 'l/J) is equivariant for the natural right action of G on U x G, i.e., forv E p-l(U), m E U, and g, hE G: -+

'l/J(v) = (m, g)

'l/J((v, h)) = (m, gh) .

As a consequence, the orbits of the G-action are exactly the inverse images p-l(m), m E M, i.e., M can be identified with the orbit space PIG. Proof We use the equivariance condition to define the action on any local trivialization (U, 'l/J) by

where on U x G we use the canonical right action of G given as (m, h) . 9 = (m, hg). In order to show that this definition is independent of the chosen trivialization, we choose two local trivializations (Ua , 'l/Ja) and (Ub, 'l/Jb). According to the definition of a general fiber bundle we have a transition function 'l/Jba : Ua n Ub -+ G. According to the definition of a principal fiber bundle this means that we have ('l/Jb 0 'l/J;; 1 ) (m, h) = (m, 'l/Jba (m) . h). Using the canonical left action of G on any product U x G defined as g. (m, h) = (m, gh), we can write this as ('l/Jb 0 'l/J;;l)(m, h) = 'l/Jba(m) . (m, h). We then compute:

'l/J;;1( 'l/Jb(V) . g)

=

'l/J;;l([(( 'l/Jb 0 'l/J;;l) 0 'l/Ja)(v)], g)

= 'l/J;;l( 'l/Jba(m) . ['l/Ja(v) . g])

=

= 'l/J;;l(['l/Jba( m) . 'l/Ja (v)]

. g)

('l/J;;l 0 ('l/Jb 0 'l/J;;l))( 'l/Ja( v) . g)

= 'l/J;;l('l/Ja(v) . g) ,

where we introduced m = p(v) = 1fl('l/Ja(V)) = 1fl('l/Jb(V)) with 1fl the projection on the IQEDI first factor and where we used that multiplication in G is associative.

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Chapter IV. Bundles

2.13 DiscussionINotation. Looking at the proof of [2.12], one sees that we have transported the natural right action of G on the (local) product U x G to p-l(U) and that the form of the transition functions of a principal fiber bundle guarantee that this local definition is compatible on overlaps and thus extends globally. We will encounter a similar phenomenon in [3.2] for vector bundles. Once we have defined the right action of G on P, we can say that each local trivialization intertwines the (global) right action of G on P with the (local) right action of G on U x G. As is usual, we will denote

2.14 Lemma. Let q : B ---+ M be a fiber bundle with typical fiber F and structure group G, whose action on F is denoted by
Proof This is actually playing around with associated bundles. The A-Lie group G has a pseudo effective smooth action on both F and G. We thus can apply the construction of an associated bundle to q : B ---+ M and the identity p = id : G ---+ G. Obviously the result is a principal fiber bundle qid : Bid,G ---+ M with structure group G, which we now denote as p : P ---+ M. Changing the roles of the typical fibers F and G, we can apply the construction of an associated bundle to p : P ---+ M and the identity p = id : G ---+ G. We IQEDI then recover the original fiber bundle q : B ---+ M.

2.15 Remark. One could say that [2.14] tells us that principal fiber bundles are a kind of universal fiber bundle in the sense that all other fiber bundles can be obtained from them by the procedure of associated bundles. More properties of principal fiber bundles can be found in chapter VII, especially in [VII.3.1]-[VII.3.6] and [VII.S.2]-[VII.S.4].

3.

VECTOR BUNDLES AND SECTIONS

In this section we specialize to vector bundles and their morphisms. We discuss the notion ofa vector subbundle in detail and show that under suitable circumstances the kernel of a vector bundle morphism is a vector subbundle. In between we show that the set ofall smooth sections ofa vector bundle is a graded C=(M)-module and that local sections can be glued together to a global section using a partition ofunity.

3.1 Definition. A vector bundle over M is a fiber bundle p : B ---+ M whose typical fiber E ~ E~ is an A-vector space and whose structure group is (can be enlarged to) Aut(E). The rank of(the vector bundle) B is the graded dimension of the typical fiber E.

157

§3. Vector bundles and sections

3.2 Discussion. Let p : B ~ M be a vector bundle with typical fiber E and let (Ua, 'l/Ja) and (Ub, 'l/Jb) be two trivializing charts for B. For m E Ua n Ub we thus have two homeomorphisms 'l/Ja : Bm ~ {m} x E and'l/Jb : Bm ~ {m} x E. Since E is a (finite dimensional) free graded A-module, we can transport this structure to Bm. Since 'l/Jb 0 'l/Ja induces an automorphism on E by definition of the structure group of a vector bundle, we can use either 'l/Ja or'l/Jb without running into contradictions. We conclude that the fibers of a vector bundle are in a canonical way (finite dimensional) free graded A-modules. We have been very careful in saying that the fibers are free graded A-modules, and not A-vector spaces. In fact, the transition functions 'l/Jba(m) belong to Aut(E), but are not necessarily smooth in the sense of [11.6.22], [111.1.27]. This means that an equivalence class of bases is not preserved by the transition functions. Which in turn means that we can not transport the structure of an A-vector space to Bm without running into contradictions. Of course 'l/Jba(m) is smooth when m has real coordinates (i.e., m E BM), hence for those points we can endow Bm with the structure of an A-vector space. But not all points in M have real coordinates.

3.3 Nota Bene. If B is a vector bundle, it is in particular an A-manifold modeled on the even part of an A-vector space and as such all points are even. On the other hand, each point lies in a certain fiber, which is a free graded A-module. Seen this way, a point can be even or odd (or neither if it is a mixture). This change of viewpoint corresponds of course to the fact that the grading of E gets lost in the identification with The fact that we define a vector bundle as a fiber bundle with typical fiber an A-vector space instead of the even part of an A-vector space will turn out to be of importance for some of the constructions of vector bundles (see [4.12]).

Eg.

3.4 Remark. Let p : B ~ M be a vector bundle with typical fiber E and let U be a trivializing atlas for B. Contrary to the discussion in [1.24], where we were rather imprecise, in the case of the vector bundle B the sets Ua x E do form an atlas for B, just because the canonical identification ~ is an atlas for E.

E Eg

3.5 Definitions. Let p : B ~ M and q : C ~ N be vector bundles with typical fibers E and F respectively. A fiber bundle map h : B ~ C inducing a map 9 : M ~ N [1.14] is called a (left linear) vector bundle morphism if the restriction to any fiber is left-linear, i.e.,

hlB", E HomL(Bm ; Cg(m)) . The map h is said to be ofparity Q if all linear maps hlB", are of parity Q. Similar definitions hold for right linear vector bundle morphisms. A (vector bundle) isomorphism is an even vector bundle morphism h : B ~ C which is at the same time an isomorphism of fiber bundles. In view of our definition of an isomorphism of fiber bundles [1.16], this requires in particular that Band C are vector bundles over the same base space M, and that 9 is the identity on M.

158

ChapterIV. Bundles

The vector bundle B is said to be trivial if it is isomorphic as vector bundle to the trivial vector bundle M x E. One should be aware that this is stronger than being trivial as a fiber bundle. The difference is that for a vector bundle not only fibers should be mapped diffeomorphic ally to fibers, but also the A-vector space structure should be preserved. In [V.3.l4] and subsequent items we will discuss this difference in more detail.

3.6 Discussion. If h : B -+ C is a vector bundle isomorphism, it follows that its restriction to a fiber Bm is an isomorphism of free graded A-modules. On the other hand, if h : B -+ C is an even vector bundle morphism (inducing the identity on the base space) such that the restriction to each fiber Bm is an isomorphism of free graded A-modules, then it is not hard to show that h is a vector bundle isomorphism (using that taking the inverse in Aut(E) is smooth). If h : B -+ M x E is a vector bundle isomorphism, then we know that B is trivial (as a vector bundle). IfU C M is a chart for M, then (U, hlp-leu)) is a local trivializing E-chart for B. Contrary to the general case as described in [1.17], this trivializing chart is Aut(E)-compatible with the trivializing charts in the vector bundle structure of B. The reason is that the restriction of h to each fiber is, by definition, linear, even and invertible, and the structure group Aut(E) of B contains all even invertible linear maps.

3.7 Discussion. The fact that fibers of a vector bundle p : B -+ M are free graded Amodules has some important consequences. A first one is that the set of (smooth) sections r( B) is never empty; it always contains the element 0 called the zero section defined by Q(m) = E Bm. Since in a trivializing chart (U,'lj;) we have ('lj;oQ)(m) = (m,O), it follows that 0 is a smooth map, i.e., indeed QE f(B). We obtain in particular a canonical submanifold Q(M) C B which is diffeomorphic to M. One often identifies M with this submanifold of B. Another consequence is that the set r(B) can be given the structure of a graded CCXl(M)-module in the following way. Let 8, t E r(B) be two smooth sections and f E CCXl(M) a smooth map. We define the section 8 + f . t by

°

(8 + f· t)(m) = 8(m)

+ f(m) . t(m)

.

If(U, 'lj;) is atrivializingchart, it follows that (8+ f· t)1j; is given by 81j; + flu' t1j; [1.20]. It thus follows from [111.1.24] and [1.20] that 8 + f· t is smooth, i.e., (8 + f· t) E f(B). We will say that a section 8 E f(B) has parity a if 8(m) E p-l(m) has parity a for all m E M. (Nota Bene. We use the parity of E, not that of Eg!) It follows from [III. 1.23-b] that f(B) is a graded CCXl(M)-module. If p : B -+ M is a trivial vector bundle over M with typical fiber E, we know that f(B) is in 1-1 correspondence with CCXl(M; E) [1.20], [1.22]. Moreover, we now know that both sets are graded CCXl(M)-modules. It is easy to check that the correspondence preserves this structure (see also [3.6]), i.e., f(B) andCCXl(M; E) are isomorphic as graded CCXl(M)-modules. But, as in [1.22], this isomorphism depends upon the trivialization.

159

§3. Vector bundles and sections

However, in most cases there will be a natural choice for a trivialization, which thus gives a natural identification between sections and functions. A particular example is the trivial vector bundle B = M x A with typical fiber A, in which case f(B) ~ C=(M).

3.8 Definition. Letp: B -+ M be a vector bundle and S E r(B). The set supp(s) C M, called the support ofs, is defined as the closure of the set of those points where s(m) is non-zero (cf. [111.5.15]).

3.9 Lemma. Let p : B

-+

M be a vector bundle.

(i) Let U o C M be open and So E fuo (B) a local section. If supp( so) C Uo is closed in M, then s : M -+ B defined by sluo = So, s == 0 outside Uo, is a smooth global section of B. (ii) Let s'" E f(B), a E J be a collection ofglobal sections such that the collection supp( s"')' a E J is locally finite. Then s = E"'EJ S'" is a well defined smooth global section of B. (iii) Let V = {V'" I a E J} be an open cover of M, s'" E fVa (B), and {p'" I a E J} a partition of unity subordinated to V. Then s = E"'EJ P"'S'" is a well defined smooth global section of B. Proof Left to the reader as an elementary combination of the arguments in [1.20], [3.7], [111.5.16], [111.5.17], and [111.5.19]. IQEDI

3.10 Proposition. Let p : B -+ M be a vector bundle of rank n and typical fiber E, and let U C M be open. Then the following three properties are equivalent. (i) B is trivial over U, i.e., p : p-l(U) == Bu -+ U is the trivial bundle. (ii) There exist n homogeneous smooth sections Si E fu(B) such that (si(m))f=l is a basis for Bmfor all m E U. (iii) There exist n homogeneous smooth sections Si E fu(B) such that (si)f=l is a basisfor the graded C=(U)-module fu(B). If'lj; : p-l (U) -+ U x E is a (potential) trivialization and if (ei)f=l is a homogeneous basis of E, then the link between (i), (ii), and (iii) is given by 'lj;-l( m, Ei Xi ei) = Ei Xi Si (m). Proof. Let (ei)f=l be a homogeneous basis of E, fixed during this proof. Now assume (i) and let 'lj; : p-l(U) -+ U x E be a trivialization. We define the homogeneous sections Si E fu(B) by si(m) = 'lj;-l(m, ei)' These sections obviously satisfy (ii) and (iii), proving the implications (i) =} (ii) and (i) =} (iii) . • To prove (ii) =} (i), we first note that the graded dimension of a free graded A-module is an invariant and thus that we may assume that the parity of Si is the same as that of ei' From this it follows that the map X : U x E -+ p-l(U) defined by

W)

ChapterIV. Bundles

is a smooth bijection whose restriction to each fiber is even. We intend to prove that its inverse X- 1 : p-l(U) -+ U x E is also smooth and a (local) vector bundle morphism. To that end, let 1/J : p-l(V) -+ V x E be a trivializing chart such that V C U. Since the Si are smooth, there exist smooth functions ai j E COO(V) such that for all m E V:

It follows that 1/J 0 X restricted to V x E is given by (m, Eixiei)

f---t

(m, Ei,jxiaij (m )ej).

Since the si(m) form a basis for Bm, the matrix aij(m) must be invertible. Since taking the inverse of a matrix is smooth [111.2.26], it follows that the inverse map X- 1 o1/J-l : (m, Eixiei) f---t (m, Ei,jxi(a-1)ij(m)ej) is also smooth. Since we can cover U by such charts V, this proves that X- 1 is smooth and at the same time that it is a (local) vector bundle morphism (because it relates to 1/J by an automorphism of E). This proves that X- 1 is a vector bundle trivialization of p-l(U) . • Finally to prove (iii) :::::} (ii), let Cv, 1/J) be a trivializing chart with V C U and let p be a plateau function around mo E V in V [111.5.21]. With this we define the smooth sections ti E fv(B) byti(m) = p(m)1/J- 1(m, ei). Sincesupp(p) eVe U, we may assume that the ti belong to fu(B). By hypothesis there exist functions ai j E COO(U) such that ti = Eiaij Sj. Writing 1/J(sj(m o)) = (mo, Eks/ek) and using the fact that 1/J(t i (m o)) = (mo, ei), it follows that ei = Ej,kaij(mo) s/ek. Hence aij(m o) is invertible, and thus the si(m o) form a basis of Bm o' In particular there are as many even IQEDI elements among the Si as among the ei (invariance of the graded dimension).

3.11 Definition. Let p : B -+ M be a vector bundle with typical fiber E and U C M open. A set of homogeneous local sections Si E fu(B), i = 1, ... , dim(E), is called a set of local trivializing sections ofB if it satisfies either [3.1O-ii] or [3.1O-iii].

3.12 Definition. Let p : B -+ M be a vector bundle with typical fiber E and let F be a graded subspace of E in the sense of A-vector spaces. A subbundle with typical fiber F of the vector bundle B is a subset C C B for which there exists a trivializing atlas { Ua , 1/Ja } of B such that 1/Ja(p-l(Ua) n C) = Ua x F. Such an atlas is called an atlas adapted to the subbundle. If C is a subbundle of B, it follows in particular that for all m E M the set Cm = C n Bm is a graded subspace of the free graded A-module Bm isomorphic to F [11.2.1].

3.13 Discussion. Let C be a subbundle of a vector bundle B with typical fiber E. Let

FeE and let U = {Ua, 1/Ja } be as specified by the definition of subbundle. Since Ua x F = (Ua x E) n (Ua x F) it should be obvious that C is also a submanifold of B. If we denote by q : C -+ M the restriction q = pic, it follows that q-l (Ua ) = 1/J;;1 (Ua x F), i.e., that (Ua, 1/Jalq-l(Ua )) is a local trivializing F-chart for the projection q : C -+ M. Since the transition functions 1/Jba are even linear maps, the same holds true for the

161

§3. Vector bundles and sections

transition functions associated to the trivializing F-charts for q : 0 ---+ M. We conclude that q : 0 ---+ M is a vector bundle with typical fiber F, justifying its name. The existence of a subbundle tells us also something about the vector bundle p : B ---+ M itself. Denote by G C Aut(E) the subgroup of automorphisms of E that preserve the graded subspace F. If (ei)f=1 is a basis of E such that (ei)~=1 is a basis for F, the matrix of an automorphism 9 of E has a block form 9 = (~~) such that 9 belongs to G if and only if c = O. Hence G C Aut(E) is an A-Lie group on its own and the canonical injectionG ---+ Aut(E) is an A-Lie group morphism. SinceU is adapted to the subbundle, the transition functions 'lj;ba associated to this atlas take their values in G. In other words, the structure group of B can be reduced from Aut(E) to G.

3.14 Proposition. Let p : B ---+ M be a vector bundle with typical fiber E, let 0 C B be a subset, let F be a graded subspace of E, and let (ei)f=1 be a basis of E such that (ei)~=1 is a basis of F. !ffor all mo E M there exists a chartU 3 mo and k homogeneous sections Si E fu(B) ofthe same parity as ei such that

Vm E U : (si(m))~=1 is a basis of Om

(3.15)

= 0 n Bm

,

then C is a subbundle of B with typical fiber F. Nota Bene: (3.15) requires/implies that all Om = 0 n Bm be graded subspaces [II.2.lJ of Bm (in the sense of free graded A-modules). On the other hand, if C is a subbundle of B with typical fiber F, then for all mo E M there exists a chart U 3 mo and a set of local trivializing sections (n in total!) Si E fu (B) of the same parity as ei satisfying (3.15). Proof If (U, 'lj;) is a trivializing chart adapted to the subbundle, it follows that the sections si(m) = 'lj;-1(m, ei) satisfy the condition, proving fue second part. For the first part, let U and (Si)~=1 be as stated. Let Cv, 'lj;) be a trivializing chart mo EVe U, and let the sections f.i E fv(B) be defined as f.i(m) = 'lj;-1(m, ei). We thus can write j j Si (m) = = 1 a i (m ) f.j for some smooth functions a i E 0= (V). Since ¢ is injective, the rank of the n x k matrix aij must be k on V. Reordering the ei if necessary, we may assume that ((si(mo))~=1' (f.i(m o))f=k+1) constitutes a basis of Bm o ' By continuity (shrinking V if necessary) we may assume that ((si(m))~=1' (f.i(m))f=k+1) constitutes a basis of Bm for all m E V. It then follows from [3.10] that the sections ((Si)~=1' (f.i)f=k+l) on the chart V define a local trivialization in a neighborhood of mo adapted to the subbundle. IQEDI

"L.7

3.16 Discussion. The interest of [3.14] is that it allows us to define subbundles of a vector bundle p : B ---+ 1\11 in terms of local sections. According to [3.14] it suffices to specify an atlas U and for each U E U local homogeneous sections S1,"" Sk E fu(B) such that (i) the number of even sections is the same for every U, and (ii) the graded subspace of Bm generated by (si(m))~=1 has total dimension k and is independent ofU 3 m. If Mis

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connected, even the requirement that the number of even sections is the same for every U can be omitted because the graded dimension is an invariant of a free graded A-module. Another consequence of [3.14] is that it says more or less that, if the local sections s1,"" Sk E fu(B) generate C, then we can find sections Sk+1, ... , Sn E fu(B) such that (Si)?=1 is a set of local trivializing sections of B. This follows by the second part of [3.14]. The only catch is that one might have to take a smaller U to find the Sk+1, ... , Sn.

3.17 Lemma. Let E and F be A-vector spaces, Qan A-manifold, and let 'lj; : Q x E --+ F be a smooth map such that for each q E Q the map v I--> 'lj;( q, v) is right linear. Then there exists a unique smooth map 'lj; : Q --+ EndR(E; F) (abuse of notation justified by the result) such that 'lj;(q, e) = 'lj;(q) (e). Proof Let (ei)?=1' (lj)~1 be bases of E and F, and let (e i )?=1' (lj)7'=1 be the associated right dual bases. We then define functions 'lj;j i : Q --+ A by 'lj;j i (q) = f j ( 'lj; (q, ei)). These functions are smooth because the maps fj : F --+ A are smooth and because the ei have real coordinates. But then we can form the smooth function 'lj; : Q --+ EndR(E; F) by 'lj;(q) = Li,j fj 0 'lj;ji(g) . e i . Since each 'lj;(q, J is right linear, it is immediate that 'lj;(q, e) = 'lj;(q)(e) for all e E E. IQEDI

3.18 Proposition. Let p : B --+ M and q : C --+ N be vector bundles with typical fibers E and F respectively, and let h : B --+ C be a homogeneous left or right linear vector bundle morphism inducing a map 9 : M --+ N such that hlB", : Bm --+ Cg(m) is surjectivefor all m E M. Then

ker(h)

= UmEM ker(hIB",

is a subbundle of B with rank(ker( h))

: Bm

= rank( B) -

--+

Cg(m))

rank( C).

Proof We will prove the right linear case, the left linear case being similar. Our strategy will be to apply [3.14] to obtain the desired result. For mo E M, let W 3 g(mo) be a trivializing chart for the bundle C with trivialization X : q-1(W) --+ W x F and let V 3 mo be a trivializing chart for B with trivialization 'lj; : p-1(V) --+ V x E such that g(V) c W (such charts exist). The map X 0 h 0 'lj;-1 : V x E --+ W x F is of the form (m, e) I--> (g(m), h(m, e)). The factthat his a right linear vector bundle morphism translates as the fact that each map h( m, _) : E --+ F is right linear. According to [3.17] this implies that the matrix elements of these right linear maps are smooth functions. More precisely, if (ed?=1 is a basis of E and (lj )}=1 a basis of F, then there exist smooth functions h j i E C=(V) such that n

(X 0 h 0 'lj;-1)(m, Lei Xi) i=1

n

k

j

= (g(m), LLiJ h i(m) Xi) .

i=1j=1

Since hlB", is surjective, the rank [11.1.9] of the n x k matrix hji(m) must be k. By reordering the basis (ei), we may assume that the k x kmatrix (hi i (mo) )L=1 is invertible.

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§4. Constructions of vector bundles

Thus there exists a chart mo E U C V such that (h j i (m) )f,j =1 is invertible for all m E U. Since inversion of matrices is smooth, there exist smooth functions gP j E C= (U) such that the matrix (gP j (m) )~,j = 1 is the inverse of the matrix (h j i ( m)) f,j = 1 . With these ingredients we define the sections Si E fu(B), k < i .::; n by k

si(m) = 1jJ-l(m,ei -

L

epgPj(m)hji(m))

p,j=1 By elementary linear algebra the (si(m))~k+l forma basis for ker(hIB",) = ker(h)nB m . They are homogeneous because h is homogeneous. We conclude by [3.14] that ker(h) is a subbundle of B. Since n = rank Band k = rank C, we automatically have rank(ker(h)) = n - k = rankB - rankC. IQEDI

3.19 Counter example. As in [III.3.30], the condition that each hlB", is surjective can not be weakened easily, as can be seen in the following example. Let F be an A-vector space of dimension 012, let M = N = Po with coordinates (~, 1]), and let B = C = M x A be the trivial bundle. Then the map h : B ---+ C, (~, 1], a) I--> (~, 1], ~1]a) is a right linear vector bundle morphism. But ker( h) is not a subbundle. Even more, except for ~1] = 0, ker(hIB(E,ry)) is not a graded subspace of B(f.,1)) (though it is a graded submodule).

4.

CONSTRUCTIONS OF VECTOR BUNDLES

We show that the various operations one can perform on A-vector spaces (quotients, direct sums, tensor products, exterior powers, homomorphisms) have their analogonfor vector bundles. Even the constructions of new linear maps out of old ones have their counter part in vector bundle morphisms. As a byproduct ofthese constructions, we explain why we took the full A-vector space as typical fiber for vector bundles instead ofonly its even pan.

4.1 Discussion. Let p : B ---+ M and q : C ---+ M be vector bundles over the same base space with typical fibers E and F. For m E M this implies that Bm is isomorphic to E and C m to F. Since these spaces are free graded A-modules, we can perform various operations on these fibers, e.g., taking the direct sum Bm EB Cm, or the tensor product Bm 0 Cm. Whence the obvious question: can we construct in a natural way out of Band C a vector bundle r : D ---+ M whose typical fiber is, say, E EB F such that we have a natural identification Dm ~ Bm EB Cm. The obvious way to do this is to define D = UmEM(Bm EBCm ). The only draw-back is that we have to indicate the structure of an A-manifold on this D. In the next constructions we will do exactly that, although in a roundabout way. We will define vector bundles with

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Chapter IV. Bundles

the appropriate typical fiber, and we will show once (in [4.5] for the direct sum case) how this constructed vector bundle relates to D.

4.2 Construction (quotient bundle). Let P : B -+ M be a vector bundle with typical fiber E, C a subbundle with typical fiber F, and U a trivializing atlas adapted to the subbundle C. Let G be the subgroup of Aut(E) that leaves the graded subspace FeE invariant. Since Gleaves F invariant, it induces an action on the quotient ElF, i.e., there exists a group morphism p : G -+ Aut(E / F). Using a basis as in [3.13] shows that p is smooth (it is given by

(~ ~)

f---4

d). We thus can construct the associated vector bundle

with typical fiber ElF. It is called the quotient bundle of B by C and denoted by BIC.

4.3 Construction (direct sums). Let Pi : Bi -+ M, i = 1, ... ,n be a finite family ofvector bundles over M with typical fibers E i . Let { Ua, 'l/Ji,ba } be an atlas and transition functions defining the bundle B i . Indeed, by taking n-fold intersections we may assume that the charts Ua are the same for all bundles. Since the maps 'l/Ji,ba : Ua n Ub -+ Aut(Ei ) are smooth, it follows that the map 11 'l/Ji,ba : Ua n Ub -+ 11 Aut(Ei ) c Aut( EBiEi) defined as (11 'l/Ji,ba)(m) = 11 'l/Ji,ba(m) [1.7.7] is smooth. From the definition of 11 'l/Ji,ba(m) we deduce the equality

on the triple intersection Ua n Ub n Uc. Hence { Ua, 11 'l/Ji,ba } defines a vector bundle over M with typical fiber EBiEi whose structure group can be reduced to 11 Aut(Ei ). This bundle, denoted by EBiBi, is called the direct sum bundle of the vector bundles B i ; it is also called the Whitney sum of the B i . In case there are only two vector bundles involved, one writes Bl EB B2 for this direct sum bundle.

4.4 Remarks. • The definition of a direct sum of two vector bundles is essentially the same as the definition of a direct product bundle. Strictly speaking we should have indexed the EB in Bl EB B2 with an M to distinguish this sum from the direct sum of A-vector spaces. However, a vector bundle will hardly ever be at the same time an A-vector space, so no confusion is possible, since a direct sum of two A-manifolds is not defined . • Since we can reduce the structure group ofEBiBi to 11 Aut(Ei ), it should be fairly obvious that each separate bundle Bi can be seen as a subbundle of EBiBi.

4.5 Discussion. Let P : B -+ M and q : C -+ M be vector bundles over the same base space with typical fibers E and F. It follows that B EB C is a vector bundle with typical fiber E EB F. In [4.1] we suggested the space D = UmEM(Bm EB Cm), with the obvious projection r : D -+ M, as a candidate for this vector bundle. Let us show that we can

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§4. Constructions of vector bundles

identify these two spaces, i.e., that we can equip D with the structure of a vector bundle over M with the same transition functions as the bundle B EB C. Let U be a trivializing atlas for both bundles at the same time with diffeomorphisms 'l/Ja : p-1(Ua) -+ Ua x E and Xa : q-l(Ua) -+ Ua x F that induce the transition functions 'l/Jba and Xba. By definition of the structure ofa free graded A-module on each fiber, the maps 'l/Ja(m) : p-l(m) -+ E and Xa(m) : q-l(m) -+ F are isomorphisms. We thus obtain an isomorphism

'l/Ja(m)

X

Xa(m) : Bm EB Cm

-+

E EB F .

Varying mover Ua gives us a bijection between 1'-1 (Ua ) = UmEuJB m EB Cm) c D and Ua x (EEB F). Declaring this to be a diffeomorphism equips 1'-1 (Ua ) with the structure of an A-manifold, compatible with the projection 1'. It remains to show that these structures coincide on overlaps Ua nUb. But on overlaps we have (by definition of the transition functions) the equality ('l/Jb(m) x Xb(m)) 0 ('l/Ja(m) X Xa(m))-1 = 'l/Jba X Xba, which is a smooth function. It thus follows at the same time that the smooth structures coincide on overlaps and that the transition functions of the vector bundle D (yes, we just proved it is one!) are exactly the same as those of B EB C. In exactly the same spirit one should interpret the fibers of BIC (assuming of course that C is a subbundle of B) as being the quotient of the corresponding fibers in Band C :

Similar remarks apply to the forthcoming constructions of tensor product bundles, exterior powers and homomorphism bundles.

4.6 Definition. Let p : D -+ M be a vector bundle with typical fiber E and let B and C be subbundles with typical fibers Fl and F2 respectively. We will say that B and C are supplements if Vm EM: Bm EB Cm = Dm, i.e., if Vm EM: Bm and Cm are supplements to each other in the sense offree graded A-modules [11.2.1], [11.3.4].

4.7 Proposition. Let p : D then

-+

M, B, and C be as in [4.6J. If Band C are supplements,

(i) if(Si)~=1 E ru(D)is a set of local trivializing sectionsforB, if (t j )}=1 E ru(D)

is a set of local trivializing sections for C, then (Si' tj) is a set of local trivializing sectionsfor D, and (ii) B EB C is isomorphic as a vector bundle to D. Proof For (i) it suffices to note that by [3.14] Vm E U (si(m)) is a basis for Bm, and that (tj(m)) is a basis of Cm, and thus (si(m), tj(m)) is a basis of Dm because Bm EB C m = Dm (which means that the map (b, c) I--> b + c, Bm EB C m -+ Dm is an isomorphism). The announced result then follows again by [3.14]. For (ii) we use the interpretation [4.5] of B EB C = UmEM Bm EB C m and define (set theoretically) the isomorphism h : B EB C -+ D by hIB",EBc", : Bm EB C m -+ Dm,

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Chapter IV. Bundles

(b, c) I--> b + c. Since Band C are supplements, h is an even fiber bundle map, an isomorphism when restricted to each fiber, and inducing the identity on M. According to [3.6] it thus suffices to show that h is smooth in order to prove the result. To prove that h is smooth, we choose local trivializing sections (Si) E fu(B) C fu(D) and (tj) E fu(C) C fu(D) around a point mo E M (by shrinking we can always assume that we have the same U). We then denote by 'Ij; the local trivialization of B associated to the (Si) [3.10], by X the local trivialization of C associated to the (tj), and by ¢ the local trivialization of D associated to (Si, tj). Using the notation UmEM'Ij;(m) X x(m) as in [4.5], we have a commutative diagram r-I(U) U ",EM

=

U Bm mEM

1j;(m) xx(m)

U

X

EB C m

1

(FI EBF2 )

(m, (II, h))

h -----.

U Dm= mEM

1

U

",EM

-----.

p-I(U)

(m)=

UxE

-----. (m,lI+h),

just because of our choice of ¢. Since we know that all maps except h are diffeomorphisms, we conclude that h is smooth on r-I(U). Since the U cover M (by varying mo), we conclude. IQEDI

4.8 Remark. If B and C are subbundles of a vector bundle p : D -+ M, one might think that, in analogy with sums of graded subspaces, we could define a sum bundle as UmEM(Bm + Cm). However, this is a very unstable definition because there is no easy way to guarantee that Bm + Cm C Dm has a graded dimension independent of m EM. And if this graded dimension is not constant, we do not get a vector bundle in our sense.

4.9 Construction (tensor products). Let Pi : Bi -+ M, i = 1, ... , n be a finite family of vector bundles over M with typical fibers E i . Let { Ua , 'lj;i,ba } be an atlas and transition functions defining the bundle B i . Since the maps 'lj;i,ba : Ua n Ub -+ Aut(Ei ) are smooth, it follows that the map 0}=n 'lj;i,ba : Ua n Ub -+ 0}=n Aut(Ei ) C Aut( 0f=1 E i ) defined as (0}=n'lj;i,ba)(m) = 0}=n'lj;i,ba(m) [1.7.12] is smooth. (Nota Bene. We only have an inclusion 0;=n Aut(Ei ) C Aut( 0f=1 E i ) because the even part of a tensor product is not the tensor product of the even parts.) From the definition of 0}=n 'lj;i,ba(m) and the fact that the 'lj;i,ba (m) are even, we deduce the equality

on the triple intersection Ua n Ub n Uc. Hence { Ua , 0}=n 'lj;i,ba } defines a vector bundle over M with typical fiber 0f=1 E i . This bundle, denoted by 0f=1 Bi (or by BI 0···0 Bn to make the order even more explicit), is called the tensor product bundle of the vector bundles B i .

§4. Constructions of vector bundles

167

4.10 Construction (exterior powers). Let p : B ---; M be a vector bundle with typical fiber E, let { Ua, 'l/Jba } be an atlas and transition functions defining the bundle B and let k be a positive integer. Since the map 'l/Jba : Ua n Ub ---; Aut( E) is smooth, it follows that the map N'l/Jba : Ua n Ub ---; Aut(N E) defined as (N 'l/Jba)(m) = N'l/Jba(m) [1.7.15] is smooth. From the definition of!\k 'l/Jba(m) we deduce the equality

on the triple intersection Ua n Ub n Uc . It thus follows that {Ua,!\k 'l/Jba} defines a vector bundle over M with typical fiber !\k E. This bundle, denoted by !\k B, is called the k-th exterior power of the vector bundle E. According to [1.7.15] we have N 'l/Jba(m) = 'l/Jba(m) and N 'l/Jba(m) = id(A). Together with N E = E and N E = A this gives N B = Band N B = M x A.

4.11 DefinitionlDiscussion. Let p : B ---; M be a vector bundle with typical fiber E. Since elements of Aut(E) are even, they preserve Ea (a E Z2); we thus obtain a smooth action of Aut(E) on the A-manifold Ea. Since the action of Aut(E) on Ea is pseudo effective [1.5], we can apply the construction of an associated bundle with the identity map p = id(Aut(E)), but with an action on different A-manifolds. The resulting fiber bundle will be denoted by p : Ba ---; M, called the degree-a part ofE. Another way to define Ba is as a submanifold of B by Ba = {b E Bib E p-l(p(b))a}. That it is a submanifold of B follows immediately from the fact that Ea is a submanifold of E ~ Eg. That it is indeed the same as the associated bundle defined previously is left to the reader. In order to study more carefully the above situation, let us denote by 'l/Jba the transition functions of the bundle B, and by 'l/Ja,ba those of the bundle Ba. These maps are all essentially the same, they act on different spaces. Since E = Eo X El in terms of A-manifolds, it is natural to look at the transition functions 'l/JO,ba X 'l/Jl,ba who act on Eo x El ~ E, as do the 'l/Jba. An elementary calculation shows that these two sets of transition functions are the same. We conclude that the product bundle Bo XM Bl is isomorphic to the original bundle E. Although the fiber bundle Ba is not a vector bundle, it resembles it very closely. The abelian group Ea is stable under multiplication by elements of AD, i.e., Ea is an Aomodule. The action of Aut(E) on Ea preserves this structure and thus the fibers of Ba have the structure of an AD-module. Copying the discussion for vector bundles shows that f(Ba) is a CCXl(M)o-module. More precisely, one can show that we have an identification r(Ba) = r(B)a, a result in complete agreement with the corresponding fact for smooth functions CCXl(M; F)a = CCXl(M; Fa), and a result which is rather obvious once we see Ba as a submanifold of E. It then follows that we can apply (adapt) the results of [3.9] to sections of Be<> in particular that local sections of Ba can be glued together to a global section by means of a partition of unity.

4.12 Discussion. Let us now explain why we defined vector bundles as fiber bundles with as typical fiber a full A-vector space E (seen as the even part of E~), instead of the

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Chapter IV. Bundles

even part of an A-vector space. The reason is that we could not have performed all the constructions of vector bundles as we did, in particular not the tensor product. Let us show the problem. As usual we take A = 1\ X where X is an infinite dimensional vector space over R with basis (ei)iEN. We then consider the A-vector space E of dimension 011, for which we have a natural identification E 0 E ~ A (if bEE is a basis, we can identify b 0 b with 1 E A). If we consider only Eo, it will not be an A-module, but only anAo-module. Hence we cannot take the tensor product over A, but only over AD. The interesting question is whether this tensor product Eo 0Ao Eo is in a natural way the even part of an A-vector space, and more in particular whether it is the even part of E 0A E. As a set Eo is isomorphic to AI, and by definition of A, the ei E 1\1 X C Al generate Al as an AD-module. It follows that the elements ei 0 ej generate the tensor product Eo 0Ao Eo (over AD). As said before, E 0A E ~ A, and in particular (E 0A E)o ~ AD is a free AD-module on one (even) generator. Now suppose that Eo 0Ao Eo is a free Ao-module on one generator gEED 0Ao Eo. Since the ei 0 ej generate Eo 0Ao Eo, there exist Aij E AD (finitely many!) such that 9 = I:ij Aijei 0 ej. If we now take n E AD to be the (exterior) product of all different e/s appearing in this sum (adding one not in that sum if there is an odd number of different ei 's), then n is not zero (X is infinite dimensional), but ng is zero. The conclusion of these computations is that in general the AD-modules (E 0 A F)o and Eo 0Ao Fo are not the same. On the other hand, if Band C are vector bundles with typical fibers Eo and Fo respectively, it is quite natural to require from a tensorproduct of bundles that the typical fiber of B 0 C is (E 0 F)o. Since this cannot be done by means of the tensor product of AD-modules, we use the full A-vector space E as fiber, take the tensor product in the category of A-vector spaces, and restrict later to the even (or odd) part.

4.13 Discussion. Of course all our definitions and constructions of fiber bundles and vector bundles are also valid in the context of R-manifolds, i.e., in the special case A = R. Moreover, the body map B maps fiber/vector bundles in the category of A-manifolds to the corresponding objects in the category of R-manifolds. But in the category of R-manifolds something special happens with vector bundles. If X = Xo EB Xl is an R-vector space of dimension plq, its separate parts Xc< can also be seen as R-vector spaces: Xo can be seen as the R-vector space Xo EB {O} of dimension plO, and Xl can be seen as the R-vector space {O} EB Xl of dimension 0lq. Since automorphisms of X are even, they respect this decomposition, and it is not hard to see that we have an identification Aut(X) ~ Aut(Xo) X Aut(X l ) (regarding Xc< as an R-vector space). From these observations it follows that if p : B --+ M is a vector bundle in the category of R-manifolds, then the Bc< are also vector bundles in this category. We are thus able to write B ~ Bo x M Bl ~ Bo EB B l , i.e., in the category of R-manifolds the homogeneous parts of a vector bundle are vector bundles in their own right and a vector bundle is the direct sum of its even and odd part. We said that R-manifolds are nothing more (nor less) than manifolds in the classical sense. We now see that the same is not true for vector

§4. Constructions of vector bundles

169

bundles: vector bundles in the category of R-manifolds are not vector bundles in the classical sense, they are a direct sum of two vector bundles in the classical sense.

4.14 Discussion. Let p : B ---+ M and q : C ---+ N be vector bundles with typical fibers E and F respectively, and let h : B ---+ C be a right linear vector bundle morphism, inducing the map 9 : M ---+ N on the base spaces. Let furthermore V be a trivializing atlas for C with transition functions Xdc and U a trivializing atlas for B with transition functions 'l/Jba such that for all Ua E U there exists a Vg(a) E V such that g(Ua) C Vg(a). (As in [2.2] we use the same name for the function on the base space as on the indices.) By definition of a fiber bundle morphism there exist smooth functions ha : Ua x E ---+ F determined by (Xg(a) 0 h 0 'l/J;;l )(m, e) = (g(m), ha(m, e)). Since h is a vector bundle morphism, the maps ha (m, e) are right linear in e for fixed m. According to [3.17] we can interpret iliem as smooth functions ha : Ua ---+ HomR(E; F). Since h is globally defined, the ha are related on overlaps Ua n Ub by (4.15) On fue other hand, given a set of smooth functions ha : Ua ---+ HomR( E; F) verifying (4.15), we can construct a right linear vector bundle morphism h : B ---+ C inducing the map 9 : M ---+ N as follows. The restriction of f to p-l(Ua) is defined as the smooth map

where 9 x ha denotes the map Ua x E ---+ Vg(a) x F, (m, e) I--> (g( m), ha(m)( e)). Using the condition (4.15) it is elementary to show that fuese local expressions of f coincide on overlaps. In this way we obtain a bijection between right linear vector bundle morphisms and sets of smooth local functions ha : Ua ---+ HomR(E; F) verifying (4.15). And obviously a similar result is valid in the left linear context.

4.16 Construction (Hom-bundles). Let p : B ---+ M and q : C ---+ M be two vector bundles with typical fibers E and F respectively. Let U be an atlas of trivializing charts for both bundles and denote by 'l/Jba : Ua n Ub ---+ Aut(E) and Xba : Ua n Ub ---+ Aut(F) their transition functions. For a fixed mE Ua n Ub we obtain an automorphism Hba(m) of HomL(E; F) defined by (4.17) Using [1.7.12], the identification HomL(E; F) ~ *E0F, and that'l/Jba(m) and Xba(m) are even (and thus left and right linear), we can identify Hba(m) with Xba(m) 0 *'l/Jba(m)-l. We thus obtain smooth maps Hba(m) : UanUb ---+ Aut(HomL(E;F)) ~ Aut(*E0F).1t is by now elementary to show that HCb(m) 0 Hba(m) = Hca(m) on the triple intersection Ua n Ubn Uc. It thus follows that { Ua, Hba } defines a vector bundle over M with typical fiber HomL(E; F). This bundle is denoted by HomL(B; C) and called the bundle of (left)

170

Chapter IV. Bundles

homomorphisms from B to C. In the same way we obtain a vector bundle HomR( B; C) with typical fiber HomR(E; F): the transition functions Hba are given by the same expression (4.17), but in the identification HomR(E; F) ~ F 0 E* they are given as Hba(m) ~ ('l/Jba(m)-l)* 0 Xba(m). If C is the trivial bundle C = M x A, the typical fiber of the bundle HomL(B; C) is HomL(E; A) ~ *E. One usually denotes this bundle HomL(B; M x A) by *B and calls it the (left) dual bundle of B. Since for the trivial bundle M x A all transition functions Xba are the identity on A, it follows that the transition functions of the dual bundle are given by *'l/Jba,l 0 id(A) ~ *'l/J/;a1 . In a similar way one defines the right dual bundle B*.

4.18 Discussion. Let p : B -+ M and q : C -+ M be two vector bundles with typical fibers E and F respectively. The name bundle of homomorphisms for the vector bundle HomL(B; C) suggests that there might be a relation with vector bundle homomorphisms h : B -+ C (which by convention induce the identity on M). We will show that there is indeed such a relation: global smooth sections S ofHomL(B; C) are in bijection with left linear vector bundle morphisms h : B -+ C. Given a trivializing atlas, we have seen in [1.20] that smooth sections s of HomL (B; C) are in 1-1 correspondence with families of smooth maps Sa : Ua -+ HomL(E; F) satisfying the relations (1.21) with the transition functions Hba' But these conditions are exactly the conditions (4.15) (in the left linear version), once one realizes that 9 is the identity and that we use the same trivializing atlas for both bundles. We thus obtain a left linear vector bundle morphism h : B -+ C. We leave it to the reader to prove that the correspondence S I--> h is a bijection. At the intrinsic level this correspondence is very easy to understand: a section S E f(HomL(B; C)) determines, for every mE M, an element s(m) E HomL(B; C)m ~ HomL(Bm; Cm), and a left linear vector bundle morphism determines a left linear map hlB", : Bm -+ Cm. The given identification S I--> h is nothing more than saying that these two are the same: s( m) = hi B",'

4.19 Proposition. Let p : B -+ M and q : C -+ N be vector bundles and let h : B -+ C be a (left/right linear) vector bundle morphism inducing the map 9 : M -+ N. Then g* q : g* C -+ M is a vector bundle with the same typical fiber as C [2.2], 9 : g* C -+ C is an even vector bundle morphism, and hind: B -+ g*C is a (left/right linear) vector bundle morphism [2.4].

4.20 Proposition. Let 9 : M -+ N be a smooth map between A-manifolds. The operation of pull-back bundle, which transforms a fiber bundle over N into a fiber bundle over M with the same typical fiber, commutes with the constructions of vector bundles given above. E.g., if P : B -+ Nand q : C -+ N are vector bundles over N, then the vector bundles g* (B EB C) and (g* B) EB (g*C) are the same (isomorphic), as are g* (B 0 C) and

(g* B)

~

(g*C).

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§4. Constructions of vector bundles

Proof Let V be a trivializing atlas on N for the bundles over N that are involved, and let U be an atlas of M such that for each Ua E U there exists a Vg(a) E V such that g(Ua ) c Vg(a). In [2.2] it was shown that the 'l/Jba = Xg(b)g(a) 0 9 form the transition functions of the pull-back bundle over M when the Xdc are the transition functions of the bundle over N. Let us take the tensor product of two bundles as typical example, with transition functions Xl,dc and X2,dc. The transition functions of g* (B 0 C) are given by (Xl,g(b)g(a) 0 X2,g(b)g(a)) 0 g, which is the same as (Xl,g(b)g(a) 0 g) 0 (X2,g(b)g(a) 0 g), which are the transition functions of the bundle (g* B) 0 (g*C). IQEDI

4.21 Discussion. This result is not really surprising when we recall the general idea behind the vector bundle constructions. Let us again take the tensor product as typical example. The tensor product bundle B 0 C is the bundle whose fibers are just the tensor products of the originalfibers: (B 0 C)m ~ Bm 0 Cm. On the other hand, the pull-back bundle g* B is a bundle whose fiber(g* B)m is exactly the fiber Bg(m)' We thus see that all these constructions are done fiberwise, making it easy to understand the commutativity stated in [4.20].

4.22 Nota Bene. Restricting a bundle p : B ~ M to a submanifold N c M is a particular instance of a pull-back. It follows that restriction commutes with the vector bundle constructions. This fact will be frequently used in chapters V and VI without further mentioning.

4.23 Proposition. Let p : B ~ M and q : C ~ M be vector bundles over the same base space with typical fibers E and F respectively. Let 'l/Jba and Xba be the transition functions with respect to a joint trivializing atlas U. Finally, let J E BHom( E; F) be a smooth morphism of A-vector spaces. IfJ intertwines 'l/Jba and Xba, i.e., iffor each m E M we have a commutative diagram

E~F 1/Jba(m)

1 E

1

Xba(m)

J

------+

F,

then it induces a mo!yhism of vector bundles J : B ~ C. If J is ofparity a, is an isomorphism, J is an isomorphism of vector bundles.

J is too; if J

Proof For each Ua E U we define the map ha : Ua ~ Homs(E; F) as being the constant J. By commutativity of the given diagram, these functions ha verify (4.15) with 9 = id(M). Hence they define a vector bundle morphism i The remaining assertions are left to the reader.

IQEDI

Chapter IV. Bundles

172

4.24 Nota Bene. A trivializing atlas is never unique, and neither is a joint trivializing atlas. It follows that the transition functions 'l/Jba and Xba in [4.23] are not unique. The condition in [4.23] thus can be read as a search for a joint tri vializing atlas such that J intertwines the transition functions. However, in most applications a natural candidate for ajoint trivializing atlas is available whi~h satisfies the conditions. But the reader should be aware that the induced identification J depends upon the chosenjoint trivializing atlas.

4.25 Discussion. Using [4.23] we can construct many (iso)morphisms of vector bundles. Since all morphisms between various constructions of A-vector spaces as given in §1.7 (see [1.8.18]) intertwine the relevant transition functions, we obtain, among others, the following isomorphisms (finite dimensional A-vector spaces are f.g.p!) of vector bundles: HomR(B; 0) ~ 00 B*, *(B EB 0) ~ *B EB *0, etcetera. We also obtain an even morphism HOillR(B; 0) 0 B --+ 0, which can be interpreted as fiberwise evaluation. Even the interchanging map ~ satisfies the requirements, so we obtain a vector bundle interchanging map 9l : B 00--+ 00 B (abuse of notation).

4.26 Construction. In §1.7 we have given a number of constructions of new linear maps out of a family oflinear maps: the sum [1.7.2], the direct sum [1.7.6], the product [1.7.7], the tensor product [1.7.12], and the exterior power [1.7.15]. We will show that we can obtain, under suitable hypotheses, similar results for vector bundle morphisms. The basic idea is easily explained. Let the Bi be vector bundles over M, let the 0i be vector bundles over N, and let hi be a vector bundle morphism between the bundles Bi and Oi. If all vector bundle morphisms involved induce the same map 9 between the base spaces, they all map fibers above a point m E M to fibers above g( m) E N. Since the restrictions are linear, we can perform our constructions of new linear maps on these restrictions. This will certainly define an abstract map. Using [4.14] one can show that the new maps are vector bundle morphisms inducing g. We will state the precise result only for exterior powers; the other cases are similar and are left to the reader (cf. [1.7.24]).

4.27 Proposition. Let p : B --+ M be a vector bundle over M, let q : 0 --+ N be a vector bundle over N, and let h : B --+ 0 be an even (left) linear vector bundle morphism inducing 9 : M --+ N. Then there exists a canonically defined even linear vector bundle h: B --+ 0 inducing g. morphism

N

N

N

Proof Let V be a trivializing atlas for C with transition functions Xdc, and let U be a trivializing atlas for B with transition functions 'l/Jba such that the conditions of [4.14] are verified, i.e., we have local smooth maps ha : Ua --+ HomdE; F)o satisfying (4.15), where E is the typical fiber of Band F the typical fiber of C. Now recall that the transition functi ons of !\k B are given by !\k 'l/Jba, and those of !\k 0 by !\k Xdc. One then can prove easily the equality N hb(m) = (N Xg(b)g(a))(g(m)) 0 (N ha(m)) 0 (N 'l/Jba) (m)-l. As in [4.10] the maps m I--> !\k ha(m) are smooth. This proves that the local maps !\k ha : Ua --+ Homd!\k E; !\k F)o define a left linear even vector bundle morphism,

§S. Operations on sections and on vector bundles

173

which we denote by !\k h. We leave it to the reader to verify that it coincides with the map IQEDI described intuitively in [4.26].

N

N

N

4.28 Remark. For k = 0 the map h : B -+ C reduces to the "trivial" map M x A -+ N x A, (m, a) f-7 (g(m), a). See also [1.7.15].

5.

OPERATIONS ON SECTIONS AND ON VECTOR BUNDLES

In §4 we discussed various constructions of vector bundles. But in §3 we showed that the set of all sections ofa vector bundle is a graded C=(M)-module, and thus we can perform analogous constructions on the modules ofsections. In this section we show that there is a correspondence between these two sets ofconstructions: they commute with the operation of taking sections. For our purpose the most important result of this section is [5.14J, which says that sections of the k-th exterior power of the left dual bundle of a vector bundle B -+ M can be interpreted as graded skew-symmetric k-linear (over C=(M)) maps from sections of B to C=(M).

5.1 Discussion. If Pi : Bi -+ M are vector bundles over M with typical fibers E i , we know that the sets r(Bi) are graded C=(M)-modules. Since C=(M) is a graded Ralgebra, we can perform our constructions of direct sums, tensor products, exterior powers, etcetera on these modules. It is thus natural to ask whether these operations "commute" with the corresponding operations on the vector bundles, e.g., whether r(Bl EB B 2 ) is isomorphic to r(B 1 ) EB r(B2)' where the second direct sum is in the category of graded C=(M)-modules. The answer to this general question is positive, although the proofs have an increasing complexity when we go from direct sums via morphisms to tensor products. To be more precise, we will construct new sections out of old ones, and these constructions will all be pointwise, i.e., the new section at m E M will depend only on the values of the old sections at m. The first problem we then have to solve is whether the new section if smooth when all the old ones are, and whether the correspondence so obtained is an even morphism of graded C=(M)-modules. But the "real" problems come when we want to prove that these identifications are isomorphisms.

5.2 Construction. Let Pi : Bi -+ M, i = 1, ... ,n be a finite number of vector bundles with typical fibers E i . If sds a section of B i , we define a section J(EBiSi) of EBi Bi by

where we have used that the fibers of EBiBi are canonically isomorphic to the direct sum of the corresponding fibers in B i . To see that this is indeed a smooth section, it suffices to

174

Chapter IV. Bundles

verify it on trivializing charts. If U is a trivializing atlas, the sections Si are represented by functions Si,a : Ua -+ E i , and the section J(EBiSi) by the function Ua -+ EBi B i , m f---4 EBisi(m). One could take this as a definition ofJ(EBisi), of course after having verified that these local functions glue together to form a global section [1.20] (which they do by definition of the transition functions of EBi Bi). Since the Si,a are smooth, their direct sum is smooth as well, proving that J( EBiSi) is a smooth section of EBi B i . Moreover, it is an elementary exercise (left to the reader) to show that J thus defined is an even morphism of graded C=(M)-modules.

5.3 Proposition. J : EBJ(Bi )

-+

f( EBi B i ) is an isomorphism of graded C=(M)-

modules. Proof IfJ(EBisi) = 0, then in all charts Ua we have EBisi,a(m) = 0, implying that the Si,a are identically zero on Ua, i.e., the Si are zero. We conclude that J is injective. If a section U of EBi Bi is represented by local functions U a : Ua -+ EBi Ei = I1 E i , we can project on the separate factors to obtain smooth functions Si,a : Ua -+ Ei such that ua(m) = (si,a(m))i=l = EBisi,a(m). By definition of the transition functions of EBi B i , these functions glue together to form global sections Si E f( Bi). Obviously J( EBiSi) = u,

IQEDI

proving surjectivity.

5.4 Construction. Let p : B -+ M be a vector bundle with typical fiber E and let C be a subbundle with typical fiber F. If S is a section of E, we define a section J(s) of E/C by

J(s)(m)

= 7r(s(m))

E

Bm/Cm

~

(B/C)m ,

where 7r denotes the canonical projection of Bm on Bm/Cm , and where we have used that the fibers of E/C are canonically isomorphic to the quotient of the corresponding fibers in E and C. To see that this is indeed a smooth section, it suffices (as for direct sums) to verify it on trivializing charts. If U is a trivializing atlas for E adapted to the sub bundle C, the section S is represented by the function Sa : Ua -+ E and the section J (s) by the function J(s)a : Ua -+ E/F, m f---4 7r(sa(m)) (abuse of notation: this 7r denotes the canonical projection E -+ E / F). (As for direct sums, we could take this as a definition of J( S ).) In a suitable basis for E[3.13] the last map is given as ignoring the last coordinates of Sa(m). Since Sa is smooth, J(S)a is smooth as well, proving that J(s) is a smooth section of E/C. Again it is straightforward to prove that J is an even morphism of graded C= (M)-modules.

f(B /C) induces an isomorphism of graded C=(M)-modules between f(B)/f(C) and r(B /C). 5.5 Proposition. The map J : r(B)

-+

Proof First note that, since C is a subbundle of E, each section of C automatically is a section of E, i.e., that r(C) is a graded submodule ofr(B). To prove injectivity, it thus

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175

suffices to show that ker(J) = r( C). But J (s) = 0 if and only iffor all m E Ua we have This is equivalent to saying that Sa takes its values in F, i.e., that S is a section of the subbundle C. To prove surjectivity, let u : M -+ B /C be a global smooth section, represented by local functions U a : Ua -+ E / F. Since F is a graded subspace of E, it follows that there exist smooth functions Ta : Ua -+ E such that U a = 7r 0 Ta. These local functions Ta represent local sections ta E ruJB), but there is no reason to assume that they satisfy condition (1.21). We thus choose a partition of unity Pa subordinated to the open cover U and define the global section s = Ea Pata [3.9]. We claim that J(s) = u, finishing the proof. For m E M we compute: J(s)(m) = 7r(s(m)) = Ea Pa(m)7r(ta(m)) = Ea Pa(m)u(m) = u(m). The last equality follows because Ea Pa(m) = 1 and the next to last equality follows because by construction all local sections ta (whenever defined) induce u when taken modulo Cm. IQEDI

o = J (s)a (m) = 7r( Sa (m)).

5.6 DiscussionINotation. As said in [5.1], the constructed identifications are all pointwise, and they work also if the initial sections are not smooth. To be more precise, let us denote by r- 1 (B) the set of all sections s : M -+ B of a vector bundle p : B -+ M, and let us denote by C-l(M) = r-l(m x A) the set of all A-valued functions on M. It follows immediately that r- 1 (B) is a graded C-l (M)-module. If we now look at [5.2], it is 1 obvious that the identification J : EBi (B i ) -+ r- 1 (EBi B i ) is an even morphism of graded C- 1 (M)-modules. It is also not hard to show that it is an isomorphism (just copy the proof of [5.3]). Since C=(M) is a subset of C-l(M), we can see this identification as a morphism of graded C=(M)-modules. [5.3] then tells us that when we restrict to smooth sections r(Bi) C r- 1 (B), we get an isomorphism onto the smooth sections r(EBi B i ) C r- 1 (EBi Bi). Obviously, a similar remark holds for quotients.

r-

5.7 Construction. Morphisms can be left or right k-linear and may eventually be graded skew-symmetric. We will take the right k-linear graded skew-symmetric maps as a typical example, leaving the other cases (such as the isomorphism r(Homs(Bl, ... , B k ; C)) ~ Homs(r(Bl)"'" r(Bk); r(C)) ) to the reader. We thus suppose that p : B -+ M and q : C -+ M are vector bundles with typical fibers E and F respectively, and we want to construct an isomorphism between r(Hom}f(Bk; C)) and Hom}f(r(B)k; r(C)). For a section ¢ ofHom}f(B k ; C) we define J(¢) E Hom}f(r(B)k; r(C)) as follows. For Si E r(B) the section J( ¢) (SI, ... , Sk) of C is defined by

Since ¢ and the Si are smooth sections, J (¢ ) (s 1, ... , Sk) is also smooth. It is straightforward to show that J(¢) is k-linear over C=(M) and graded skew-symmetric. At a higherlevel, J is left-linear over C=(M) in ¢ and even. Before we can prove that J is an isomorphism, we need some preparations. As in [5.6] we denote by r- 1 (B) the set of all sections of B and by C-l(M) the set of all A-valued functions on M.

176

Chapter IV. Bundles

5.8 Lemma. Let p : B

-+

M be a vector bundle with typical fiber E, and let q : C

-+

M

be a vector bundle.

r-l(C) is a right k-linear graded skew-symmetric morphism of graded COO (M)-modules, thenforsectionssi E r(B) andm E M arbitrary, the value i!!(Sl, . .. , sk)(m) E Cm depends only upon the values si(m). (ii) !fin (i) we replace r(B) by r-l(B) andCOO(M) by C-l (M), then the conclusion still holds. (iii) Ifi!! : r(B)k -+ r-l(C) is a right k-linear graded skew-symmetric morphism of graded CXJ(M)-modules, then there exists a unique right k-linear morphism of graded C-l(M)-modules W : r-l(B)k -+ r-l(C) such that WIr(B)k = i!!. (i) Ifi!! : r(B)k -+

Proof • The proof of (i) breaks into two steps. Let Si and s~ be two sets of sections that coincide on an open set U 3 m. If p is a plateau function around m in U [111.5.21], we have the global equalities p' Si = P . s~ (because supp(p) C U). Since p is even, p(m) = 1, and i!! is k-linear over COO(M), we compute:

i!!( Sl, ... ,Sk) (m) = p( m)k . i!!( Sl, ... , Sk)( m) = i!!(pSl, ... , pSk) (m)

= i!!(ps~, .. . ,ps~)(m) = i!!(s~, ... , sU(m)

.

This proves that the value of i!!(Sl,"" sk)(m) does not depend upon the particular Si, provided they coincide on an open neighborhood of m. To prove that it only depends upon the values si(m) we assume that Si and s~ are two sets of sections such that Si (m) = s~ (m). Let (U, 'l/J) be a tri vializing chart for B, let p be a plateau function around min U, and let V cUbe an open set such that plv == 1. If (ej)';=l is a basis of E, we define local sections fj E ru(B) by fj(m') = 'l/J-l(m',ej)'

E j fj . S{. Since the support of p is contained in U, the support of the s{ (which is by definition contained in U) is contained in the support of p, and thus supp (s~) is closed in M, hence the s{ can be regarded as smooth functions on M. By the same argument Pfj can be seen as a global smooth section of B [3.9]. We thus obtain the equality of global sections Hence there exist smooth functions s~ E COO(U) such that (psi)lu =

U sing the k-linearity of i!! we compute by the first step

where

£il ... i k (Xl,

(from PSj =

... ,Xk) is a function on Ak obtained by shifting the coefficients s~j

Ei e i J -

1

sJij

)

out of i!! by k-linearity. These functions can be inductively

defined by £i(X) = X and £i1, ... ,ik+l (Xl"'" Xk+l) = 1t"'(Xl) . £i2, ... ,ik+l (X2,"" Xk+l) with a = E( eiJ + ... + E( eik+l)' Obviously the value of £il ... ik (si\ ... , s1k ) at m depends only upon the values of the s;j at m. Since the same is true for the sections s~, it

177

§S. Operations on sections and on vector bundles

follows from the equality Si (m) = s~ (m) that ( S1, ... , Sk)( m) = ( si, ... , sU( m) as claimed . • To prove (ii), we can copy the proof of (i), except for the fact that the functions s{ need not be smooth. However, there is a much shorter proof. Suppose that Si and s~ are sections that coincide at m. Then we can define the (non-continuous) function p by p( m) = 1 and p( m') = 0 for all m' =I m. Then PSi = ps~ and hence linearity over C-1(M) proves (ii) . • To prove (iii) we start with uniqueness. So we suppose that W exists, and that Si are arbitrary sections. If (U, 'lj;), V, p, and sj i are as above (and note, the sj i need not be smooth), we know that

L W(P~i,,···, p~ik)(m) . £i, ... ik (si',···, Skk) = L (P~i,,··· ,p~iJ(m) . £i, ... ik (si',···, Skk) ,

W(S1,"" sk)(m) =

where the last equality is by the assumption WIr(B)k

=

. This proves that the element

W(S1,"" sk)(m) is completely determined by , hence it must be unique. On the other hand, we can use this formula to define W. More precisely, for sections Si : M -+ B and a point mE M we define W(S1,"" sk)(m) by

It remains to show that this is well defined, right k-linear over C-1(M), and graded skew-symmetric. To that end we note that by (i) the right hand side does not depend upon the specific choice for p, and hence neither does it depend upon U or V. An elementary but tedious computation using the explicit expression for £i, ... ik shows that it is also independent of the choice of the trivializing sections ~i (two different choices are related by a matrix of smooth functions, which can be brought inside
5.9 Proposition. J: r(Hom}f(B k ; C))

-+

Hom}f(r(B)k; r(C)) is an isomorphism of

graded C=(M)-modules. Proof Since we know that it is an even morphism of graded C=(M)-modules, it suffices to show that it is bijective. We will do this by constructing its inverse. This means that for E Hom}f(r(B)k; r(C)) we have to define a ¢ such that J(¢) = . For mE M the value of ¢(m) should be a right k-linear graded skew-symmetric map B~ -+ Cm. We can define this map by providing the images ¢(m)(b 1, ... , bk) with bi E Bm. Wedefine these by (5.10) where the Si E f(B) are such that si(m) = bi. It follows from [5.S-i] that this is well defined. The only problem is that there need not exist smooth sections satisfying the

Chapter IV. Bundles

178

condition Si (m) = bi (if m has real coordinates, Si (m) has real coordinates, but bi need not have real coordinates). However, in order to define a k-linear graded skew-symmetric map, we need not define it on the whole of B~, but only on a generating subset, and even a basis would suffice. Since the values SiC m) certainly generate B-m when Si varies over the smooth sections of B, (5.10) indeed defines a unique k-linear graded skew-symmetric map on Bm. We do not obtain a contradiction (the values si(m) are certainly over-complete) because Ii> itself is k-linear and graded skew-symmetric. To prove that this ¢ : M ---+ Homlf(B k ; C), m f---+ ¢(m) E Homlf(B~; Cm) is smooth, it suffices to show that in a trivializing chart (U, 'IjJ) the associated function ¢-.p : U ---+ Homlf(E k ; F)) is smooth. Now this function is smooth if and only if its coefficients with respect to a basis are smooth. If we use (U, 'IjJ), V, and p as in he proof of [5.S], these coefficients are given by ¢-.p (m) (eil , ... , eik) (or more precisely, by the coefficients of this vector with respect to a basis of F), and these are given on the neighborhood V (see above) by the formula

which is manifestly smooth.

5.11 Discussion. Let us introduce some names for spaces of right k-linear graded skewsymmetric morphisms: Homlf(r- 1(B)k; r-1(C)) will contain those that are linear over C-l(M), Homlf(r(B)k; r-l(C)) and Homlf(r(B)k; r(C)) those that are linear over C=(M). For any 1lf E Homlf(r-1(B)k; r-l(C)) we can take its restriction to r(B)k : Ii> = 1lflrCB)k E HomRk(r(B)k; r-l(C)). [5.S-iii] shows that the map 1lf f---+ Ii> is a bijection. On the other hand, Homlf(r(B)k; r(C)) is obviously a graded submodule of HomRk(r(B)k; r-1(C)). We conclude that we can interpret Homlf(r(B)k; r(C)) as a graded submodule of Homlf(r-1(B)k; r-1(C)). But the reader should be aware that the first space concerns morphisms linear over C= (M) and that the second one concerns morphisms linear over C-l(M). Looking at the definition of 'J in [5.7], it is obvious that it defines an identification r-1(Homlf(B k ; C)) ---+ Homlf(r-1(B)k; r-l(C)). Looking at the proof of [5.9], it is not hard to show that this is an isomorphism of graded C-l(M)-modules. The result of [5.9] can now be interpreted as saying that if we restrict attention to smooth sections in the source space, then we get an isomorphism onto the graded submodule Homlf(r(B)k;r(C)) C Homlf(r-1(B)k;r-1(C)). An interesting consequence is the following. Suppose we have an arbitrary section ¢ of Hom (Bk; C) and suppose that under the identification 'J it maps r(B)k to r( C), i.e., it maps smooth sections of B to smooth sections of C. Then ¢ must be smooth.

If

5.12 Definition. If h : B ---+ C is a (left or right linear) vector bundle morphism of bundles over the same base space, we define an induced map h* : r( B) ---+ r( C) by h* (s) = h 0 s. For a section S E reB), the section h*(s) E r(c) is usually called the pushforward of

§5. Operations on sections and on vector bundles

179

S by h. If h is left/right linear of parity a, the same is true for h* (which "obliges" us to write (( S II h*)) = soh in the left linear case). Also, if h is an isomorphism, so is h*. This applies especially to the isomorphisms given in [4.2S], which gives, for example, an isomorphism r(HomdB; C)) ~ r(*B 0 C).

5.13 Discussion. Let p : B ----+ M and q : C ----+ M be vector bundles over M. In [4.18] we argued that sections of HomdB; C) are in bijection with left linear vector bundle morphisms B ----+ C. With theidentification r(HomdB; C)) ~ Homdr(B); r(C)) and the push forward we can describe this bijection in a roundabout way. For a left linear vector bundle morphism h : B ----+ C we obtain a left linear morphism of graded C= (M)-modules h* E Homdr( B); r( C)). Under the identification of this space with r(HomdB; C)), we obtain an element J-1(h*) E r(HomL(B; C)). We leave it to the reader to verify that this is exactly the section of HomdB; C) we have defined in [4.18]. An interesting consequence of all these isomorphisms is that any morphism r( B) ----+ r( C) of graded C= (M)-modules is necessarily of the form h* for some vector bundle morphism h : B ----+ C.

5.14 Discussion. Using the bundle isomorphisms Homtk(Bk; C) ~ HomdN B; C) ~ *(N B) 0 C ~ N (*B) 0 C, the induced isomorphisms (by push forwards) on the modules of sections, and [S.9], we obtain an isomorphism between r(l\k (*B) 0 C) and Homtk(r(B)k; r(c)). If we take for C the trivial bundle C = M x A, we thus obtain the isomorphism

(5.15) and in particular for k = 1 :

r(*B)

~

*r(B)

and similarly

r(B*)

~

r(B)* .

One of the consequences of [S.9] for these particular examples is the following (see also [S.l1]). Let w : M ----+ I\k *B be any section and let Si : M ----+ B be smooth sections. If the function m f---+ L(sl(m), ... , sk(m))w(m) is smooth for all possible choices of the

Si,

then w must be smooth.

5.16 Construction. Let P : B ----+ M and q : C ----+ M be vector bundles with typical fibers E and F respectively. If S is a smooth section of Band t a smooth section of C, we define a section J( S 0 t) of B 0 C by

J(s 0 t)(m) = sCm) 0 t(m) E Brn 0 C m

~

(B 0 C)rn ,

where we have used that the fibers of B 0 C are canonically isomorphic to the tensor product of the corresponding fibers in Band C. Using arguments similar to those used for direct sums and a quotient, one can easily show that this defines indeed a smooth section and that J thus defined is an even morphism of graded C=(M)-modules.

180

Chapter IV. Bundles

5.17 Proposition. J: reB) 0r(C) modules.

----+

r(B0C) is an isomorphism of graded C=(M)-

Proof An elementary but tedious verification shows that we have the following commutative diagram:

reB) 0r(C) [1.8.5]1

reB 0 C)

~

~

r(*(B*)) 0 r(C)

[1.8.5]

r(*(B*) 0 C)

[5.14]1 ~

*r(B*) 0 r(c)

r r

~ ------t

[L7.11]

Homdr(B*); r(C))

~ ~

[1.8.8]

f(HomdB*; C)) ,

[5.9]

in which two arrows are not yet known to be isomorphisms. In [7.25] we will prove that for any vector bundle P : B ----+ M the graded C=(M)-module r(B) is f.g.p, and thus by [1.8.8] the identification *r(B*) 0 r(c) ----+ Homdr(B*); r(c)) is an isomorphism. Hence the last arrow must also be an isomorphism. IQEDI

5.18 Discussion. If we have a finite family Pi : Bi ----+ ]'\11 of vector bundles with typical fibers E i , it is by now obvious how to construct an identification J : 0ir(Bi) ----+ r( 0iBi) : if the Si are sections of B i , J (S1 0 ... 0 S k) is defined by

Since we know that this identification is an isomorphism of graded C=(M)-modules when k = 2, we can use induction and associativity of the tensor product to prove that it is an isomorphism for all kEN.

5.19 Construction. Let P : B ----+ M be a vector bundle with typical fiber E. If S1, are smooth sections of E, we construct a section J(S1 1\ ... 1\ Sk) of I\k E by

•.. , Sk

By now standard arguments show that this defines a smooth section and that the map N reB) ----+ r(N B) is an even morphism of graded C=(M)-modules.

J:

5.20 Proposition. J modules.

N reB)

----+

r(N B)

is an isomorphism of graded C=(M)-

181

§6. The pull-back of a section

Proof A rather tedious verification shows that the map 'J can also be written as the following series of isomorphisms/identifications: [1.8.5]

Nr(B)

~

[5.14]

N r(*(B*))

[1.7.16]

N *r(B*)

-----'>

*(N r(B*))

[5.9]

[1.5.5]

~ [1.5.5]

r(Homtk((B*)k; M x A))

[1.8.16]

[1.8.5]

r(N B).

In [7.25] we will show that r(B) is f.g.p [7.25], and thus by [1.8.16] the identification ----+ *(N r(B*)) is an isomorphism. IQEDI

N *r(B*)

6. THE

PULL-BACK OF A SECTION

In this section we introduce operations which are crucialfor differential geometry: the exterior or wedge product ofsections and the contraction ofa section ofa bundle with a section of the k-th exterior power of its dual bundle. We also introduce the notion ofthe pull-back ofa section and we show how this relates to the various operations discussed before.

6.1 Discussion. Let p : B ----+ M be a vector bundle with typical fiber E. In [1.5.8] we defined the wedge product as a bilinear map I\k E x 1\£ E ----+ I\kH E, factoring in a linear E ----+ I\kH E. It is straightforward to show that this map intertwines map I\k E 0 the corresponding transition functions [4.23], giving us an even bundle morphism I\k B 0 1\£ B ----+ I\kH B. Now taking the push forward of this morphism acting on sections, we can form the even bilinear morphism of graded C=(M)-moduJes

N

Tracing the various morphisms immediately shows that this map is given by the pointwise formula 'J t\ (a,;3) (m) = (a( m)) 1\ (;3( m)). In view of this pointwise identification we will denote this bilinear map 'J t\ : B) x B) ----+ B) also by just a wedge product: 'Jt\(a,;3) == a 1\;3. Since the original map for A-vector spaces (or better, family of maps, since they depend upon the powers k and £) satisfies the relation A 1\ B = (_l)(€(A),€(B))+kf.B 1\ A, it follows immediately from the fact that this identification is pointwise that the same is true for the induced map on sections as described above. It follows that the graded C=(M)-moduJe EBkEZ B) can be given the structure of a (Z x Z2-graded) C=(M)-algebra. Combined with the identifications N r(B) ----+ B), which is also pointwise, we thus obtain a morphism of graded

r(N

r(N

r(N+£

r(N

r(N

182

Chapter IV. Bundles

commutative Coo (M)-algebras 00

00

J : /\r(B) == E9N r(B) ---., E9r(N B) , k=O

where [5.20] shows that it is in fact an isomorphism. Note that we carefully did not write r(EBk /\k B), as EBk /\k B is not (because of the infinite sum) a vector bundle in our sense. For2t-graded A-modules E we have defined in [1.6.16] (see also [1.6.21]) a contraction operator [ : E 0/\k *E ---., /\k-1 *E such that, for a fixed e E E, the operator [(e) is a left derivation of /\ *E. It is elementary to verify that this map [ satisfies the conditions of [4.23] and thus induces a morphism of the corresponding bundles. Taking sections, we can construct the morphism

[ : r(B) x r(N *B) ---., r(B) 0 r(N *B) ---., r(B 0

N *B)

---., r(N-1 *B) .

There are two justifications for using the same symbol [ for this morphism on sections. In the first place, as the reader can check, this operation is given for s E r(B) and IP E r(N *B) by the pointwise formula

[(s)1P == [(s 01P) : m

f-*

[(s(m))IP(m) E N-1 *Bm .

It follows easily that we obtain an even morphism of graded Coo (M)-modules 00

[ : r(B) ---., DerdE9r(N *B)) . k=O

On the other hand, we could have applied [1.6.16] directly to the graded Coo (M)-module

*r(B) itself and have obtained a contraction operator [ : r(B) ---., Derd/\ *r(B)). We leave it (again) to the reader to show that these two contraction operators coincide under the identification /\ *r(B) ~ /\ r(*B) ---., EB~o r(N *B), i.e., that we have a commutative diagram

/\ *r(B)

1 Now let

S1, ... , Sk

.( s) -----+

.(s) -----+

/\ *r(B)

1

EB~o r(N *B) . EB~=o r(N *B) E r(B) be arbitrary, then we obtain, by composition, a map

On the other hand we have the evaluation map

We leave it to the reader to check (as always using that all identifications are pointwise) that these two maps coincide under the identification r(N *B) ~ HOffiLk(r(B); Coo(M)) (5.15) (see also [1.7.21]).

§6. The pull-back of a section

183

6.2 Discussion. Since all our identifications are pointwise, they commute with restrictions to submanifolds (and thus in particular with restrictions to open subsets). Let us elucidate this by a few examples, in which N c M is a submanifold. Let Band C be vector bundles over M and let s E reB) and t E r(C) be smooth sections. Then the identification in [5.2] satisfies J(sIN EB tiN) = (J(s EB t))IN ,

obviously because for mEN we have sIN(m) = sCm) and tIN(m) = t(m). In a similar way, if ¢ E r(HomFf(B k ; C)) ~ r(N B* 0 C) and Si E reB) are smooth sections, then, using the identification J from [5.7], we have

As a final example, consider ¢ E

r(N B*)

and

S

E reB) smooth, then

6.3 Definition. So far we have always considered bundles over the same base space, but now we will look at bundles over different base spaces. Let p : B ---+ M and q : C ---+ N be vector bundles and let h : B ---+ C be a right linear vector bundle morphism inducing the map 9 : M ---+ N on the base spaces. For each m E M we have a right linear map hm = hlB~ : Bm ---+ Cg(m)' and thus we can form its right dual h;" : C;(m) ---+ B;". However, the collection of all h;" does not define a vector bundle map C* ---+ B* unless 9 is bijective: it is only defined on fibers of the form Cg(m), missing fibers of C* if 9 is not surjective, it could be multiply defined if 9 is not injective, and it does not induce a well defined map N ---+ M (unless g-1 exists and is smooth). On the other hand, if we take a section r : N ---+ C* of the right dual bundle of C, we can form a section (( r I h * > : M ---+ B* of the right dual of B by the formula

(rllh*))(m) = (r(g(m)) Ilh;")) = r(g(m)) ohm: Bm

(6.4)

---+

Cg(m)

---+

A,

or equivalently by the commutative diagram h*

B;" 3 ((r(g(m)) I h;,,> ~ r(g(m)) E C;(m)

((TW»I m

IT ~

gem).

9

The section (( r I h*)) is called the pull-back of r by h. By varying r we obtain a map 1 1 h* : (C*) ---+ (B*) which is called the pull-back map. It follows easily from the definition that h* is additive and verifies for any ¢ : N ---+ A :

r-

r-

(¢. rllh*)) = (¢og). (rllh*> .

Chapter IV. Bundles

184

This looks like a left linear map (the reason why we wrote it as (( r II h*))), but not quite: r-1(c*) is a graded module over C-l(N), while r-1(B*) is a graded module over C-1(M). This is also apparent from the factthat on the right we have to multiply by ¢ 0 9 instead of just by g. We now show that if the section r is smooth, i.e., r E r( C*), then the pull-back (( r II h *)) is also smooth. For that we evaluate on a smooth section s E r (B) according to [5.14], which gives

«rllh*))(s): m f-* (<
= r(g(m))(h(s(m))).

Since this is smooth when s and r are smooth (g and h are smooth from the beginning), we conclude by [5.14] that (( r II h*)) is smooth. We thus have shown that h* maps r(C*) to r(B*). If h is left linear instead of right linear, we can construct in a similar way a pull-back map *h : r-1(*C) ---+ r-1(*B) by the formula

*h(r)( m) = *(h m)( r(g(m))) = hm <> r(g(m)) = r(g(m)) 0 hm : Bm or equivalently by its action on a section s E

r- 1 (B)

---+

A,

:

«sll*h(r))): mf-* Hs(m) II h)) Ilr(g(m)))) . It can be shown easily that *h is additive and verifies for any ¢ : N

*h(a. ¢)

=

---+

A:

(*h(a)) . (¢o g) .

And just as for the right linear case, *h maps smooth sections to smooth sections, i.e., *h : r(*C) ---+ r(*B). It should be noted that, although h* and *h have many properties that resemble the properties of right and left dual maps, some properties are not shared. In particular a bi-dual for vector bundle morphisms does not exists, simply because h* and *h are not morphisms of modules over the same ring. The fact that we used a notation as if they were dual maps will never leads to confusion because a vector bundle morphism simply is not a morphism of modules.

6.5 Remark. Using the construction of pull-back bundles, the pull-back map can be decomposed into a standard pull-back map to the pull-back bundle, and a part involving bundles over the same base space. In the case of pull-back bundles, one can "pull back" any section, not only those of the dual bundle (restriction is the typical example). And then, when one is in bundles over the same base space, one can apply the dual map. Although such a picture is aesthetically very pleasing, we will never need it, so we leave it to the interested reader to pursue this line of thought on his own.

§6. The pull-back of a section

185

6.6 Discussion. In [6.2] we argued by examples that all identifications commuted with restriction. For the pull-back map, although also pointwise in a certain sense, this is not precisely the case because the bundles are not over the same base space. Nevertheless, some kind of commutativity still holds for pull-back maps. To be more precise, let p : B ---+ M and q : C ---+ N be vector bundles and let h : B ---+ C be a right linear vector bundle morphism inducing 9 : M ---+ N. Now suppose that M c M is a submanifold. If we denote (slight abuse of notation) by hi if the restriction of h to B Iif (the restriction

of the bundle B to M), then it follows immediately from the definition (6.4) that for any section T E r (C*) we have (hlif)*T = (h*T)lif . ~

On the other hand, if N verification shows that

c N

~

is a submanifold such that g(M) C N, then again a direct

where on the left we interpret h as a right linear vector bundle morphism from B to

CI N

(which is allowed because g(M) C N; if one really wants to be precise, one should use [Y.2.17] and [Y.2.18] to show that this map is smooth). This result is due, of course, to the fact that the pull-back map is essentially the collection of dual maps h";" defined on fibers of the form Cg(m)' It is in these two senses that the pull-back map commutes with restrictions.

6.7 Discussion. There are two interesting special cases to consider for the pull-back map. The first concerns the case when the vector bundles Band C are over the same base space M. So let h : B ---+ C be a right linear vector bundle morphism (which induces the identity on M!), i.e., for each m E M we have a right linear map Bm ---+ Cm. We thus can form the right dual map (h m)* : (Cm )* ---+ (Bm)*. The collection of all these right dual maps forms a left linear vector bundle morphism f : C* ---+ B*. A careful look at the definition of the pull-back shows that <(J" I h*)) is the composition of (J" E r( CO) with this f. Comparing this with the definition of the push forward map [5.12], we see that in case the bundles are over the same base space, the pull-back map is exactly the same as the push forward of the dual map. Another way to see this phenomenon is to play around with the various identifications. For A-vector spaces E and F we can identify HomR(E; P) with HomL(P*; E*) via HomR(E; P) ~ P 0 E* ~ *(P*) 0 E* ~ HomL(P*; E*), an identification which is explicitly given by taking the right dual. In [4.18] we proved that right linear vector bundle morphisms are in bijection with sections ofHomR (B; C). Since our identifications remain valid at the vector bundle level [4.25], we thus have an identification of a right linear vector bundle morphism B ---+ C with a left linear vector bundle morphism C* ---+ B*. And again the isomorphism is given by taking the right dual, in agreement with the previous discussion. The second interesting case is when both bundles are the trivial bundle with fiber A. A right linear vector bundle morphism h : M x A ---+ N x A which induces the map

Chapter IV. Bundles

186

---+ N must necessarily be of the form hem, a) = (g(m), f(m)a) for some smooth function f : M ---+ A. On the other hand, sections of M x A and its dual are just functions on M, the same being true for N. Moreover, evaluation of a dual "function" on a normal "function" is just pointwise multiplication of those functions. An elementary computation then shows that

9:M

((O"llh*)) = (O"og). f , where we identified sections with functions. In the applications of this result, the function representing the right linear vector bundle morphism h, usually is identically 1, i.e., h( m, a) = (g( m), a), in which case the formula simplifies to <0" II h*)) = 0" 0 g. With a similar argument a left linear vector bundle morphism is given as h( m, a) = (g(m), af(m)), and the pull-back map is given by *h(O") = f . (0" 0 g) == f· (g <> 0").

f,

6.8 Discussion. Once we know of the existence of the pull-back map, we might recall that we also have constructions of new morphisms out of families of old ones, both for morphisms of graded COO(M)-modules (in §1.7) and for morphisms of vector bundles (see [4.26] and [4.27]). The natural question is: how do these constructions behave under taking the pull-back map? The answer is that they behave as one would expect (see also [1.7.24] and [1.7.25]), except that this expected behavior requires some of the identifications we met before. [6.9] gives the precise result for exterior powers; similar results for the other constructions are left to the reader.

6.9 Proposition. Let p : B ---+ M be a vector bundle over M, let q : C ---+ N be a vector bundle over N, and let h : B ---+ C be an even (left) linear vector bundle morphism inducing 9 : M ---+ N. Then the map *(I\k h) corresponds to the map I\k (*h) under the r(*B) ---+ r(*(N B)) and 'Jc : r(*C) ---+ r(*(N C)), i.e., identifications 'JB : we have a commutative diagram

N

N

N r(*C) Jc

N Ch)

-----+)

1

N r(*B)

lJB

r(*(N C)) ----;) r(*(N B)) . *(Nh)

Proof Let us first start to analyze in more detail the identification 'J B, which decomposes into two parts. The first part is the map 'J : I\k r(*B) ---+ r(N *B), which sends O"k 1\ ... 1\ 0"1 E r(*B) to the section J(O"k 1\ ... 1\ O"d given in any point mE M by J( O"k 1\ ... 1\ O"d( m) = O"k (m) 1\ ... 1\ 0"1 (m). We then use (1.7.19) to identify this section with a section of * (I\k B). More precisely, we have

N

where 'J,\ is the map 'J defined in [1.7.16]. We added the subscript 1\ to distinguish this map from the other identifications.

§7. Metrics on vector bundles

187

We are now ready to prove the commutativity of the diagram. For Ti E r(*C) we have

(JB 0 N (*h))(Tk

1\ ... 1\

Tl)(m)

= JI\(*h(Tk)(m) 1\ ... 1\ *h(TI)(m)) = JI\(Tk(g(m)) 0 hm 1\ ... 1\ Tl(g(m)) 0 hm) = JI\(Tk(g(m)) 1\ ... 1\ Tl(g(m))) 0 N hm = Jc(Tk 1\ ... 1\ TI)(g(m)) 0 (N h)m =(*(Nh)oJe)(Tkl\···I\TI).

IQEDI

6.10 Discussion. Let p : B

----+ M be a vector bundle over M, let q : C ----+ N be a vector bundle over N, and let h : B ----+ C be an even (left) linear vector bundle morphism Band *B are isomorphic (see [4.25]), inducing 9 : M ----+ N. Since the bundles * we can interpret the pull-back map *(I\k h) as a map

N

*(N h) : r(N *C)

N

----+

r(N *B) .

Now let a E r(N *C) and (3 E r(N *C) be arbitrary, then we can form the wedge product a 1\ (3 E r(l\k H *C) according to [6.1]. Using that the identifications J Band Je of [6.9] are isomorphisms, it follows immediately from [1.7.15] and [6.9] that we have the equality

*(NH h)(a 1\ (3) = *(N h)(a)

1\

*(N h) ((3) .

An other way to prove this equality without using that J B and Je are isomorphisms is the following pointwise argument. In [6.1] we have seen that (a 1\ (3)g(m) = ag(m) 1\ (3g(m) , and in [6.3] that (*(N h)a)(m) = (*(N h)m) (ag(m») = (*(N hm)) (ag(m»). And then we obtain immediately

*(NH hm)(a 1\ (3)g(m) (NH *hm)(ag(m)

1\

(3g(m»)

(N *hm)ag(m) 1\ (N *hm)(3g(m) (*(N h)(a) 1\ *(N h)((3))(m) , where in the first line we have used the identification of I\k+£ *Cg(m) with * I\kH Cg(m).

7.

METRICS ON VECTOR BUNDLES

In this section we introduce the notion ofa (pseudo) metric on an A-vector space and we show its canonicalform. We then use a metric to define the notion ofan orthogonal complement to a subspace ofan A-vector space. These results can be implemented on vector bundles to show that a metric on a vector bundle can be used to define a supplement to a given vector subbundle. This in turn we use to show that the module ofsections ofa vector bundle is afinitely generated projective module over the ring of smooth functions on the base space.

ChapterIV. Bundles

188

7.1 Definition. We define the complexified ring A C as the A-vector space A2 with multiplication (a, b) . (a', b') = (a a' - bb', ab' + ba'). Introducing as usual the notation 1 == (1,0), i == (0,1) E AC, an element (a, b) E A C can be written as (a, b) = a + ib. On A C we define an involution (complex conjugation) by

a + ib = a - ib. A direct verification shows that this involution is a ring morphism. Moreover, the body map B : A C == A2 ---+ R 2 can be interpreted as a surjective ring morphism B : A C ---+ C, which respects complex conjugation.

7.2 Remark. In the (mainly physics) literature one can find a different definition of complex conjugation, which involves an internal operation on A (changing the order in the exterior powers of A ~ 1\ X). In our context this internal operation has a disadvantage: either it is not a smooth map on A in the sense of [III. 1.20], or, redefining A as the subset of A C on which complex conjugation is the identity, A is not a ring.

7.3 Definition. Let E be a free graded A-module. An even bilinear map 9 : Ex E is called apseudo metrict on E ifit satisfies:

---+

AC

(i) for all homogeneous e, fEE: gee, I) = (-l)(€(e)I€(f»)g(f, e); (ii) Bg : BE x BE ---+ C is non-degenerate; (iii) for all homogeneous e, fEE: B(g(e, I)) = (-l)(e(e)I€(f))B(g(e, I)).

7.4 Discussion. Let us investigate the conditions in more detail. The first one is the natural definition of symmetry (of a hermitian metric), the sign appears because we changed the order of the elements e and f. The second condition is the natural definition of nondegeneracy. The third one is a strange one: the sign is not justified by a change of order of the elements e and f; apart from this sign, it says that Bg should take its values in R. Using condition (i), condition (iii) can be rephrased as Bg( e, I) = Bg(f, e), which says that Bg is real symmetric, whereas condition (i) says that 9 itself should be hermitian symmetric. And of course the strangeness did not disappear: e and f are interchanged without an additional sign.

7.5 Lemma. Let E be afinite dimensional free graded A-module of dimension n = plq, and let (ei)r=l be an ordered homogeneous basis. If 9 : Ex E ---+ A C is a pseudo metric tOur free graded A-modules are essentially "real" in the sense that they are in particular vector spaces over R, but not (necessarily) over C. It thus will be strange to see a definition of a (pseudo) metric in the sense of vector spaces over C. However, I challenge the reader to find a "natural" definition of a "real" metric such that a linear combination of metrics with positive coefficients is again a metric.

§7. Metrics on vector bundles

189

on E, it defines a matrix 9ij in M(plq, A C ) by

9ij=g(ei,ej)~(~ ~), with P of size p x p and T ofsize q x q. The matrices P, Q, R, and T verify P ij = P ji , Tij = -Tji' Rij = Qji' Moreover, BP and -iBT are non-degenerate symmetric R-valued matrices. Conversely, given matrices P, Q, R, and T verifying these properties, then the bilinear map 9 : E x E ---+ A C defined by g(2:i Xi . ei,

2: j ej . yj) = L

.. xi . gij . yj

'J

(note the order in the coefficients!) is a pseudo metric on E. Proof The properties of P, Q, R, and T follow immediately from the first condition on a pseudo metric. The properties of BP and -iBT follow from this and the last two conditions. The converse is a direct (tedious) verification, using that the coefficients xi and yj are A-valued and not AC-valued. And note that among the properties of the matrices P, Q, R, and T is the property that the entries of P and T are even and that the entries of Q and R are odd (g represents an even bilinear map). IQEDI

7.6 Discussion. By definition, a pseudo metric on an A-vector space E can be seen as an element ofHom(E 0 E; AC)o. Using our identifications [1.8.8], this space is isomorphic either to (*(E 0 E) 0 AC)o (when we see it as left linear), or to (A C 0 (E 0 E)*)o (when we see it as right linear). Using the identifications * (E 0 E) ~ *E 0 *E and (E 0 E)* ~ E* 0 E* and using the dual bases (ie) of *E and (e i ) of E*, we find that a pseudo metric is represented by j 9 = 2:ij 9ij 0 e 0 ei = 2:ij je 0 ie 0 gij ,

where we interpret the gij as elements of the free graded A-module A c. The order of the indices of gij in these formulae is a consequence of our definition of the identification given in [1.7.12]. That we obtain the same coefficients in both expressions is a direct consequence of the fact that 9 is even; it can also be understood in terms of the identity 'I(ie) = (_l)(€(ei)I€(ei»)e i [11.4.5]. Let us now investigate in more detail how the matrix elements of 9 depend upon the basis (ei) of E. So let (ei) and (Ij) be two (ordered homogeneous) bases of E related to each other by a matrix

Ai j

~ (~ ~)

E

M(plq, A) as Ii =

2:i ei Aij

(note the choice of

order). Invariance of graded dimension tells us that the entries of A and D are in Ao and those of Band C in AI. If we denote by mate(g) and mat! (g) the matrices of 9 with respect to the bases e and f respectively, then an elementary calculation shows (7.7)

_ct) Dt .

e

mat (g).

(AC D B) '

where (as usual) the superscript t denotes the ordinary transpose of a matrix, i.e., interchanging the roles of row and column index.

190

Chapter IV. Bundles

7.8 Proposition. Let 9 be a pseudo metric on a finite dimensionalfree graded A-module E, then there exists a basis in which the matrix representation of 9 is given as

mat (g) = (So

+0 iX

0)

iS1 + Y

,

where So and Sl are diagonal matrices with only ± 1 on the diagonal, and where X and Yare skew-symmetric matrices (i.e., whose transpose is their own negative) with entries in No C Ao. Proof If M is a symmetric non-degenerate matrix with entries in R, a standard result of linear algebra tells us that there exists an invertible matrix T with real entries such that Tt . M . T is a diagonal matrix with only ±l on the diagonal. Applying this to a pseudo metric g, it follows from [7.5] that there exists a basis f, linked to the basis e by a real valued A (i.e., A and D real valued, Band C zero) such that

mat f (g) = (So

(7.9)

zt+ X

where So and Sl are diagonal matrices with only ±l on the diagonal (and thus S;l = Si), and where X, Yand Z are matrices with nilpotent entries (in A C ). The change of basis from e to f can be obtained by the Gram-Schmidt orthogonalization process on Bg. Our goal is to prove that we can find a basis in which Z is zero and in which X and Y have a special form. The idea is to apply a Gram-Schmidt type of procedure. The problem is that a change of basis has to be done with A-valued matrices, and that mat(g) is AC-valued. And indeed, if we were allowed to use AC-valued matrices A to change a basis, we could easily eliminate X, Y, and Z completely. In order to obtain our goal, we define for any square matrix N with even nilpotent entries a new matrix (1 + N)-1/2 as a power series in N with coefficients defined by the power series expansion of the real valued function x f-* (1 + x) -1/2. This is well defined because a matrix with nilpotent entries is nilpotent and thus any power series is finite. We also introduce three special types of coordinate changes: type I type II : type III:

( AC DB) (~ ~) (cA DB) ( ido idB)

(cA

DB)

id ( C

BA = 1,

BD = 1,

0)

id

The last ingredient we will need are the subsets N(k) c A C which consist of all finite sums/differences of products of at least k entries of the matrices X, Y, and Z in (7.9). Since the entries of these matrices are nilpotent (and because there are only finitely many of them), it follows that there is a kEN such that N(k) = {O}. Although we will frequently change the basis while keeping the general form (7.9) of mat(g), the subsets N(k) are defined once and for all in terms of the initial (sub)matrices X, Y, and Z.

§7. Metrics on vector bundles

191

We will denote by Re and 1m the operations that give the real and imaginary part of an element in A C ; we will say that X (respectively Y or Z) is homogeneous if either Re(X) or Im(X) is zero (and remember that the entries of these matrices are already homogeneous with respect to the dichotomy even-odd). It should be clear that a priori these matrices are not homogeneous. We now start our analysis by noting that initially the matrix Z has entries in N( 1). Our strategy will be to change bases such that the matrix Z will belong to N(k) with steadily increasing k. To simplify notations, we will say that Z has order k if its entries belong to N(k). Our first claim is that we may assume that Re(X) and Im(Y) are zero (and thus X and Yare homogeneous). This result can be accomplished by a type I change, using A = So . (1 + Re(X) . SO)-1/2 and D = Sl . (1 + Im(Y) . Sd- 1/ 2. In order to prove the result, one should use that products of power series in Re(X) . So commute, that So' (Re(X) . So)£ = (So' Re(X))£ . So, and that So and Re(X) are symmetric (i.e., the same as their transpose). Apart from the result on X and Y, it should be noted that Z transforms to At Z D, and thus that the order of Z and the eventual homogeneity of Z do not change under this change of basis. The next claim is that, if Re(X) is zero, then we may assume that Re(Z) is zero. To obtain this result, we perform a type II change with B = - So . Re( Z), which transforms Z to Z - Re(Z) - X So Re(Z). This change of basis will destroy the homogeneity of X and Y. Moreover, ifIm(Z) were zero before, it will be non-zero after, but of at least one order higher than before. The third claim is that, if Im( Y) is zero, then we may assume that Im(Z) is zero. This can be obtained by a type III change with C = S1 . (Im(Z))t, which transforms Z to Z - ilm(Z) - Im(Z)S1Y. As before, this destroys the homogeneity of X and Y. And as before, if Re(Z) were zero before, it will be non-zero after, but of at least one order higher than before. We now are in business for our induction. Applying the first and second argument, we may assume that Re(Z) is zero. Ifwe then apply, in this order, the first, the third, the first, and finally the second argument, we again have a Z with Re(Z) zero, but with its order increased by at least 2. Since N(k) = {o} if k is sufficiently large, it follows that after a finite number of basis changes, we obtain a matrix representation of the pseudo metric 9 of the form (7.9) with Z being zero. Applying once more the first argument, we also may assume that Re(X) and Im(Y) are zero. IQEDI

7.10 Discussion. Once we have the notion of a pseudo metric 9 on a finite dimensional free graded A-moduleE, we would like to define the notion of the orthogonal p.L of a graded subspaceP c E. Anaivedefinitionwouldbep.L = {e EEl Vf E P: g(e,!) = 0}.1t is easy to show that this p.L is a graded submodule of E, but unfortunately, even in simple cases, it need not be a graded subspace. For instance, consider E = A2 of dimension 210, letP c E be the graded subspace generated by (l,0), and let the pseudo metric 9 be

-~n i~) for some even non-zero nilpotent N. An immediate computation learns that p.L = { (0, b) 1 b E A, bn = O}. This

given in the basis (l, 0), (0,1) by the matrix (

n E

192

ChapterIV. Bundles

is indeed a graded submodule, but it is not itself a free graded A-module, i.e., it is not a graded subspace. Since we still want to create new graded subspaces out of old ones by means of the notion of orthogonal, we need another definition of orthogonal. Before we can state, in [7.15], a definition that works, we need some preparations.

7.11 Definition. Let 9 be a pseudo metric on E. A basis (ei)f=l is called an orthonormal basis with respect to 9 if the matrix of 9 with respect to the basis e has the form given in [7.8].

7.12 Lemma. Let 9 be a pseudo metric on a finite dimensional free graded A-module E, and let (ei)f=l be an orthonormal basis with respect to g. /f(

~ ~) describes a change

of basis such that the new basis is again orthonormal, then necessarily Band C are zero. Proof If the old and the new basis are orthonormal, then especially the Z part of mat (g) in the new basis must be zero. According to (7.7), this is given by the equation

At. (So

+ iX) . B -

ct. (iS1

+ Y)

.D = 0 .

Splitting this equation into its real and imaginary parts, we obtain the two equations

At. So' B = ct. y. Dand At. X· B

=

ct. Sl' D. Since (~~) describes a change

of basis, it is invertible. Since it is also even, this implies that A and D must be invertible. We then obtain the equation C t = At . X . B . D-1 . Sl and thus

B = (So' X) . B . (D- 1 . Sl . Y . D) . Iterating this equation gives us the equality (7.13) Since X and thus So' X is nilpotent, there is a kEN such that (So' X)k zero. The equation for C t then tells us that C also is zero.

= 0, i.e.,

B is IQEDI

7.14 Definition. A pseudo metric 9 on a finite dimensional free graded A-module E is called a metric on E if the restriction ofBg to BEo and the restriction of -iBg to BEl are positive definite. This definition makes sense because in terms of a basis, these restrictions are represented by the matrices BP and -iBT [7.5]. Since these are symmetric and R-valued, we can require that they are positive definite. Moreover, under a change of basis these matrices change to BAt. BP . BA and -iBDt . BT . BD (7.7), and thus the notion of being positive definite is independent of the choice of a basis. Note that a pseudo metric 9 is a metric if and only iffor an orthonormal basis the matrices So and Sl of [7.8] are the identity.

§7. Metrics on vector bundles

193

7.15 Definition. Let 9 be a metric on a finite dimensional free graded A-module E and let F be a graded subspace of E. It is immediate that the restriction of 9 to F defines a metric on F (the same is in general not true for a pseudo metric for which the restriction to a graded subspace might be degenerate). Let (ei)7=1 be an orthonormal basis of F. We define the orthogonal complement F.L C E by (7.16)

s(ei) = 0 =} Re(g(e, ei)) = 0 s(ei) = 1 =} Im(g(e,ei)) = 0

}.

7.17 Lemma. The subset F.L C E is a graded subspace of E, independent of the orthonormal basis (ei)7=1 of F. Moreover, it is a supplement to F. Proof It is immediate from the definition of F.L that it is a graded submodule of E. According to [7.12] two orthonormal bases are related in such a way that even (respectively odd) vectors in the new basis are a linear combination of the even (respectively odd) vectors in the old basis. This implies that F.L is independent of the specific orthonormal basis for F. To finish the proof, it thus suffices to prove that we can extend the basis (ei)7=1 for F to a basis (ei )r=l for E such that (ei )r=k+l is a basis for F.L . Since F is a graded subspace of E, we can extend the orthonormal basis for F to a basis for E [11.3.7]. Let this basis of E be ordered in such a way that within the subsets (ei )f=l and (ei)r=k+l the even ones come before the odd ones. In this basis, the matrix of 9 looks like

o il + *+ i *+ i

* * *

* + i* * + i* *l< + *l< +

i i

* + i* ) * + i*

*l<

*l<+~*l<

*l<

*l< +

'

2 *l<

where ** denotes an arbitrary matrix with entries in A and where * denotes an arbitrary matrix with entries in N c A. We now modify the vectors ei>k by the formula (7.18) It is obvious that el, ... , ek, ek+l, ... ,en still form a basis for E and then (using that g( ej, ej) = 1 for j :S k and ej even and g( ej, ej) = i for j :S k and ej odd) a direct computation shows that in this basis 9 looks like

o (7.19)

il

+*

* *

From this matrix form of 9 it is immediate that the (ei)r=k+l form a basis for F.L.

IQEDI

194

Chapter IV. Bundles

7.20 Definition. Let p : B ---+ M be a vector bundle over an A-manifold M. A (pseudo) metric on B is a smooth section 9 of the bundle A C 0 (B 0 B)* (where A C denotes the trivial bundle M x A C ---+ M), such that for each mE M the value gem) is a (pseudo) metric on the finite dimensional free graded A-module Bm = p-1(m).

7.21 Proposition. Let p : B there exists a metric on B.

---+

M be a vector bundle over M with typical fiber E, then

Proof Let E be the typical fiber of B, let U = { Ua I a E J} be a tri vializing atlas for the bundle A C 0 (B 0 B)*, and let {Pa} be a partition of unity subordinated to U. According to [1.20] we have an identification ru" (A c 0 (B 0 B)*) ~ Coo(Ua; A C 0 (E 0 E)*). Moreover, on the finite dimensional A-vector space E we have a more or less obvious

choice go for a metric: the one represented by the matrix

(~ i~)'

Since the metric go

C

has real coordinates, the constant local section ga : Ua ---+ A 0 (E 0 E)* given by ga (m) = go is smooth. We thus can form the global section 9 = 2:a Pa . ga [3.9-iii]. Since the Pa take their values in Ao, not A~, conditions (i) and (iii) of a pseudo metric are automatically satisfied on each Bm. Remains to prove that the restrictions Bg to B(Bm)o and -iBg to B(Bmh are positive definite. But for each mE M, Bg is a finite sum of terms (Bpa)' (Bga) (the Pa are a partition of unity, thus only finitely many of them are non-zero at a time). Since each of the ga is a metric, a finite sum with positive coefficients of positive definite forms is again positive definite. And thus g( m) is a metric. IQEDI

7.22 Proposition. Let p : B ---+ M be a vector bundle, let 9 be a metric on B, and let C c B be a subbundle [3.12J. Then there exists a well defined subbundle C.L C B which is a supplement to C [4.6J. Proof For each m E M, gem) is a metric on the finite dimensional free graded Amodule Bm and C m is a graded subspace of Bm. We thus can define C.L as a set by (C.L)m = (Cm).L. We "only" have to show that this subset C.L is a subbundle and a supplement to C. To prove that, we will reconsider the proofs of [7.8] and [7.17] in order to check that all basis changes depend in a smooth way on the point m. In doing so, we will use a different kind of degree than that used in [7.8]. If I : U ---+ A is a smooth function, we know that I decomposes as I = 2:i Ii ~i in the notational shorthand of [111.3.17]. We will say that I has degree d if I = 2:i>d Ii ~i. From this definition it is obvious that if I has degree d 1 and 9 has degree d2 , then their product f . 9 has degree d 1 + d2 . And then the details. Since C is a subbundle of B, there exists a triviaJizing atlas U and, for each U E U, there exist local trivializing sections 81, ... ,8 n E ru (B) [3.14] such that 81, ... ,8k are local trivializing sections for C. Following [7.8] we want to modify the local sections 81, ... , 8k so that the 81 (m), ... , 8k (m) form an orthonormal basis of Cm for all m E U. The first step to take is to get them into the form (7.9). This is done by a standard Gram-Schmidt process applied to Bg( m), which is described by an upper triangular matrix whose entries

195

§7. Metrics on vector bundles

are rational functions in the matrix entries of Bg( m). It follows that we can perform this operation on our smooth sections SI, ... , Sk and obtain new smooth trivializing sections for C. Since in our notational shorthand Bg = go, it follows that go is a constant diagonal matrix with 1 or i on the diagonal. It follows that the entries of the matrices X and Y are smooth functions on U of order at least 2, and the entries of Z have at least order 1. Rence the power series definition of a type I coordinate change necessarily breaks of after the q /2-th term (q being the odd dimension of M). This implies that all three types of basis changes preserve smoothness of the local sections. Moreover, a type II or III change will increase the degree of (the entries of) Z by at least 1, and thus after q steps we have found local sections (Si)~=1 such that they form an orthonormal basis at each Cm. Up till now we have not touched the local sections (Si)r=k+l of B. We now modify these sections according to (7.18), i.e., we replace the vectors ei in (7.18) by the elements si(m). Since the functions m f---+ Re(g(m)(si(m), sj(m))/g(m)(sj(m), 8j(m))) are smooth, the new sections (8i)r=k+l are again smooth local sections of B. According to [7.17], they form a set oflocal trivializing sections for C1-. We conclude that C1- is indeed a subbundle and a supplement to C. IQEDI

7.23 Theorem. [ORV, Ch 1.1, Thm 1] Let M be an R-manifold and Uan open cover of M. Then there exists a dEN and an open cover V = {Vij I i, j EN, j :S d} such that (i) for all V E V there is an U E U such that V C U, (ii) i =f. if = } Vij n Vi, j = ¢.

7.24 Corollary. Let p : B ----+ M a vector bundle over an A-manifold M. Then there exists a finite number of global smooth homogeneous sections SI, . .. ,Sr E r( B) such thatfor all m E M the SI (m), ... , Sr( m) generate thefree graded A-module Bm. Proof Let U = { Ucr. I a E J} be a trivializing atlas for B. It follows that BU is an open cover of the R-manifold BM. Since B is a bijection between open sets in M and open sets in BM, there exists an open cover V = {Vi j I i, j E N, j :S d} of M such that BV has the properties mentioned in [7.23] with respect to BU. But then V has these properties with respect to U. Now define the open sets WI, ... , Wd as Wj = UiEN Vij. Since U is a trivializing atlas, there exist local trivializing sections s1, ... , s~ E ru" (B). By [7.23-i] there exists for each (ij) at least one a( ij) such that Vij c Ucr.Cij). We thus can define the

local tri vializing sections

s{, ... , s~ E r Wj (B) by

S{ restricted to Vij = s~(ij) This is a correct definition because the

Vij

restricted to Vij

.

are disjoint for fixedj by [7.23-ii]. Finally, let

Pj be a partition of unity subordinated to the (finite) open cover WI, ... , W d . The global

sections Pj . s{ are homogeneous, finite in number (there are r = nd of them) and at each point they generate Bm because there is at least one B Pj (m) different from zero, and then the Pj . s{ with this j form a basis of Bm. IQEDI

196

Chapter IV. Bundles

7.25 Theorem. Let p : B ----+ M be a vector bundle over an A-manifold M. Then reB) is afg.p graded C=(M)-module. Proof Let S1, ... , Sr E reB) be homogeneous global sections as in [7.24] and let F be the A-vector space generated by the abstract elements Si. To distinguish the abstract basis of F from the actual sections, we will denote the former as sf. We then define a vector bundle morphism ¢ : M x F ----+ B from the trivial bundle M x F to B by

Since the si(m) generate the fiber Bm, it follows easily that ¢ is a smooth surjective even bundle morphism. By [3.18] we deduce that C = ker( ¢) is a subbundle of M x F. By [7.22] we can find a subbundle C.L of B such that C EB CJ. = M x F. By definition of C, the restriction of ¢ to C.L is a vector bundle isomorphism between C.L and B. We thus obtain the sequence of isomorphisms [5.3]

r(C) EB reB) ~ r(c) EB r(c.L) ~ r(c EB c.L) ~ reM x F) ~ C=(M; F) , from which we deduce that reB) is f.g.p (because C=(M; F) is a finitely generated free IQEDI graded C=(M)-module).

8.

BATCHELOR'S THEOREM

In this section we prove Batchelor's theorem which says that for any A-manifold one can find an atlas such that the transition functions have a special form: even coordinates in one chart depend only upon the even coordinates in another chart and odd coordinates in one chart depend linearly on the odd coordinates in another chart (and on the even coordinates in the other chart). This can be rephrased as saying that an A-manifold is a vector bundle over an R-manifold in the ordinary sense and that smooth functions correspond to sections ofthe exterior algebra ofthis ordinary vector bundle.

8.1 Remark. Theorem [8.2] could have been stated in §IIIA and proved in §III.5. However, the interpretation [8.8] requires the notion of a vector bundle, so we postponed it till now.

8.2 Theorem (Batchelor [Ba]). IfM is an A-manifold, then there exists an atlas U = {tpa: Ua ----+ Oa I a E I} such that the transition functions (Xb,';b) = (tpb 0 tp;;1)(xa,';a) are of the form xi

.;~

=

=

fi(x a )

Lgk(Xa)'';~ . k

197

§8. Batchelor's theorem

Proof In this proof we will use extensively [111.2.23], [111.2.27] and the notational shorthand introduced in [111.3.17]. The general idea of the proof is that given any atlas U = { 'Pa : Ua ----+ Oa I a E I}, we will find a new atlas [j with the same sets Ua, Oa and Oab, but with different maps ipa such that the transition functions are given as claimed in the theorem. We divide the proof in two steps. In the first step we eliminate by induction the ~-dependence in the transition functions of the even coordinates. In the second step we eliminate the higher powers of ~ from the transition functions of the odd coordinates. To start, we thus assume that all transition functions of the even coordinates are given by

(8.3) The induction starts trivially at k = 1; we will show that we can increase k by 1, thus eliminating the ~ dependence. The idea is to change the coordinates x such that they absorb the term with i = k. Since 'Pba1 = 'Pab, it follows from [111.3.18] that 'Pba,O : O::'bo. d ----+ Ob~o.d has 'Pab,O as smooth inverse. We apply 'Pab,O to (8.3) and we invoke [111.2.27], using that 2:i>k 'Pba,2i(X a )' (~a)2i is nilpotent. We deduce the existence of smooth functions Xba,2i, i ?:.-k defined on O::'bo. d such that

The term of order 2k interests us most and can be computed as

°

where 'P~~ is (the equivalent of) the function jCl) in (111.2.28). We next compute a relation between various xs, needed to prove that we indeed will increment k by 1. For a point (x a , ~a) E Oab n Oac one has

Xc

= 'Pca,O(Xa + LXca,2i(Xa)' (~a)2i) = 'Pcb,O('Pba,O(Xa + LXca,2i(Xa)' (~a)2i)) i~k

Xc

i~k

= 'Pcb,O(Xb(Xa,~a) + LXcb,2i(Xb(Xa,~a))' (~b(Xa,~a))2i)

.

i~k

Applying 'Pbc,O, substituting (8.3) for Xb (xa, ~a ) ,expanding into powers of ~a and retaining only (~a)2k, we obtain the equality

(8.4)

Xcb,2k('Pba,O(X a ))' ('Pba,l(X a )' ~a)2k

+ 'Pba,2k(X a )' (~a)2k

= 'P~~~o(Xa) . Xca,2k(X a ) . (~a)2k .

With these preparations, we can introduce a coordinate change that eliminates the (~)2k in all transition functions at once. To do so, let Pa be a partition of unity subordinated to the cover U, and define the functions Pa : Oa ----+ Ao as Pa = Pa 0 'P;;-l. We then note

198

Chapter IV. Bundles

that if (J" is a smooth function defined on Oab, then Pb . (J" is a smooth function defined on Oa (supp(Pbloab) C Oab is closed in Oa). We thus can define the diffeomorphisms 'l/Ja : Oa ----+ Oa, (Xa, ~a) f-* (Ya, ~a) by (8.5) c

That 'l/Ja is indeed a diffeomorphism follows from [III.3.l8]. With these diffeomorphisms we create new charts <Pa : Ua ----+ Oa for M by <Pa = 'l/Ja 0 'Pa; they also form an atlas for M. To see how the transition functions
c

where h.o.t. stands for higher order terms in ~a. We now compute the transition function

'Pba :

c

c

c

c

=

'Pba,O(Ya)

+ LPc('Pca,o(Ya))'

(XCb,2k('Pba,O(Ya))' ('Pba,1(Ya)'

~a)2k

c

In view of(8.4), we may conclude that we have indeed incremented k by 1. By induction we thus may conclude that there exists an atlas in which we have eliminated the ~-dependence of the x transition functions. To eliminate the higher powers of~ from the transition functions of the odd coordinates, we proceed again by induction, assuming now that the transition functions are given by

(8.6)

199

§8. Batchelor's theorem

Since we know that the matrix ~ba,l is invertible [111.3.18], we can find smooth functions Xba,2i+ 1, i 2': k defined on O:,;o.d such that

~b = ~ba,l(Xa)' (~a + LXba,2i+l(Xa)' (~a)2i+l)

.

i?:.k The leading term is easily computed as

Xba,2k+l(X a ) . (~a)2k+1 = ~ab,l(~ba,O(xa)) . ~ba,2k+l(Xa) . (~a)2k+1 . We then compute a compatibility relation between various X on Oab n Oae :

~e = :Pea,l(X a )' (~a

+ LXea,2i+1(Xa)' (~a)2i+1) i?:.k

~e = :Peb,l(~ba,O(Xa))' (~b(Xa,~a)

+ LXeb,2i+1(~ba,o(Xa))' (~b(Xa,~a))2i+1)

.

i?:.k Substituting (8.6) for ~b(Xa, ~a), multiplying by ~be,l (~eb,O( ~ba,O(Xa))), expanding into powers of ~a and retaining (~a)2k+1, one obtains the compatibility relation

Xeb,2k+l(~ba,O(Xa)) . (~ba,l(Xa) . T}a)2k+1

(8.7)

+ ~ba,2k+1(Xa) . (T}a)2k+l

= ~ba,l (x a )Xea,2k+l (xa) . (T}a)2k+l .

As for the x coordinates, we now use the partition of unity phism 'l/Ja : Oa ---+ Oa, (x a , ~a) f---+ (x a , T}a) by

T}a

p and define the diffeomor-

= ~a + LPe(~ca,O(Xa))' Xea,2k+1(X a )' (~a)2k+l

.

e

As for the x coordinates, it is easy to check that the inverse relation is given by

~a = T}a - LPe(~ea,o(Xa))' Xea,2k+1(X a )' (T}a)2k+l

+ h.o.t ..

e

Still following the line of argument used for the x coordinates, we define new maps ipa : Ua ---+ Oa by ipa = 'l/Ja 0 ~a. These maps define a new atlas for which we calculate fuetransitionfunction (xa,T}a) f---+ (Xb,T}b) = (~ba,O(Xa),T}b(Xa,T}a):

T}b = ~b

+ LPe(~eb,O(Xb))' Xeb,2k+l(Xb)' (~b)2k+1 e

e

e

e

=

+ ~ba,2k+l(Xa) . (T}a)2k+l + h.o.t. ~ba,l(Xa)' T}a + LPe(:Pea,o(Xa))' (Xeb,2k+l(~ba,O(Xa))' (~ba,l(Xa)' T}a)2k+1 e

200

Chapter IV. Bundles

In view of (8.7), this means that we have incremented k by 1, and thus by induction we can assume that we have eliminated all higher powers of the odd coordinates in the transition functions of the odd coordinates. IQEDI

8.8 Discussion. Batchelor's theorem has a ramification which transcends the practical applications of it. Recall that the functions and can be interpreted as real valued are the components of the transition smooth functions on BO ab . More in particular, the we can interpret as components of a functionBipba : BO ab ----+ BO ba . Thefunctions matrix valued function '1fJba from BO ab to real valued matrices of size q x q, where q is the number of odd coordinates of the A-manifold M. Changing slightly our viewpoint, the ~ Bipba are the coordinate changes in the R-manifold BM and the functions functions g~ ~ '1fJba become matrix valued functions on BUa n BUb c BM. But we have more: since the '1fJba are derived from the original ipa, they satisfy the condition

P

9k P 9k

P

In view of [1.24] this means that we have constructed a vector bundle 7r : V M ----+ BM with typical fiber Rq ~ Rq EB {O}, all within the category of R-manifolds. On the other hand, let 7r : V M ----+ BM be a vector bundle over BM (of dimension p) with typical fiber Rq ~ Rq EB {O}. Such a vector bundle can be described by coordinate changes Bipba for BM and transition functions '1fJba. If E is an A-vector space of dimension plq, we can define Ua = B-l(BUa ) C Eo, and then the pairs (Bipba, '1fJba) define coordinate changes for an A-manifold M modeled on E, coordinate changes that already have the property given in Batchelor's theorem. Obviously this construction is the "inverse" of the construction that associates to an A-manifold M a vector bundle 7r : V M ----+ BM in the category of R-manifolds. But there is more to this correspondence. Associated to M we have the R-algebra C=(M) of smooth A-valued functions on M. Given the vector bundle 7r : V M ----+ BM, we can construct the exterior algebra bundle /\ V M ----+ BM, the sections of which also form an R-algebra r(/\ V M). We claim that these two R-algebras are isomorphic. If f is a smooth function on M, then for any chart ipa : Ua ----+ Oa we obtain a smooth function f 0 ip;;l : Oa ----+ A which can be written as q

(foip;;l)(x,~)

=L

On the other hand, any section S : BM ----+ /\ V M is represented locally by a function Sa : BOa ----+ /\ R q. If (Vi) 1 is a basis of R q, such a function can be written as

i=

We thus can identify the localfunction f 0 ipa with the local section Sa. Using an atlas as in Batchelor's theorem [8.2J, it is elementary to show that these local identifications glue together to a global identification.

§8. Batchelor's theorem

201

We can summarize this discussion by saying that any A-manifold can be described as a vector bundle over an ordinary R-manifold, and that smooth functions on the A-manifold are just the sections of the exterior power of the ordinary vector bundle. In other words, A-manifolds can be studied entirely in the category of R-manifolds.

8.9 Discussion. In [S.S] we have been a little vague about the kind of correspondence we have constructed between A-manifolds and (ordinary) vector bundles. In the first place, the construction of the vector bundle V M ---+ BM depends upon the choice of the atlas having the properties of Batchelor's theorem [S.2]. It is elementary to show that another atlas will give us another vector bundle, but one that is isomorphic to the first. An intrinsic map thus should associate to an A-manifold an isomorphism class of vector bundles over BM. Going the other way, the A-manifold associated to a vector bundle depends upon the actual vector bundle, but its diffeomorphism class is independent. We thus can conclude that we have a bijection between diffeomorphism classes of vector bundles over R-manifolds and diffeomorphism classes of A-manifolds. The natural reflex is to say that this bijection is functorial. However, there is a "small" problem: a functor not only maps objects to objects, but also morphisms to morphisms. And as soon as the odd dimension is greater than 1, there seems to be no natural way to associate a morphism of vector bundles to a morphism of A-manifolds.

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Chapter V

The tangent space

In this chapter we introduce the standard machinery of differential geometry of A-manifolds. Differential geometry can be seen as the interplay between analysis and geometry, or as the passagefrom local to global. The first step in this interplay is the definition of the tangent bundle of an A-manifold and the interpretation that the module of sections of the tangent bundle is isomorphic to the module of left derivations of the ring of smooth functions on the A-manifold. The next step is the generalization of the derivative of a function, which leads to the definition of the tangent map, defined on the tangent bundle. Yet another passage from local to global is the definition of the flow of a vector field, which generalizes the (local) existence of solutions of differential equations. Another globalization is Frobenius' theorem which discusses the integrability of a distribution (roughly speaking a system of vector fields) and the existence of leaves (maximal integral manifolds). Dual to these results concerning the tangent bundle is the theory of differential forms, i.e., sections of (the k-th exterior power of the dual of) the tangent bundle. Here one finds the exterior derivative, the Lie derivative, and the de Rham cohomology. However, we make no attempt to integrate differential forms. Using the flow of a vector field, we also have a kind of reverse passage from global to local. Ifwe let the flow of a vector field X act on a vector field Y, we can derive with respect to the flow parameter and we obtain the commutator [X, Y]. If we let the flow act on a differential form w, then the derivative with respect to the flow parameter yields the Lie derivative ,C( X)w. With a few exceptions, all results as well as their proofs are standard. The first exception concerns the integration of vector fields. Only homogeneous vector fields are considered for integration, and we show that only those having auto-commutator zero admit a flow. The second exception concerns Frobenius' theorem. We show (as expected) that a subbundle of the tangent bundle is involutive (a foliation) if and only if it is locally generated by some of the i . Due to our approach to smooth functions (constant functions must take real values), this does not imply that through every point passes an integral manifold (leaf of the foliation). The existence of leaves is only guaranteed through points that have real coordinates.

a

203

204

1.

Chapter V. The tangent space

DERIVATIONS AND THE TANGENT BUNDLE

In this section we define the tangent bundle TM of an A-manifold M directly in terms of transition functions with respect to an atlas. We then show that the smooth sections of TM (vector fields) correspond to derivations of the ring of smooth functions C= (M). This correspondence is used to define the commutator of two vector fields.

1.1 Definition. Let M be an A-manifold modeled on an A-vector space E and let U = { 'Pa : Ua ----+ Oa I a E I} be the atlas of all charts for M (i.e., the differentiable structure). U sing notation as in [111.4.9] we have the even functions 'Pba : 0 ab ----+ 0ba, and associated to them their Jacobians Jac('Pba) : Oab ----+ Aut(E). And recall that in terms of a basis (ei) of E the Jacobian is given as Jac( 'Pba) (x) = 2:i,j ie 0 (8i (x)) ej. We thus can define the functions 'l/Jba : Ua nUb ----+ Aut(E) by 'l/Jba = Jac( 'Pba) 0 'Pa' It follows from the chain rule that these functions satisfy the conditions of [IV. 1.24]. The vector bundle so obtained is called the tangent bundle 7r : T M ----+ M; its typical fiber is E. For a point mE M the fiber of TM above m is usually denoted by TmM, instead of by (T M)m. Moreover, if X E ru (T M) is a local section of TM, the value X (m) is also denoted by Xm or by Xlm.

'Pt

1.2 Discussion. It is obvious that we do not really need the whole differentiable structure of M to define the tangent bundle TM, an atlas is sufficient. The reason to use the whole differentiable structure is to have a definition which is manifestly independent of the choice of an atlas. Since an atlas is sufficient, it follows immediately that if 0 c Eo is open, its tangent bundle is justthe trivial bundle TO = 0 x E with projection p = 7r 0 : 0 x E ----+ O. More generally, if U c M is an open subset of an A-manifold, the induced smooth structure on U consists of all intersections Ua n U for charts Ua of M. It follows that the tangent bundle TU of U is (isomorphic to) the restriction of the vector bundle TM to the open set U C M, and thus ru (T M) = r(TU) (see also [IV. 1.13] and [IV. 1.19]).

1.3 Discussion. Let M be an A-manifold and consider the graded commutative R-algebra

C=(M) and the R-vector space of its right linear derivations DerR(C=(M)). We can turn DerR(C=(M)) into a graded C=(M)-module by the following procedure. For f E C=(M) and DE DerR(C=(M)) we define f· D by

v9 E C=(M) : (f. D)(g) = f . (D(g))

.

We leave it to the reader to verify that this indeed turns DerR(C=(M)) into a graded C=(M)-module. What we intend to do is to show that there is a natural identification between the C=(M)-modules DerR(C=(M)) and r(TM).

205

§1. Derivations and the tangent bundle

1.4 Proposition. Let 0 be open in M, f,g E C=(O), and DE DerR(O). IfU is open in 0 and flu = glu, then (Df)lu = (Dg)lu. Proof Without loss of generality we may assume that D is homogeneous and that f and 9 are homogeneous of the same parity. For any m E U, let P be a plateau function around min U [111.5.21]. It follows that p . (f - g) = O. Since D is a derivation, we obtain

0= D(p· (f - g))(m)

= (Dp)(m)· (f(m) - gem)) + (-l)(.,(D)I,,(p»)p(m). ((Df)(m) - (Dg)(m)) . Since f(m)

= gem)

and p(m)

= 1 we deduce that (D f)(m) = (Dg)(m).

IQEDI

1.5 Proposition. Let U C 0 c M be open and DE DerR(C=(O)). Then there exists a uniqueDu E DerR(C=(U)) suchthatVf E C=(O): (Df)lu = Du(flu)· Proof The main problem of this proposition is that not every smooth function on U need be the restriction of a smooth function on O. So let 9 E C= (U) and m E U be arbitrary and let p be a plateau function around min U [111.5.21]. It follows that pg is a well defined smooth function on O. We now define Du 9 by

(Dug)(m)

(1.6)

=

D(pg)(m) .

To see that this is well defined, independent of the choice for p, suppose pi has the same properties as p. It follows that pg and pig coincide in a neighborhood of m, and thus by [104] D(pg)(m) = D(p'g)(m). By the same argument we have that (Df)lu = Du(flu) for any f E C=(O). Uniqueness of Du follows in the same way. Suppose Du exists. The functions 9 and pg coincide in a neighborhood ofm, and thus by [104] Du(pg)(m) = Du(g)(m). But then by Du(flu) = (Df)lu we have Du(pg)(m) = D(pg)(m), and thus Du must be defined by (1.6). Finally to prove that Du is a derivation, suppose that D and f, 9 E C=(U) are homogeneous. The functions f 9 P and f 9 p2 coincide on a neighborhood of m and thus

Du(fg)(m) = D(fgp)(m) = D((fp)(gp))(m) = D(fp)(m)· (gp)(m) + (_l)(.,(D)I"(f») (fp)(m) . D(gp)(m) =

(Du f)(m) . gem)

1.7 Corollary. Let V cUe 0

Dv = (Du)v.

+ (_l)(c(D)I"(f») f(m)

c M be open and D

E

. (Dug)(m) .

DerR(C=(O)).

IQEDI

Then

206

Chapter V. The tangent space

1.8 Proposition. Let E be an A-vector space of dimension plq and let 0 C Eo be open. If A satisfies (C [q+ll) (or ifq = 0), then DerR(Coo(O)) is a free graded Coo(O)-module generated by the 1 :S i :::; p + q. If Al = {a}, then DerR(Coo(O)) is afree graded COO (O)-module generated by the 1 :::; i :::; p.

ai,

ai,

Proof Let us first assume that A satisfies (C [q+ll), then we know from [111.3.3] and [111.3.6] that all 1 :S i :::; p + q exist and are derivations. For the independence of the (ai ), suppose that D = 2:i D i . ai = a E DerR(Coo(O)) with Di E Coo(O). Denoting by (xi) the even and odd coordinate functions associated to a basis (ei) of E, we obtain a = Dxi = Di, i.e., the (ai ) are indeed independent. To prove that the (a i ) generate DerR(Coo(O)), let D E DerR( Coo(O)) be arbitrary, for which we define functions Di E Coo(O) by Di = DXi. We claim that D = 2:f,!i Di ·ai . To prove the equality we change notation and denote by x the even coordinates of a point and by ~ the odd ones. Let f = 2: ~il ... ~ik . fit ... ik (x) be an arbitrary smooth function on O. Since D and 2:f,!i Di . ai are both derivations, it is immediate that D(~il ... ~ik) = (2:f,!i Di . ai)(~il ... ~ik). To show the equality, it thus suffices to show that both derivations act the same on functions of even coordinates only. To do so, note first that the identity 1 . 1 = 1 and the fact that a derivation is R-linear imply that D f = a whenever f is a constant function (with a real value!). Now let f depend upon even coordinates only and let Xo E BO be arbitrary, then (by condition (Ai) in [III. 1.20]) there exist an open U C 0 and local functions ¢i depending upon even coordinates only such that \Ix E U: f(x) = f(x o) + 2:f=1 (Xi - x~) . ¢i(X, x o). Since Xo has real coordinates, f(x o) is a real constant and ¢i(X, x o) is a smooth function of x on U. Thus, using [1.5], we find

ai,

p

=

(Du (i)x i - x~) . ¢i(X, x o)) ) (xo,~) i=1 p

=

P

2:)DuXi) (xo, ~) . ¢i(X o, x o) + 2:)x~ - x~) . (DU¢i)(Xo,~) i=1 p

i=1 p+q

= L:Di(xo,~)' (ad)(xo,~) = (L:D i . ad)(xo,~) i=1

,

i=1

where the last equality follows from the fact that f does not depend upon the odd coordinates, i.e., ad = a whenever i > p. Since a smooth function is completely determined by its values in points with real even coordinates and arbitrary odd coordinates, we have shown that D f = 2:f,!i Di . ad also holds for functions of even coordinates only. Ifwe now assume that Al = {a}, then again according to [III.3.3] and [111.3.6] the 1 :::; i :::; p exist and are derivations. Their independence is proven as for the previous case. That they are generating is also proved in the previous case, once one realizes that, when Al = {a}, any smooth function on 0 automatically is a function of the even coordinates IQEDI only.

ai,

21J7

§1. Derivations and the tangent bundle

a

1.9 Remark. The fact that we obtain D = 2:i Di . i in that order is a consequence of the fact that we write x - Xo to the left of ¢(xo, x) in the definition of smooth functions and the fact that we use right derivations.

1.10 Discussion. With [1.8] we have proved the difference between the generic case in which A satisfies (C [q+ll) (with q the odd dimension) and the special case A = R (or more generally Al = {O}). In the first case DerR(C=(O)) has dimension plq, whereas in the second case it has dimension p ~ piO. This confirms the idea that in the real case A = R we do not have partial derivatives in the odd directions. Of course there is a kind of no man's land in between where Al -=I=- {O} but where A does not satisfy (C [q+ll). In order to avoid that we enter this no man's land (by using too large odd dimensions), we required A to satisfy (C [=l).

1.11 Remark. An alternative approach to the tangent bundle could be to use [1.8] as a kind of definition in the following sense. For 0 C Eo an open subset, its tangent space TO is defined as TO = 0 x F with F an A-vector space of the same graded dimension as DerR(C:xJ(O)). This requires that DerR(C=(O)) is a free graded C=( O)-module, which in tum puts a condition on A. If A = R or if A satisfies (C [=l) it follows from [1.8] that this is the case for all possible dimensions of E. We could go one step further and define the differential dimension of 0 as the (graded) dimension of Der R (C= (0)), provided of course that Der R (C= (0)) is a free graded C= (O)-module. In this way the even part Eo of an R-vector space E of dimension plq would have differential dimension plO, in accordance with the fact that Eo is completely determined by p even (real) coordinates.

1.12 Discussion. Our next purpose is to construct an isomorphism between r(T M) and ---+ Oa I a E I} be an atlas and f : Ua ---+ A arbitrary. Since by definition f is smooth if and only if f 0 ipa is smooth on Oa, we deduce from [1.8] that DerR( C:xJ (Ua)) is a free graded C=(Ua)-module generated by the derivations aai defined by

DerR(C=(M)). Let {ipa : Ua

(1.13) Defining the (local) coordinate functions x~ on Ua by x~ = Xi 0 ipa == ie 0 ipa, it follows immediately that aai(X~) = of. This shows that the derivations aai can be interpreted as the partial derivatives ax~ with respect to the coordinates x~ on Ua. To construct the isomorphism, let DE DerR(C=(M)) be arbitrary. On each chart we have Da == DUa = 2:i D~ . aaiwith D~ E C=(Ua) defined by D~ = Da(x~). We thus can construct a family of functions Sa : Ua ---+ E by

208

Chapter V. The tangent space

We now want to apply the result of [IV. 1.20] to deduce that this family defines a global section SD E r(TM). Therefore we have to prove that ((Sa(m) II7/!ba(mH = sb(m) for m E Ua n Ub Although 7/!ba (m) is even, we use the left linear character because the Jacobian is left linear in general. Substituting the explicit expressions, this amounts to proving that 2: i D~(m) . (ai ~ba)(~a(m)) = Db(m) (use that the Jacobian is given by Jac(~ba)(X) = 2:i,j ie 0 (ai ~ba(x)) ej [111.3.8]). We thus compute on Ua n Ub :

j j Dl(m) = Db(x o~b)(m) = Da(x o~b)(m) = L i (Dax~)(m) . aai(X j 0 ~b)(m) = Li D~(m). ai(x j O~bO~~l)(~a(m))

= Li D~(m). (ai~ba)(~a(m)) . In the first line we have used that on Ua nUb the derivations Da and Db both coincide with DUanub; to obtain the second line we applied Da = 2:i D~ . aai. Since this is exactly what we needed to prove, we conclude that we have defined a map D f---+ SD, DerR(C=(M)) ---+ r(TM). Since the parity of the aai is the same as of the ei, it should be rather obvious from the construction that this map is an even morphism of C=(M)-modules. It thus remains to show that it admits an inverse. If S E r(T M) is a global section, it defines, according to [IV. 1.20], a family of maps Sa : Ua ---+ E, Sa(m) = 2:is~(m)· ei satisfying ((sa(m)ll7/!ba(mH = sb(m), i.e., 2: i s~(m) . (ai ~~a)(~a(m)) = sb(m), on intersections Ua nUb. With these, we define a derivation Ds E DerR(C=(M)) by the formula (1.14) Since the aai belong to DerR(C=(Ua)), the derivation property of Ds follows immediately. It only remains to be shown that Ds! is a globally well-defined smooth function, i.e., we have to show that these local expressions coincide on Ua nUb' We therefore compute

(L i S~ . aad)(m) = L

i

s~(m) . ai(J 0 ~~l)(~a(m))

= Li s~(m) . ai(J 0 ~bl O~ba)(~a(m))

= L . . s~(m) . (ai~t)(~a(m)) . aj(f ~bl)(~ba(~a(m))) ',J

0

= L . sb(m). aj(JO~bl)(~b(m)) = (L . Sb' abjJ)(m) . J

J

To obtain the third line we have applied the chain rule (beware the order!). This calculation shows that D s is a well defined derivation. Since it is rather obvious that S f---+ D s is inverse to D f---+ SD, we have proved [1.15].

1.15 Proposition. The map D f---+ SD from DerR(C=(M)) to reT M) and the map f---+ Ds in the other direction are isomorphisms of graded C= (M)-modules, inverse to

S

each other.

209

§1. Derivations and the tangent bundle

1.16 DiscussionlDermition. Sections of TM are usually called vector fields on M; we thus have proved that there is an isomorphism between smooth vector fields on M (elements of reT M)) and derivations of C=(M). Given that we can interpret smooth vector fields as derivations, it is customary to introduce some notation to reflect that fact. Let 'l/Ja : 7r- 1 (Ua) ---+ Ua x E denote the formal local trivialization Of7r-I(Ua ) C T M. Then we define the elements aailm == ax~ 1m E TmA1 by

a

-1

aailm = -a·Im = 'l/Ja (m,ei)' x~

Since the free graded A-module structure on T mM is defined by means of 'l/Ja, it follows that the vectors aailm form a basis ofTmM. It also follows that the coefficients of a vector X E TmM with respect to the basis aailm are exactly the same as the coefficients of the vector 'l/Ja(X) with respect to the basis ei of E. We thus will often confuse the vectors aailm in TmM with the vectors ei in the corresponding trivialization. As a particular application, it follows that for any local section X : Ua ---+ TMlu a , not necessarily a smooth one, there exist functions X~ : Ua ---+ A such that X(m) = 2:i X~(m) . aailm. For the moment the a~i 1m (or aai 1m or axi 1m) are just formal symbols, but when a vector field is seen as a derivation, they are exa~tly the partial derivations aai with respect to the (local) coordinates x~ == ie 0 'Pa defined in (1.13), as follows immediately from (1.14). Now let (Ua,'l/Ja) and (Ub,'l/Jb) be two trivializing charts, then there exist transition functions 'l/Jb 0 'I/J;;1 : Ua x E ---+ Ub X E, (m,v) f--+ (m,((vIIJac('Pba)('Pa(m)». In particular (see [III.3.11]) =

L .ai(ie

0

'Pb 0 'P;;1) ('Pa(m))

ej =

)

L .(aaiXb)(m) .

ej .

)

a

It follows that for the abstract vectors aailm, bj 1m E T mM we find the relation

a

aXb

a

-a .Im=~-a.(m)·-Im. x' ~ x' ax) a

jab

1.17 Remark. In the literature one can also find another approach to define the tangent space which encodes in a certain way the derivation property directly. The idea is to consider, for a fixed mE M, maps Om : C=(M) ---+ A satisfying the derivation property om(f· g) = (om(f)) . gem) + (_1)(€(8~)1€(f)) f(m)· (Om(g)). As in [1.4] one can show that if f and 9 coincide in a neighborhood U of m, then omf = omg. For a fixed m E M the set of all such Om is a graded A-module which is taken as the definition of the tangent space at m EM. One then has to show that it indeed defines a vector bundle. In our approach, each Xm E T mM defines a map Om on smooth functions defined in a neighborhood U ofm by f f--+ 2:iXi (ax~f)(m), where we used a local coordinate system (x~) around m in which the tangent vector takes the form Xm = 2:i Xiaail m (1.14). This satisfies the condition for a map Om because the partial derivative operators aai are derivations [III.3.6].

210

Chapter V. The tangent space

1.18 Discussion. In [III. 1.24] and [IIIA.21] we have seen that C=(M) is a graded Ralgebra. From §1.6 we then learn that DerR(C=(M)) is a graded R-Lie algebra with even bracket [X,Yj = XoY - (_l)(o(X)to(Y»). YoX. By the isomorphism between Der R (C= (M)) and reT M) we can transport this bracket to vector fields. Although DerR(C=(M)) is also a graded C=(M)-moduJe [1.3], it is not a graded C=(M)-Lie algebra. The relation between the module structure and the Lie algebra structure is expressed in [1.19]

1.19 Lemma. For all homogeneous IE C=(M), X, Y E DerR(C=(M)) we have

[X,I' Yj

=

(X!)· Y

+ (_l)(o(X)to(f») I·

[X, Yj.

Proof A derivation is determined by its action on functions g, so we compute

[X, I· Yj(g) = X(UY)(g)) - (_l)(o(X)to(fY» UY) . Xg) = XU· Y g) - (_l)(o(X)to(f»)+(o(X)to(Y») I· Y(Xg)

= (X!)· (Yg) + (_l)(o(X)to(f») I· X(Yg) - (_l)(o(X)to(f»)+(o(X)to(Y»

=

((X!) . Y

I· Y(Xg)

+ (_l)(o(X)to(f»)[X, Y])(g)

1.20 Lemma. Let X and Y be vector fields on M, and let U E DerR(C=(U)) and [X, Yjlu = [Xlu, Yluj.

.

c

IQEDI

M be open.

Then

Xlu

Proof Since TU is the restriction of TM to U, we have X Iu E ru (T M) ~ r(TU) ~ DerR(C=(U)). Uniqueness in [1.5] proves the second part. IQEDI

1.21 Lemma. If Xlu a = 2:i X! . 8ai and Ylu a = 2:i Y; . 8ai are the local expressions of homogeneous vector fields X and Y, the local expression of [X, YJ is given by

(1.22)

[X, Yjlua = L:(X! . (8ai Yj) -

(_l)(o(X)to(Y») .

Yd . (8aiX~)) . 8aj .

2,)

Proof Since on Ua we have [X, Yjlu a =

[Xlu a, Ylualx~

2: j ([X, Yjluax~). 8aj , we compute

Xlu a(yluax~) - (_l)(o(X)to(Y») . YluJXluax~) = Xlu a(Yj) - (_l)(€(X)to(Y») . Ylu a(X~) =

= 2:( X~ . (8ai Yj) -

(_l)(o(X)to(Y») .

Yd . (8aiX~)) .

IQEDI

2ll

§2. The tangent map and some standard applications

1.23 Nota Bene. If X is an even vector field, obviously [X, X] = 0, but the operation f--+ X(X(J)) is not a derivation ofC=(M) (unless X = 0). On the other hand, if X is an odd vector field, we have (globally) [X, XlI = 2X(X1), and the local expression for [X, X] gives us:

I

This is not necessarily zero, but it shows that the operation I f--+ X (X (J)) is a derivation ofC=(M) (see also [1.6.8]). As an example in which [X, X] is not zero, consider an A-vector space E of dimension 012, and use (odd) coordinates (e, on the A-manifold M = Eo. The odd vector field X = at;' + ae on M has auto-commutator [X, Xl = ae, which is manifestly not the zero vector field.

(e ,e)

ee .

e)

2e .

1.24 Definition. Let M be an A-manifold and X E r(TM) ~ DerR(C=(M)) a smooth vector field on M, i.e., X acts on smooth functions I : M ---+ A. If F is an A-vector space, we can extend this action to functions I : M ---+ F in the following way. Let (Ej )j=l be a homogeneous basis for F, then associated to a function I : M ---+ F we have its component functions Ii = I 0 jE : M ---+ A, i.e., I (m) = 2: j Ii (m) Ej. Moreover, I is smooth if and only if all Ii are smooth. We now define the function X I : M ---+ F as (X 1) (m) = 2: j (X Ii) (m) Ej. This is a straightforward generalization of (111.3.9). Moreover, it is independent of the choice of the basis (in its equivalence class) because the vector field X acts as the zero operator on (real) constants (the matrix elements of a basis change).

2.

THE TANGENT MAP AND SOME STANDARD APPLICATIONS

In this section we define the tangent map T I : T M ---+ TN associated to a smooth map ---+ N. Once we have the tangent map, we can define immersions and embeddings and their relation to submanifolds, as well as some of their elementary properties. We also show the behavior of the tangent map with respect to the product of A-manifolds and smooth maps between them, which provides some useful identifications. At the end of this section we define the notion of vector fields related by a smooth map and we show that this relation preserves brackets.

I :M

2.1 Construction. Let M be an A-manifold modeled on E, N an A-manifold modeled on F and let I : M ---+ N be a smooth map. As before (especially in [111.3.8]) we denote by (ei) a basis of E and by (Ej) a basis of F. Our purpose is to construct a smooth even (left linear) vector bundle morphism T I : T M ---+ TN which induces I. Let V = {Xa : Va ---+ Pa } be an atlas for Nand U = { «Ja : Ua ---+ Oa} an atlas for M such

212

Chapter V. The tangent space

that for each Ua E U there exists a Va E V such that I(Ua) C Va' Using [IVA.l4] we define the map T I by the set of maps ha : Ua ----+ HomdE; F),

ha = Jac(Xao/o~;;l)o~a. That this set satisfies the conditions (IVA. IS) is a direct consequence of the chain rule. And thus T I is a well defined left linear (even) vector bundle map, called the tangent map. In terms of charts Oa x E and Pa x F for M and N respectively, the tangent map takes the form

TI

: Oa

E

----+

Pf(a) x F

(X, v)

f--*

((Xao/o~;;l)(x), (vIIJac(Xao/o~;;l)(x)))).

X

Now that we know how TI is defined, let us determine its expression in terms of the basis 8 ai l m for TmM introduced in [1.16]. Let U a and Va be charts around m and I(m) respectively as above, then we know that a vector Xm = 2:i X~ 8ai l m E TmM corresponds to the vector v = 2:i X~ ei in the local trivialization Ua x E. The image (( Xm II T I)) then is represented in the trivialization Va X F by

((vllJac(Xa 0 I 0 ~;;l)(~a(m))))

=

.

. 8(~-1<>I<>Xa<>jE) L:X~ a 8x i (~a(m)) . Ej

E

F,

t,)

where we used the ML-matrix representation of the Jacobian as determined in [111.3.8]. If we denote by Y~ = jE 0 Xa == Xa <> jE the (even and odd) coordinates on Va, it follows that It = jE 0 Xa 0 I == 1<> Xa <> jE are the local coordinates of I in the chart Va, and that Ej E F corresponds to 8aj l f (m) E Tf(m)N. Using (1.13), we thus find

(LX~8~i 1m liT!)) = LX~. ~~~ (m)· a

i

a

i,i

8 j If(m) 8Ya

.

Now this is the expression in the special preselected charts Ua and Va of the chosen atlases, so how does it look in arbitrary charts Ub around m and V(3 around I (m)? Using the results of [1.16], we find j

k

8 c

( Lxt 8~i 1m liT!)) = L xt, ~:~ (m) . 8/~ (m) . 8Y~ (f(m)) . 88 c . 8xa Ya Y(3 .

b

t

.. k ,t:."

1,,),

b

It looks like we can apply the chain rule to combine these three matrices into the single one (8 xt I~)( m), except for the fact that the arguments do not really conform to the conditions of the chain rule (twice the same argument m is suspicious). So let us have a more detailed look, using the formal definitions for the derivations 8ai etcetera:

L j,k

.

k

8 c

8xb

8x a

8Ya

8x~ (m) . 8/~ (m). Y~ (f(m)) =

213

§2. The tangent map and some standard applications

Written this way, we have the official partial derivatives chain rule does make perfect sense and gives us indeed

axi and ayj

of §II1.3 and here the

We thus obtain at the same time that our intuitive feeling about the chain rule was correct, and that we have for the tangent map the general expression

(2.2)

valid in any chart Ub around m and V,6 around f(m). Since the formal definition of the tangent map is rather awkward to use, it is (2.2) that is used in practice as if it were the definition. It also justifies to a certain extent the fact that we defined the tangent map as being left linear: the form of (2.2) is ((X aliT!)) = (a1) a, i.e., we do not have to change any order.

x

2.3 Remark. Comparing the definition of the tangent map with the definition of the transition functions for the tangent bundle shows that the latter are exactly the tangent maps of the transition functions between the charts.

2.4 Notation. If f : M ---+ N is a smooth map between A-manifolds, and m E M a point, we have a left linear map TfIT~M : TmM ---+ Tf(m)N. It is customary to denote this map by Tmf, i.e., Tmf == T fIT~M' According to the definition ofT f, the left linear map Tmf is essentially the Jacobian lacf(m). If X is a vector field on M, Xm belongs to TmM. With the notations we have introduced, we thus can write «XmIITm!)) = «XIITm!)) = ((XmIIT!)) E Tf(m)N. However, the notation «X II T f)) does not indicate a point in TN, it denotes the map M ---+ TN, m f-> ((XIIT!))(m) = ((XmIITm!)), a map which we can also denote as X oTf or as Tf oX.

2.5 Nota Bene. If f : M ---+ N is a smooth map between A-manifolds, then for m E M we have defined the left linear map Tmf : TmM ---+ Tf(m) N. But this map is also even and thus right linear as well. Hence for any X E TmM we may write Tmf(X) for (( X II Tmf)). The left linear notation has the advantage that we do not have to interchange the symbols X and f in computing the result (see (2.2)), whereas the right linear notation has the advantage of tradition which says that the function symbol should be written before its argument upon which it acts. As it turns out, the left linear notation is used quite systematically in this chapter, whereas in the next two chapters the right linear notation is slightly favored.

214

Chapter V. The tangent space

2.6 Proposition. Let f : M ---+ Nand 9 : L thenT(Jog) = (T1) 0 (Tg).

---+

M be smooth maps between A-manifolds,

Proof According to the construction of the tangent map, it suffices to verify this formula on a local chart. But on a local chart it is just the chain rule. IQEDI

2.7 Remark. With the notation T mf for the restriction of the tangent map to the fiber TmM, the above chain rule for tangent maps takes the form Te(Jog) = Tg(e)f 0 Teg. If we want to stress the left linear character of the tangent maps, we could write this as Te(g 01) = Teg 0 Tg(e)f (see also [111.3.8]).

2.8 Corollary. If f : M

---+

N is a diffeomorphism, T f : T M

---+

TN is a diffeomorphism.

2.9 Definition. Let f : M ---+ L be a smooth map between A-manifolds, let f. E BL be arbitrary, and define N = f- l (f.) eM. The point f. E BL is called a regular value of f if for all mEN the (left) linear map T mf : T mM ---+ TeL is surjective.

2.10 Lemma. If £ E BL is a regular value for f : M submanifold of M (see also [III.5.IOJ).

---+

L, then N

=

f- l (f.) is a

Proof Suppose M is modeled on the A-vector space E of total dimension nand L on the A-vector space F of total dimension k ::::; n. Let 'IjJ : V ---+ P c Fo be a coordinate chart around f. E V and let ¢ : U ---+ 0 c Eo be a chart around some point mEN such that f(U) C V (such charts exists by continuity of f). By assumption the map 'IjJ 0 f 0 ¢-l : 0 ---+ P has rank dim L at ¢(m). According to [111.3.30] there exists an open set 0' C 0 containing m and a diffeomorphism 9 : 0' ---+ 0" c Eo such that ('ljJo fo¢-l 0 g-l)(x l , ... ,xn) = (x!, ... ,xk). Since 9 is a diffeomorphism, go ¢ : U' = ¢-l (0') ---+ 0" is a coordinate chart around m. Since f. has real coordinates, translation over 'IjJ( f.) in Fo is smooth. We thus may assume without loss of generality that 'IjJ(f.) = (0, ... ,0). Hence N n U' is described by the equations xl = ... = xk = 0 with (xl, ... , xn) the coordinates on U'. Since mEN was arbitrary, we conclude that N is a IQEDI submanifold of M.

2.11 Definition. A smooth map f : N ---+ M is called an immersion if for all n E N the map Tnf : TnN ---+ Tf(n)M is injective. It follows that f is an injective immersion if and only ifT f : TN ---+ T M is injective. An injective immersion is called an embedding if it is a homeomorphism onto its image.

2.12 Lemma. Iff: N ---+ M is an injective immersion, it is an embedding if and only if ---+ BM is an embedding.

Bf : BN

§2. The tangent map and some standard applications

215

Proof If f is an injective innnersion, then obviously Bf = flBN : BN ---+ BM is an injective immersion of R-manifolds (because BT f = T flBT M)' We thus have to verify that the induced topology on f (N) coincides with that of N if and only if the induced topology of f(BN) coincides with that of BN. But that is a direct consequence of the fact that B induces a bijection between open sets of M (respectively N) and those of BM (respectively BN) [111.4.12]. IQEDI

2.13 Remark. At first sight [2.12] looks natural. On second thoughts however, it has a strange aspect: the map f : BN ---+ BM ignores everything about the odd coordinates and nevertheless it determines a result on f : N ---+ M. The secret is that the odd coordinates constitute a vector bundle over the body [IV.S.S]. The condition that f is an injective innnersion then says more or less that the odd coordinates of N form a kind of sub bundle of the odd coordinates of M. The problems of an embedding are thus concentrated in the body parts.

2.14 Lemma. Let M and N be A-manifolds and let i : N ---+ M be a smooth map. !ffor some n E N the map Tni is injective, then there exists a neighborhood U ofn such that flu : U ---+ M is an embedding. !f Tn i is bijective (which implies dim M = dim N), then there exists a neighborhood U of n and a neighborhood V of i( n) such that i : U ---+ V is a diffeomorphism. In particular, if i is an immersion between A-manifolds of the same graded dimension, it is a local diffeomorphism and an open map. If i is also injective, it is a diffeomorphism onto its (open) image i(N) eM. Proof • If Tni is injective, it is of maximal rank, and then the first part follows from [1I1.3.30-ii] . • If Tni is bijective, we deduce from the inverse function theorem [111.3.23] the existence of U and V with the required properties . • Ifi is an immersion, Tni is injective for all n E N. But since Nand M are supposed to have the same graded dimension, it must be bijective [II.2.6-iii]. Hence we can apply the previous result to show that i is an open map and a local diffeomorphism. If i is also IQEDI injective, the second result follows immediately.

2.15 Nota Bene. The first part of [2.14] is a purely local result. It is by no means true that an immersion (not even an injective immersion) is automatically a global embedding, although it is so locally. An explicit counter example is given in [2.19].

2.16 Proposition. Let N be a connected submanifold of M and i : N ---+ M the canonical injection. Then i is an embedding and Ti(TN) is a subbundle ofTMIN' Conversely, if i : N ---+ M is an embedding, then i(N) is a submanifold ofM and the canonical injection i : N ---+ i(N) is a diffeomorphism when i(N) is given the A-manifold structure of [Ill. 5.4 ].

216

Chapter V. The tangent space

Proof Let us use the atlas for the A-manifold N introduced in [111.5.4] with charts

'ljJa : Va ---+ Pa C Fo induced from charts on M t.pa : Ua ---+ Oa C Eo by 'ljJa = t.palva , Va = Ua n N, and Pa = Oa n F. Since N is connected, we may assume that the graded subspace FeE of definition [111.5.1] is the same for all these charts. We already know that i : N ---+ M is injective and a smooth homeomorphism onto its image. In terms of the given charts Ti is given as Ti : Pa X F ---+ Oa X E, (x, v) f--+ (x, v), and thus i is also an immersion, proving that i is an embedding. Denoting by 7rN : TN ---+ Nand 7rM : T M ---+ M the canonical projections, it follows from the commutative diagram i 0 7rN = 7rM 0 Ti that Ti(T N) c T MIN' In terms of the above charts, TMIN has charts Pa x E and Ti(TN) is the subset described as Pa X F. It follows that Ti(T N) is a subbundle ofT MI N and that the given atlas is adapted to the subbundle[IV.3.12]. Conversely, let i : N ---+ M be an embedding and let no E N be arbitrary. If Uo is a chart of M around mo = i(n o), i-I(Uo) is open in N and thus there exists a chart 'IjJ : V ---+ P of N around no with V C i-1(Uo). Since i is an embedding, there exists a chart t.p : U ---+ 0, U c Uo around mo such that i(V) = i(N) n U. Again because i is an embedding, the map I = t.p 0 i 0 'IjJ-l : P ---+ 0 has rank n = dim Fat 'IjJ(no) E P. Applying the second part of [III. 3.30] we conclude that there exists a chart in which i(N) is described by equations of the type ie(t.p(m)) = 0 [111.5.2], and thus i(N) is asubmanifold. Using the same charts, it follows immediately that i : N ---+ i(N) is a bijective immersion when i(N) is equipped with the A-manifold structure given in [111.5.4]. The result now IQEDI follows from [2.14].

2.17 Corollary. Let N C M be a submanifold. Then the A-manifold structure on N given in [Ill. 5.4J is the unique structure such that the canonical injection i : N ---+ M is an embedding. Proof Let us denote by N' the set N endowed with another A-manifold structure. If the canonical injection i' : N' ---+ M is an embedding, it follows from the second part of [2.16] that the identity map j : N' ---+ N is a diffeomorphism, i.e., that N' = N as an IQEDI A-manifold.

2.18 Corollary. Let M, N, and L be A-manifolds, let i : N ---+ M be an embedding, and let I : L ---+ M be a smooth map such that I (L) c i( N). Then there exists a unique smooth map 10 : L ---+ N such that i 0 10 = I. Proof A direct combination of [111.5.8] and the second part of [2.16].

2.19 Counter example. The condition that i : N ---+ M is an embedding in [2.18] can not be weakened easily to injective immersion, as can be seen from the following example. Let N = L = AD, let M = A6, and define the maps i : N ---+ M and I : L ---+ M by i(t) = (t(l + t 4)-1, t 3(1 + t 4)-I) and I(x) = (x 3(1 + x 4)-1, x(l + x 4)-I). Both have as their image the figure eight, and their parameters are related by t = x-I, which shows

217

§2. The tangent map and some standard applications

that J can not be factorized through N. On the other hand, we will see in [6.8] a situation in which the embedding condition is not needed.

2.20 Discussion. The direct product M x N of two A-manifolds M and N is defined as the set of ordered pairs. When doing computations with tangent vectors it is often very useful to be able to see tangent vectors to M x N also as ordered pairs: of tangent vectors to M and N respectively. This boils down to a canonical identification T(M x N) ~ T M x TN given in [2.21 ]. Once we know that tangent vectors to M x N can be seen as pairs of tangent vectors, it becomes interesting to see how a (special kind of) tangent map behaves on these pairs. This is done in [2.22] and [2.23], yielding identifications T(f, g) ~ (TJ, Tg) and T(f x g) ~ T J x T g. All these identifications will be used in the sequel without further mentioning.

2.21 Proposition. For A-manifolds M and N, the map (T7rM' T7rN) from T(M x N) to (TM) x (TN) given by X f-* « X I T7rM )), <X I T7rN » is an isomorphism ofvector bundles. Proof Using the notation of [2.1], {Ua x E} is an atlas for TM, {Va X F} one for TN, and {Ua x Va} one for M x N. It follows that {(Ua x Va) X (E x FH is an atlas for T(M x N) and that {(Ua x E) x (Va X FH is an atlas for (TM) x (TN). With respectto these two atlases, the map (T7rM, T7rN) is given by (( m, n), (v, w)) f-* (( m, v), (n, w)). From this observation the result follows. Those who want to go into the details should use the fact that for the transition functions tpba X X(3a of M x N [III.S.ll] the Jacobian is given by Jac(tpba x ¢(3a) = Jac(tpba) x Jac(¢(3a) E EndL(E x F) = EndL(E EB F). IQEDI

2.22 Corollary. Let J : L ----+ M and 9 : L ----+ N be smooth maps between A-manifolds L, M, and N. In the identification T(M x N) ~ TM x TN of[2.2l] the map T(f,g) (see [lIl.5.12]) is given as (TJ, Tg). Proof To get the expression ofT(f, g) in terms of the given identification, we just have to compose with the identification map (T7rM, T7rN). But (T7rM, T7rN) 0 T(f, g) = (T7rM o T(f,g),T7rN o T(f,g)) = (T(7rM ° (f,g)),T(7rN ° (f,g)) = (TJ,Tg). IQEDI

2.23 Corollary. Let J : M ----+ Nand 9 : K ----+ L be smooth maps between A-manifolds. In the identifications T( M x K) ~ T M x T K and T( N xL) ~ TN x T L of [2.21 j, the tangent map of J x 9 : M x K ----+ N x L (see [11/.5.14]) is given as T(f x g) ~ TJ x Tg, i.e., we have a commutative diagram T(M x K)

T(fxg) )

TMxTK

T(N x L)

) TNxTL. TfxTg

218

Chapter V. The tangent space

Proof Since the identification maps are bijections, it suffices to verify that the maps

(T f x Tg) 0 (T7r)I!' T7rK) and (T7rN' T7rL) 0 T(J x g) are the same. But that is immediate IQEDI from the equalities 7rN 0 f x 9 = f 0 7rM and 7rL 0 f x 9 = go 7rK.

2.24 Construction. If X : M ----+ T M and Y : N ----+ TN are vector fields, we have the product map X x Y : M x N ----+ TM xTN defined by (X x Y)(m, n) = (X(m), Y(n)). Using the identification TM x TN ~ T(M x N) of [2.21], the map X x Y becomes a vector field on M x N. If Y is the zero vector field Y = Q, the vector field X x Q can be interpreted as the vector field X extended to the product M x N (by being independent of N). Using that we can add vector fields, the general case can be written as X x Y = X x Q + Q x Y. If it is clear from the context that the vector field X on M has to be seen as being extended to M x N as X x Q, one abbreviates X x Q as just X. With this shorthand, one thus can write X x Y = X + Y.

2.25 Definition. If f : M ----+ N is a smooth map and X and Y vector fields such that T foX = Yo f, i.e., such that we have the commutative diagram

TM~TN (2.26)

M

------ N, f

then we will say that X and Yare related by f. Sometimes the vector field Y is called the push forward of X and denoted as Y = f*X. We will not do so because it suggests that Y is determined by X, which is not the case. Even the existence of a vector field Y that is related to X by f is not guaranteed.

2.27 Lemma. Let f : M ----+ N be smooth map, X a vector field on M and Ya vector field on N. The vector fields X and Yare related by f if and only if for all smooth functions 9 E C=(N) we have the equality (Y g) 0 f = X(g 0 1 ) . Proof Let Va be a chart for Nand U a a chart for M such that f(Ua ) C Va. Using the local expressions for the vector fields Xlu a = X~ . 8ai and Ylv" = 2: j Yi ·8aj , commutativity of the diagram (2.26) translates as

2:i

(2.28) where the x~ and y~ denote the local coordinates on Ua and Va. Commutativity of the diagram and the chain rule then imply for 9 E C= (N) that (Y g) (J (m)) = 2: j Yi (J (m)). (8aj g)(J( m)) = 2:i X~( m) . (8ai (g 0 1))(m) = (X(g 0 1 ) )(m). To prove the equivalence in the other direction, we restrict the domain of definition to Ua and substitutefor gone of the local coordinates y~ to obtain immediately (2.28).IQEDI

219

§3. Advanced properties of the tangent map

2.29 Proposition. Let 1 : M ----+ N be smooth map, Xi vector fields on M and Yi vector fields on N (i = 1,2). If Xi andY; are related by 1, then [Xl, X2] and [YI , Y2 ] are related by f. Proof Since the relation is additive in both Xi and Yi, we may suppose that these vector fields are homogeneous. Since T 1 is even, it follows that Xi is of the same parity as Yi. We now apply [2.27] and compute

[Xl, X 2 ](g 0 1) = X I (X 2(go 1)) = Xl ((Y2g)

0

= (YI (Y2g) -

3.

1) -

(_1)(c(X 1 )lc(X 2») . (_1)(c(X )lc(X 2») . 1

(_1)(c(Ytll c (Y2») .

X2(XI(go 1)) X2((Y1 g) 0 1)

Y2(YI g)) 0 1

=

([YI , Y2 ]g) 0

1 . IQEDI

ADVANCED PROPERTIES OF THE TANGENT MAP

In this section we discuss in some detail the relation between the tangent map T 1 and the derivative of Ifor maps taking values in an A-vector space. Using the tangent map, we generalize the notion of the (directional) derivative of a map with values in an A-vector space to maps with values in a vector bundle. This allows us to show that a vector bundle is trivial as a vector bundle if and only if either its even or its odd part is trivial as a fiber bundle. We then define the tangent map of a map which is itself not necessarily smooth, but which can be seen as being part of a smooth family of maps. As a first application of this generalized notion of tangent map we prove a generalization of the fact that a function is constant if and only if its derivative is zero.

3.1 Discussion. Let M be an A-manifold, 1 : M ----+ A a smooth map and X a smooth vector field on M. With these ingredients we can define a new smooth map XI: M ----+ A as well as a smooth map X 0 T 1 : M ----+ T A. A superficial glance at the definitions of these maps shows that they look very much alike. On the other hand, the notion X 0 T 1 also makes sense for a smooth map 1 : M ----+ N to an arbitrary A-manifold N, whereas XI is no longer defined in such a general setting. However, in particular cases we will need, and we will still can indeed, make sense out of XI. In order to prepare this generalization of the action of a vector field on a function and in order to study in more detail the link between X 1 and X 0 T 1, we start with studying (2.2) in more detail. So let E and F be A-vector spaces, let 0 c Eo be open, and let 1 : 0 ----+ Fo be a smooth map. The tangent spaces of 0 and Fo are the trivial bundles TO = 0 x E and T Fo = Fo x F, and thus the tangent map is a map T 1 : 0 x E ----+ Fo x F. Let (xi) be the coordinates of a point x E 0 with respect to a basis (ei) of E, and let vi be the coordinates of a (tangent) vector vEE with respect to the associated basis (axi). Similarly we denote by yj the coordinates of a point y E Fo with respect to a basis (fj) and by w j the coordinates of a (tangent) vector w E F with respect to the associated basis (ayj). Now let X be a vector

220

Chapter V. The tangent space

field on 0, which we can identify with a map X : 0 ----+ E, i.e., X(x) = 2: i X i (x)8 x i with (smooth) functions Xi : 0 ----+ A. According to (2.2), the map X 0 T f is given by

In this expression, fj = jf 0 f is a homogeneous A-valued function of parity c(yj), and 2:;Xi(x)8x ifj(x) is the function (Xfj)(x), i.e., the action of the derivation X on the function Ii. If we now remember that the basis (8 y j) is essentially the same as the basis (fj) [1.16], it becomes very tempting to write the right hand side as (Xi)(x), in which the action of X on the Fo-valued function f should be interpreted component wise [1.24]. If we are careful, this can indeed be done.

3.2 Lemma. Let M be an A-manifold, F an A-vector space, f : M ----+ Fo a smooth map, and X a smooth vector field on M. In the trivialization T Fo ~ Fo x F, the map X oTf: M ----+ TFo is given as (X o Ti)(m) = (i(m), (Xi)(m)) E Fo x F ~ TFo,

i.e., X oTf

~

(i,Xi).

Proof In [3.1] this was proved in case M is an open subset of Eo. The generalization to arbitrary A-manifolds follows immediately from (1.14).

IQEDI

3.3 Example. Let ¢ E B HomL(E; F) be a smooth linear map. We define the function f E C=(Eo; Fo) as f = ¢IEo. According to [111.3.14] the Jacobian of f is ¢. In the trivializations TEo = Eo x E and T Fo = Fo x F, the tangent map T f is given by

Tf: (x,v) E Eo x E

f-*

(i(x),¢(v)) E Fo x F,

i.e., T(¢IEo) = (¢ X ¢)IEoxE [III.S.14].

3.4 Discussion. In [3.2] we saw that if we know the map f : M ----+ Fo, then the map X 0 T f is completely determined in a very natural way by the map X f : M ----+ F. But X f makes sense also when f takes values in the full A-vector space F, not only in its even part. One thus might think that again X f determines the map X oT f. To investigate this idea, we suppose that f : M ----+ F is a smooth map. As an A-manifold, F is modeled on the A-vector space F~ with basis (fj, Ej) [III. 1.26]. We thus can write

f(m) = L j

fj(m)fj = L(i1(€j)(m)fj

+ ff-€(€j)Ej)

,

j

where f~ denotes the homogeneous part of parity a of the A-valued function fj. In terms of the homogeneous functions fer. : M ----+ Fer. these functions are given by the formula and

221

§3. Advanced properties of the tangent map

Denoting by (yj, fl) the coordinates with respect to the basis (fj, fj), we find

((X(m) liT!))

~(

= ~ j

. a

(((Xlo)(m) Pf))8)' If(m) y

+ «Xh)(m) Ill.f >auaj If(m) ) , y

whereas (X I)(m) is given as 'L,j (X Ji)(m)fj. Obviously these two objects are not the same. Still, there is a link. In order to investigate this link in detail, we introduce, in analogy with the projection 7r : F~ ----+ F defined in [III. 1.26], the left linear map 7rf) : T F = F x F~ ----+ F defined as

7rf)(v

. a = L' . (w +TI:P)f" L (w .ayla +vp-)) &yl. 1

'.

1 _.

1

1

1

It then follows immediately that 7rf)((X(m) liT!)) = (XI)(m). Although 7rf) is not injective, there are several situations in which knowledge of X I is sufficient to reconstruct (( X ( m) I T I))· In the first place if I is an even function, in which case all (( 11 I jf)) are zero, and thus <X (m) I T I)) = 'L,j (( (X I) (m) I jf > yj . It also happens if I is an odd function

a

(with all ((fo I jf)) zero), in which case we have <X(m) I T!)) = 'L,j <(X 1)( m) Pf)) 8yj. A third case is when X is a homogeneous vector field of parity c(X). In that case we have

((X(m) liT!)) = L((((XI).,(X)(m)lljf> ayj

+ <(X/h-.,(x)(m)lljf)) 8yj)

.

j

The conclusion is that ifeither X or I is homogeneous of known parity, then X I determines the vector <X (m) I T I)) completely, i.e., we have proven [3.5]. That these conditions cannot easily be weakened can be seen in the following example. Let I : A ----+ A be the Then map I(x,~) = x +~, i.e., the identity, and let X be the vector field X = obviously XI = O,but ((X liT!)) = X -=I- o.

ax - at;.

3.5 Lemma. Let M be an A-manifold, F an A-vector space, I : M ----+ F a smooth map, and X a smooth vector field on M. Then the map X I : M ----+ F is the same as the map X <> T 1<> 7rf) : M ----+ T M ----+ T F ----+ F. Moreover, ifeither X or I is homogeneous, then X <> T I is completely determined by X f.

3.6 Discussion. A little thought will reveal that there is no hope to define something like X I for a map I : M ----+ B with values in an arbitrary A-manifold B. However, if B is a vector bundle p : B ----+ N over some A-manifold N and if the vector field X behaves properly with respect to the bundle structure, then indeed can one define a new function X I : M ----+ B. The proper behavior of X is that it does not see the base manifold N, but only the fibers of B, which are A-vector spaces. The idea is as follows. Since any vector bundle is locally trivial, we assume for simplicity that B = N x F is the trivial bundle with fiber the A-vector space F. Any

222

Chapter V. The tangent space

map f : M --. B decomposes as a couple f = (g, h) with 9 = 7rN 0 f : M --. N and h = 7rF 0 f : M --. F (see [111.5.12]). If X is a vector field on M, we still do not know what X 9 should be, but X h is defined. So the idea is that if X is such that X 0 Tg == X 0 T(f 0 7rN) is the zero map, then we can define X f as being the couple (g, Xh). If N consists of a single point, the first entry in the product N x F is always the same, so we can ignore it. If we do so, f becomes the map h and X f becomes the map Xh, i.e., we recover the old definition of the action of the vector field X on the F-valued function h. As an example of the setting in which we will need this generalization, we mention the case M = AD X N, B = TN the tangent bundle, f : M --. B of the form f(t, n) E TnN, and X = Since f(t, n) E TnN for all t E AD, X 0 T(f 0 7r) = (with 7r : B = TN --. N the bundle projection), and thus we will be able to define ad : AD x N --. TN. In order to make all this more precise, we need to generalize the map 7rf) : T F --. F for an A-vector space F to the vector bundle setting.

°

at.

3.7 Construction. Let p : B --. M be a vector bundle with typical fiber F over an A-manifold N modeled on the A-vector space E. We thus have the smooth vector bundle map Tp : T B --. TN. Since TbP is surjective for all b E B, we can apply [IV.3.1S] to conclude that ker(Tp) is a subbundle of TB over B. We intend to construct a surjective vector bundle map 7rf) : ker(Tp) --. B such that the special case with N a single point reduces to the previously defined 7rf). The tangent space of a local trivialization U x F of the vector bundle B is given by T(U x F) = TU x T F = U x E x F x F~. Since p is described as p(n, v) = n, the map Tp is described as Tp(n, e, v, 1) = (n, e). The subbundle ker(Tp) is thus described as the points (n, 0, v, 1). Two (local)trivializations Ua x F and Ub x F are related (on Ua n Ub)by (n,va) f-* (n,7/!ba(n)(v a)), with 7/!ba(n) E Aut(F). This implies that the associated trivializations of TB are related by (3.8)

(n, e, v, 1)

f-*

----

(n, e(n, v, e, 1), 7/!ba(n)(v), 7/!ba(n)(f)) ,

where ~) E Aut(F~) is the natural extension of 7/! to F~ defined as follows. Let (fi) be a basis for F, and let (h f i ) be the associated basis of F~. Then for 7/! E Aut( F) we define:(f E Aut(F~) by

This is indeed even, and it preserves the graded subspaces F and Fol of F~ = FEB Fol [111.1.26]. If, as before, 7r denotes the (left linear) projection F~ --. F of [III.1.26], it is obvious from the definition of:(f that 7r o:(f = 7/! 0 7r. The tangent vector e( n, e, v, f) in (3.S) depends linearly on e and f; it picks up tangent vectors in the direction of e because 7/!ba (n)v depends upon n. We now are ready to define the left linear map 7rf) : ker(Tp) C T B --. B. On the local trivialization U x {o} x F x F~ we define itby (3.9)

7rf): (n,O,v,1)

f-*

(n,7rf)(v, 1)) = (n,7r(f)).

§3. Advanced properties of the tangent map

223

Our discussion on how two different trivializations are related proves that this is a globally well defined smooth map. And obviously, if M consists of a single point we recover the initial definition of 7rf).

3.10 Definition. Let p : B ---+ N be a vector bundle with typical fiber F, let f : M ---+ B be a smooth map, and let X be a smooth vector field on M such that X 0 T f belongs to ker(Tp), i.e., for all m EM: ((X(m) IITf 0 Tp> = O. Then we define the map X f : M ---+ B by

Xf = X

0

Tf

0 7rf) •

This makes sense because X oT f takes values in ker(Tp) which is the domain ofdefinition of 7rf). According to [3.5] this definition coincides with the old one in case N reduces to a single point, i.e., B = F an A-vector space.

3.11 Discussion. In order to see that [3.10] fulfills our idea of [3.6], we consider a local trivialization U x F of the bundle p : B ---+ N. In this trivialization the map f restricted to a suitably chosen V c M is given as f( m) = (p(J (m)), h( m)) for some smooth map h : V ---+ F. In other words, in the trivialization U x F the map f is given as the couple (po f, h). According to [2.22] the tangent map Tf is given as (T(J 0 p), Th). Since X 0 T f takes values in ker Tp, we have X 0 T f = (Q, X 0 T h). And then (3.9) and [3.5] tell us that on the trivialization U x F we have

(3.12)

Xf == X oTf 0

7rf)

= (J 0 p, X 0 Th 07rf)) == (J 0 p, Xh) .

We conclude that the formal definition of X f in the bundle setting indeed gives us what we expected in [3.6].

3.13 Lemma. Let f : M ---+ B be a smooth map, p : B ---+ N a vector bundle, and X a vector field on M such that X 0 T f takes values in ker Tp. If either X or f is homogeneous (f taking values in Ber.), then the map X f = X 0 T f 0 7rf) completely determines the map XoTf. This means in particular that for all m EM: (X!)(m) = 0 ifandonly if (X oTf)(m) == ((XmIITJ)) = O.

Proof This is an immediate consequence of [3.5] and the local expression (3.12) for Xf. IQEDI

3.14 Discussion. We now have sufficient material to attack the question of triviality of vector bundles. Let p : B ---+ M be a vector bundle with typical fiber the A-vector space F. Ifwe suppose that B is trivial as a vector bundle, we have a vector bundle isomorphism h : B ---+ M x F. Since the restriction of h to a fiber Bm is an even linear map, it preserves the parity, and thus the restriction of h to the subbundle Ber. [IVA.11] gives us a diffeomorphism h : Ber. ---+ M x Fer.. We conclude that the subbundles Ber. are trivial as fiber bundles.

224

Chapter V. The tangent space

3.15 Proposition. Let p : B ----+ M be a vector bundle with typical fiber F. Then B is trivial as a vector bundle ifand only ifeither Bo or Bl is trivial as a fiber bundle. Proof In view of [3.14] we only have to prove that if Ber. is trivial, then B is trivial as a

vector bundle. For simplicity we only treat the case a = 0, the case a = 1 being similar. So let us suppose that 9 : M x Fo ----+ Bo is a fiber bundle isomorphism. This implies that the restriction of 9 to a fiber {m} x Fo is a homeomorphism onto the fiber (Bm)o, but there is nothing that says that this should be linear (in whatever sense). Let (Xi) be coordinates on Fo with respect to a basis (fi) for F. Then we define vector fields Vi on M x Fo by Vi(m, x) = axi. In terms of a local trivialization U x F of B, the map 9 is described by g( m, x) = (m, g( m, x)) with 9 : U x Fo ----+ Fo a smooth map. It follows that Vi 0 Tg is given in this trivialization as (3.16)

((ViI(m,xdTg~ =

L

agJ

a

axi (m,x) ax j

•

j

Since Bo is a submanifold of B, its tangent bundle T Bo is a subbundle of T BIBa [2.16]. From (3.16) it is obvious that ((Vil(m,x) I Tg)) lies in kerTp [3.7]. We thus can use the map 7r{) to define the smooth sections Si : M ----+ B by

i.e., si(m) = (Vig)(m,O) [3.10]. We claim that these sections satisfy the condition of [IV.3.10-ii], and thus that B is trivial as a vector bundle by [IV.3.10-i]. First we note that the vector fields Vi are homogeneous and everywhere independent, and thus, because T 9 is a diffeomorphism, the tangent vectors (Tg 0 Vi)( m, 0) are homogeneous (of the same parity as fi) and independent vectors in Tg(m,O)BO C Tg(m,O)B. Since 9 takes values in Bo, the vectors (Tg 0 Vi)( m, 0) do not have components in the directions associated to the overlined vectors in F~. But the projection 7r : F~ ----+ F is an even bijection when restricted to the graded subspace generated by the non-overlined basis vectors (see [111.1.26]). It follows that the vectors (7r() 0 T 9 0 Vi) (m, 0) are independent and homogeneous of parity c(fi)' Since Bm is isomorphic to the A-vector space F, an independent set of the correct number of vectors is also generating, i.e., a basis. We conclude that the sections Si satisfy [IV.3.1O-ii], and thus that B is trivial as a vector bundle by [IV.3.10-i]. In the odd case (a = 1), the only difference worth mentioning is that there the vectors (Tg 0 Vi)( m, 0) lie completely in the graded subspace generated by the overlined vectors. And then the restriction of 7r : F~ ----+ F to that graded subspace reverses the parity of these vectors. IQEDI

axi

3.17 Counter example. Let p : B ----+ M be a vector bundle with typical fiber F. If B is trivial as a vector bundle, we have seen that then Ber. is trivial as a fiber bundle. Applying the body map, we deduce that BB ----+ BM is trivial as an R-vector bundle and that BBer. ----+ BM are also trivial as a fiber bundle. Now let F be of dimension 111 with basis fo, fl (fer. of parity a), and let M = GS 1 be the A-manifold without odd dimensions whose underlying R-manifold is the circle.

225

§3. Advanced properties of the tangent map

We cover M by two charts U± given by U+ ~ {() E AD I 0 < B() < 27r} and U_ ~ {() E AD I -7r < B() < 7r}. The change of charts ip_+ : U_ ---+ U+ is given by () 1-* () if B() > 0 and () 1-* () + 27r is B() < O. Over M we define the vector bundle p : B ---+ M with typical fiber F by the transition functions 'I/J-+ given by () 1-* (~ ~ ) if B()

> 0 and() 1-* (~1 ~1) ifB() < O.

This vector bundle is essentially twice (in the 10 direction and in the h direction) the Mobius bundle over the circle. We claim that this bundle is trivial as a fiber bundle, but not trivial as a vector bundle. To prove that it is not trivial as a vector bundle, we consider the fiber bundle BBo ---+ BM. This is a fiber bundle over the circle S1 with typical fiber R ~ BFo given by the transition functions (f;-+ (()) = sign () E Aut(R), i.e., the Mobius bundle, which is not trivial. Hence B can not be trivial as a vector bundle. On the other hand, let us denote by (x,~, y, 7]) the even and odd coordinates in F ~ Fg according to (x + ~)10 + (y + 7])h = xlo + 7]h + {fo + yf1' Then we can define a fiber bundle isomorphism h : M x F ---+ B in the trivialization U+ by

h((), x,~, y, 7]) = ((), cos(()/2)x + sin(()/2)y, cos(()/2)~ + sin(()/2)7], - sin( () /2)x

+ cos( () /2)y, -

sin( () /2)~

+ cos( () /2)7])

.

We leave it to the reader to show that this can indeed be extended to a global smooth diffeomorphism. Since it obviously respects the fibers, it is a trivialization of B as a fiber bundle. This example shows that a vector bundle which is trivial as fiber bundle need not be trivial as a vector bundle. The obstruction lies in the fact that the group Aut(F) is more or less incapable of mixing even and odd basis vectors. This is seen most clearly when we look at a matrix representation of B Aut(F), which decomposes as Gl(p, R) x Gl( q, R) if the dimension of F is plq.

3.18 Remark. In the special case A = R, a vector bundle p : B ---+ M is trivial as a vector bundle if and only if both Bo and B1 are trivial as fiber bundles. The reason is that for generic A (satisfying (C [=l)) the subbundle Bo still contains information about B1 in the form of the odd coordinates with respect to the odd directions, while for A = R one loses this information. This is confirmed by the fact that for A = R the group Gl(plq, R) decomposes as the direct product Gl(p, R) x Gl( q, R), whereas for generic A there is a cross-over between the even and odd directions in the form of odd matrix elements.

3.19 Discussion. Let M, N, and Q be three A-manifolds and 1 : M x N ---+ Q a smooth map. For a fixed m E M we define the map 1m : N ---+ Q by 1m (n) = 1(m, n). According to [III.1.23-g], if m has real coordinates, then 1m is smooth. On the other hand, if m does not have real coordinates, there is no reason to assume that 1m will be smooth (in general it will not be). Thus we can not speak about the tangent map T 1m for an arbitrary fixed m E M. Since it is often desirable to do so anyway, we will circumvent this problem as follows. We define the generalized tangent map T 1m : TN ---+ TQ by

226

Chapter V. The tangent space

Since T f is even and left linear on any fiber T( m, n) M x N, T f m is even and left linear on any fiber TnN. But for general m it will not be a smooth map. However, if we do not look at a fixed value of m, but see it as a variable, we get a map M x TN ---+ TQ. Using the identification T(M x N) ~ T M x TN [2.21], we can see this map as the composition of the smooth maps Q. x id(T N) : M x TN ---+ T M x TN and T f. It follows that the family of maps Tfm with varying m E M is smooth. ---+ Q by fn( m) = f(m, n). In a similar way wecanfixapointn E N and define And then we define the generalized tangent tangent map T f n by

in : Ai.

the maps f m, this is an even (left) linear map when restricted to a fiber T mM, but T f n need not be smooth. However, when we see n as a variable, we get a smooth map T M x N ---+ TQ (the restriction ofT f to TM times the zero section), which is smooth. More generally, let 0 C M x N be an open set and f : 0 ---+ Q a smooth map. For a fixed m E M we define the set Om C N by 0 n {m} x N = {m} X Om. It is an open subset of N on which we have a map fm : Om ---+ Q defined by fm(n) = f(m, n). As before, if m does not have real coordinates there is no reason to assume that f m is smooth, but we can define a generalized tangent map T f m : TOm ---+ TQ by the formula (( Yn I T f m> = (( (Q.m, Yn ) I T f)). To show that the family of tangent maps T f m is smooth, we need a more careful discussion about the domain of definition of this family. Since we will not need it, we will not go into these details. A~...f0r

3.20 Lemma. Let M, N, and Q be A-manifolds of which N is connected, and let ---+ Q be smooth. Then the following three assertions are equivalent.

f :M x N

(i) There exists a smooth map 9 : M ---+ Q such that f = go 7rM. (ii) There exists a collection of vector fields X on N such that at each n E N the Xn generate TnN and such that (Q. x X) 0 T f = O. (iii) \f(m, n) EM x N: Tnfm = O.

In the case Q is an A-vector space, the condition (Q. x X) 0 T f = 0 in (iO can be replaced by (Q. x X) f = O. Proof. (ii) <=} (iii) By definition we have HQ. x X)(m,n) IITn = <Xn IITnfm>, hence if Tnfm = 0 then (Q. x X) 0 Tf = O. On the other hand, if the Xn generate TnN, it follows that (ii) implies that Tnfm is zero on a generating set, i.e., Tnfm = O. • (i) <=} (ii) Since T f = Tg 0 T7rM and (Q. x X) 0 T7rM = 0, we have immediately the implication (i) =} (ii). To prove the reverse implication we proceed as follows. For each (m, n) E M x N we choose a chart Wm,n of Q around f(m, n) and then, by continuity of f, charts Um,n of M around m and Vm,n of N around n, Vm,n connected, such that f(Um,n x Vm,n) C Wm,n' It follows that U = {(Um,n x Vm,n I (m, n) EM x N} is an open cover of M x N with the property that for each U x V E U there exists a chart W such that f(U x V) C W. Wenowfirstprove a local result. For U x V E U and We Q such that f(U x V) C W, we choose coordinates Xi on U, yj on V, and zk on W. For any vector field X on N its local

§4. Integration of vector fields

expression is given as Xlv becomes

227

= 2: j Xj(y) ayj. And then the condition (Q x X) <>Tf = 0

Since we have this equation for vector fields X that generate at each point the tangent space, we conclude that we have (ayj fk)(x, y) = O. By [111.3.15] we conclude that there exist functions gtxv : U ----+ A such that fk luxv = gtxv 0 7ru. Combining the functions gtxv into a single function guxv : U ----+ W, we conclude that for each U x V E U there exists afunction gux v : U ----+ Q such that flux v = gux v 0 7r M lux v· Once we have the local result, we have to glue it together to get the global result. For each m E M we define Urn = {U X V E 1m E U}. Obviously Urn covers {m} x N which is connected. For Ui x Vi, U2 X V2 E Urn such that Vi n V2 i:- ~ we must have gUI x VI (m) = gU2 X V 2 (m), and thus by connectedness of N, there exists g( m) E Q such that\fU x V E Urn: guxv(m) = g(m).t In this way we obtain afunctiong: M ----+ Q. But for any U x V E U and any m E U we have gux v( m) = g( m) and thus 9 is smooth. Moreover, glu 0 7rMluxv = guxv 0 7rMluxv = fluxv and thus 9 0 7rM = f· • If Q is an A-vector space and if (ii) holds, then by [3.5] obviously (Q x X)f = O. On the other hand, if (Q x X) f = 0 holds for vector fields that span the tangent space of N at each point, we can assume that it holds for all vector fields X on N. It thus holds in particular for all homogeneous vector fields, which span TN. For these we can apply [3.5] in the other direction to conclude that (Q x Y) <> T f = 0 for a set of vector fields on N which generates TN (i.e., all homogeneous vector fields). But then it also holds for the original set of vector fields X. IQEDI

3.21 Corollary. Let Nand Q be A-manifolds, N connected, and let smooth. then the following three assertions are equivalent.

f :N

----+

Q be

(i) f is constant. (ii) Tf=O. (iii) There exists a collection of vector fields X on N such that at each n E N the Xn generate TnN and such that X <> T f = O.

In the case Q is an A-vector space, the condition X <> T f

= 0 in (iO

can be replaced by

Xf=O.

tThis is an often neglected exercise in point set topology. Let X be a connected topological space, U an open cover of X, and Xo E X an arbitrary point. If C(xo) denotes the set of x E X for which there exists a finite sequence Uo , . .. , Un E U such that Xo E Uo, x E Un, and Ui nUi+l of- ¢, then C(xo) = X.

228

4.

Chapter V. The tangent space

INTEGRATION OF VECTOR FIELDS

In this section we investigate how to integrate homogeneous vector fields: even vector fields should have an even time parameter, odd vector fields an odd one. It turns out that for even vectorfields the results are the same as in the ordinary case: there exists aflow on an open connected subset of time times the manifold, and this flow has the usual properties of a i-parameter group of local diffeomorphisms. For odd vector fields a flow exists if and only if its auto commutator is zero. If that is the case the flow is defined on the whole of the (odd) time times the manifold.

4.1 Definitions. An interval is a connected subset I of Aa; it is called even if a = 0, odd ifa=l. Let Ie Aa be an open interval, to E BI a real number, and U an open set in M. An open subset W of I xU c Aa x M is called a wave in I x U around to if {to} x U c W and if W n Aa x {m} is an open interval for all m E U. In particular, a wave is connected if U is. A border point for W is a point (tl, md E (BI) x U \ W such that J x {mt} n W -=I=- ¢ for any open interval tl E J c Aa. Existence of a border point requires that W is not the whole of I x U.

4.2 Lemma. There exists only one odd open interval: Al itself. An even open interval I is of the form I = {a E Ao I Te < Ba < Tr }for some Te, Tr E R U {-oo, oo}.

4.3 Definitions. Let X be a homogeneous vector field of parity a on M. A local flow for X is a smooth map ¢ : W ----+ M defined on a wave W in I x U (U eM, I an interval) around to E I satisfying the condition (4.4)

at oT¢ = ¢oX .

or equivalently

at

In this condition t denotes the (canonical) coordinate on Aa and is the shorthand for x Q. [2.24]. The coordinate t is usually the vector field on Aa x M officially denoted by and X called the time coordinate. Condition (4.4) can be stated in words as saying that are of parity a. are related by ¢; it makes sense because both X and An initial condition at time to for a local flow ¢ : W ----+ M for X is a smooth function f : U ----+ M such that ¢(to, m) = f(m) for all mE U. Note that we need that to is real to be sure that ¢(to, J is a smooth function on {to} x U. The flow ¢x : W X ----+ M (without the adjective local) for the homogeneous vector field X is a local fl ow defined on a maximal wave W X in Aa x M around 0 and satisfying the initial condition id(M) at time t = O. A homogeneous vector field X is said to be integrable if it admits a flow. We will show that this is the case if and only if [X, X] = 0, a condition automatically satisfied for even vector fields.

at

at

at

229

§4. Integration of vector fields

4.5 Discussion. We start our investigations for even vector fields X only and thus assume that we have an even time parameter t E Ao. To see more clearly what condition (4.4) amounts to, we assume that we have a local flow ¢ : W ---+ M defined on a wave in I x U around to. We assume furthermore that U is a chart and that there exists a chart V such that ¢(W) c V. Let (x,~) be local (even and odd) coordinates on U and let (y, TJ) be local coordinates on V. Using the notational shorthand introduced in [111.3.17], we write (y, TJ) = ¢(t, x,~) with

and From this we deduce that
IIT¢ > == (at 0 T¢)(t, x,~) is given by

Using the same shorthand, we write the tangent vector X(y, TJ) as

2) 'F+ a (~ 2- 1) 'F' a ~X2i+1(Y)'TJ'+

~ X(Y,TJ)= ( ~X2i(Y)'TJ' i

y

Substituting (y, TJ) = ¢(t, x, ~), using T¢ 0 at = obtain the following two sets of equations:

L

a::i(t,x)

·e

i

TJ

i

x

0

¢ and expanding as in [111.2.27], we

=

i

L

a¢~;+1 (t,x). ei+l =

i

Comparing the various powers of ~ gives us the equations

(4.6.a) (4.6.b) (4.6.c) (4.6.d)

p2::1

230

Chapter V. The tangent space

where the Pi are expressions in the Xj and (Pk with k < i. For arguments sake we now assume that we have at time to E f an initial condition f : U ~ V, f(x,~) = (y, TJ), which we expand into powers of ~ as y = 2: i hi(X) . i and TJ = 2:d2i+l(X)' i + 1 . On W n {to} x U, the requirement
e

e

Finding a

0 takes its values in BV, the others can take their values "anywhere." Since the Pi do not involve the 0 (they are of the form y' = ay + b), the fact that these
4.7 Lemma. Let W be a wave in I x U around to with U a chart and I an interval, let V be another chart for M, and let f : U ~ V be a smooth function. There exists a local flow

4.8 Lemma. Let Wo be a wave in fax Uo around to with Uo an open set in M and fa an interval, and let f : Uo ~ M be a smooth function. There exists a local flow
§4. Integration of vector fields

231

Our first claim is that W is not empty. For each m E Uo we choose local charts mEUm and Vm and an open interval to E 1m such that'IjJ is a local flow on BIm x BUm with values in BVm and with initial condition Bf at time to. According to [4.7] there is a unique local flow
4.10 Remark. It follows from [4.8] that (

4.11 Examples. • Let M = Ao be the A-manifold with (global) coordinate x on which Since there are no odd coordinates, we we define the even vector field X (x) = x 2 .

ax.

Chapter V. The tangent space

232

only have to solve the equation ottO (t, x) = (
- tx

• Now let E be an A-vector space of dimension 111. On M = Eo with coordinates (x, ~) with respect to some basis we define the even vector field X (x,~) = eX ·8x + x~ . 8t;. In this case we have a
7)t(t,x) = Xl(
x - In(l - eX) ) (TJy) (exp((x+1)t+(e-X-t)ln(1-teX))~ t

=

Wx =

{(t,x,~)

I B(te X) < 1} .

• Finally, let Ebe an A-vector space of dimension 112. On M = Eo with coordinates (x, 6, 6) we define the vector field X(x, 6, 6) = (~x3 +x66)· 8 x +6. 8 6 -6· 8t;2' which is even. Since we have two odd coordinates, we have a
8~o (t, x) = ~(<po(t, x))3

8~1 (t, x) =

X 1 (
(~1 ~).
8
= ~(<po(t, x) )2
= ~ (
using the previous equation.

Careful analysis, using the initial conditions
(V1:

Y) (

TJl

TJ2

-

tx2

+

(cost

- sin t

J~;_~:::~3

66)

sint)(6) cos t

6

= 0,

,Wx={(t,x,~)IB(tx2)<1}.

§4. Integrati on of vector fields

233

4.12 Lemma. Let X be an even vector field on M and ¢x : Wx ----+ M its flow. If ¢x(t,m) is defined, then ¢x(-t,¢x(t,m)) is defined; if ¢x(t,¢x(s,m)) is defined, then ¢x (s + t, m) is defined; ifs and t have the same sign and if¢x (s + t, m) is defined, then ¢x(t,¢x(s,m)) is defined.

Proof The statements say something about the domain W X . Since W X is completely determined by its body BWx == BWBx C R x BM, the result follows from the fact that these statements are true for ordinary vector fields on R-manifolds (e.g., [HS, §8.7]).t

IQEDI

4.13 Lemma. If both ¢x(t o, ¢x(so, mo)) and¢x(so+t o , mo) are defined, they coincide.

Proof Continuity of ¢x and openness of Wx imply the existence of open intervals So E Is and to E It and an open set m E U c M such that both terms are defined for all (t, s, m) E It x Is x U. Moreover, the existence of both terms implies that ¢x (so+O, mo) and ¢x (so + to, mo) are defined as well as ¢x(O, ¢x(so, mo)) and ¢x(to, ¢x (so, mo)). Since Wx is a wave, we may assume that 0 E It. Now define two maps IPi : It x Is x U ----+ Ao x M by

IP 1 : (t,s,m) IP2: (t,s,m)

f---+

(s+t,m)

f---+

(s+t,¢x(s+t,m))

f---+

(t,s,¢x(s,m))

f---+

(s

+ t, ¢x(t, ¢x(s, m)))

.

U sing the given decomposition and the fact that ¢ X is the flow of X, it is not hard to show that both IPi satisfy the relation TIPi 0 at = (as x X) 0 IPi [2.24]. We conclude that both IPi are local flows for the vector field (as x X). Since they have the same initial condition IQEDI at time t = 0, they coincide.

4.14 DiscussionlDefinition. In the context of R-manifolds, the above results are usually stated by saying that the family of maps {¢x,d, defined by ¢x,t(m) = ¢x(t, m), forms a I -parameter group of local diffeomorphisms of M; local because ¢ x, t need not be defined on the whole of M. In the context of A-manifolds such a formulation is not possible because for non-real t the map ¢X,t is in general not smooth. However, the group property as well as the local diffeomorphism property can be stated in a remarkably simple statement about a global diffeomorphism. It follows from [4.12] that the smooth map

IPx: Wx

----+

Wx

(t,m)

f---+

(-t,¢x(t,m))

tPor ordinary vector fields, these results follow from the fact that n is defined as the collection of all maximal integral curves. And then the statement that n is open needs a proof. In our context we cannot speak of individual integral curves (they are not smooth). Hence the approach inwhich n is automatically open. But then it seems not to be easy to prove the results of [4.12]. I thus "cheat" and rely upon the classical result.

234

Chapter V. The tangent space

is well defined. It then follows from [4.13] that IPx is a diffeomorphism satisfying the equality IPx 0 IPx = id(Wx). In this way the fact that ¢X,-t = (¢X,t)-l is captured in the statement (IP x) 2 = id(Wx) about the global diffeomorphism IP x. Since we will use this IP X sometimes, we will call it the global flow of X to distinguish it from the flow ¢ X without adjective. In the same spirit we can capture the group property. We define Wx c Ao x Wx as Wx = {(s, t, m) I (t, m) E Wx & (s, ¢x(t, m)) E Wx }. It follows from [4.12] that the smooth map

(s,t,m)

f--*

(-s-t,s,¢x(t,m))

is well defined. It then follows from [4.13] that IPx is a diffeomorphism satisfying the equality IPx 0 IPx 0 IPx = id(Wx ). Taking s = 0 in the relation (IPX)3 = id(Wx ) and using the initial condition for ¢x gives us (again) that ¢X,-t = (¢X,t)-l. This plus the general case of (IPX)3 = id(Wx ) gives us the group property ¢X,8 0 ¢X,t = ¢x,s+t.

4.15 Discussion. So far we have discussed the case of even vector fields. Let us now turn our attention to odd vector fields. To give an odd flavor to this discussion, we will denote the odd time parameter by T rather than by t, in accordance with our general principle to denote odd coordinates by greek letters. In order to investigate the existence of a flow ¢ for an odd vector field X, we start looking at condition (4.4) in a local chart. Let U and Vbe local charts such that ¢(Al xU) C V. Because we have an odd time parameter, we will use a completely different approach. We denote by x the (even and odd) coordinates on U, by y the coordinates on V and we expand y = ¢(T,X) as y = ¢o(x) + T' ¢l(X) . It follows that ((aTI(T,X) IIT¢))

== (aT <>T¢)(T,X) is

given by

Writing X(y) = Xe(Y) . ay, substituting y = ¢(T,X), and expanding into powers of T gives us

X(y) = (Xe(¢o(X))

+ T' ¢l(X)' a~e(¢o(X)))

. :y .

We conclude that condition (4.4) translates as the equations (4.16)

aXe (

(

¢l(X)' ay ¢ox))=O.

It now becomes very important to realize that X is supposed to be an odd vector field. Substituting the first equation of (4.16) in the last one, one should recognize (see [1.23]) the local expression (up to a factor 2) of the coefficients of the commutator [X, X] at the point ¢o(x) = ¢(D, x). Moreover, since 0 always lies in the odd open interval Al (it is

§5. Commuting flows

235

the only real value in AI), we can add the initial condition ¢(O, x) = f(x). The whole set of equations, initial condition plus (4.4), thus becomes equivalent to the set of equations

¢O(x)

= fex)

¢l(X)

=

Xc(J(x))

[X, XJ(J(x))

=0

.

This discussion is summarized in [4.17].

4.17 Proposition. Let X be an odd vector field on M and let f : M ---+ M be smooth. There exists a (unique) local flow ¢ : Al x M ---+ M for X with initial condition f at time o ifand only if [X, XJ = 0 on im(J). In particular, the flow ¢x of an odd vector field on M exists if and only if [X, XJ = o. If this condition is satisfied, it is unique and W X = Al X M. It has the following properties. (i) ForanyopensetU eM: ¢X(AI xU) C U. (ii) \:IT,O" E Al \1m EM: ¢X(T, ¢x(O", m)) = ¢x(O"

+ T, m).

Proof Property (i) follows from the facts that open sets are saturated with nilpotent vectors and that the initial condition is the identity f(x) = x. One way to prove property (ii) is to use a local chart and to make an explicit calculation, judiciously using [X, X] = O. Another way is to copy the proof of the case for even vector fields. Consider the two smooth maps i : Al x Al X M ---+ Al X M :

l : (T, 0", m) 2 : (T, 0", m)

f-* f-*

(0" + T, ¢X(O" + T, m)) (0" + T, ¢X(T, ¢X(O", m))) .

As in the even case, one shows that both maps satisfy Ti 0 aT = (aa x X) 0 i and i(O,O",m) = (0", ¢x(O", m)). We deduce from the uniqueness of flows that the i coincide. IQEDI

4.18 Discussion. If X is an integrable odd vector field on M, the local expression for its flow is essentially given by the simple formula

¢X(T,m)=m+T·Xc(m) , where we obviously confuse the coordinates of a point m with the coordinates of a tangent vector Xm = Xc(m) . Unlike the case for even vector fields, for odd vector fields we do not have a domain problem: ¢x, if it exists, is defined on the whole of Al x M. As for even vector fields, we can form the global flow X : Al x M ---+ Al X M, X(T,m) = (-T,¢x(T,m)) wi th the property ( x) 2 = id. Likewise we can construct the global diffeomorphism >': : Al X Al X M ---+ Al X Al X M with the property (>.:)3 = id.

am.

236

5.

Chapter V. The tangent space

COMMUTING FLOWS

In this section we show that two integrable homogeneous vector fields commute if and only if their flows commute. As a byproduct we show that if X is an integrable homogeneous vector field with flow 1>t and Yan arbitrary vector field, then the commutator [X,y] equals atT1>-tY at t = O.

5.1 Definitions. Let X and Y be two homogeneous integrable vector fields on an Amanifold M with flows 1> X and 1>y respectively. We say that the flows commute if we have the equality 1>x (t, 1>y(s, m)) = 1>y(s, 1>x(t, m)) for all points (t, s, m) for which both sides make sense. It is the purpose of this section to show that the flows commute if and only if the vector fields commute, i.e., [X, Y] = O. The proof of this statement requires some preparatory lemmas, some of which have an interest on their own.

5.2 Lemma. Let M and N be two A-manifolds. Let I be a homogeneous open interval, W a wave in I x M around to for some real value to E f. Let Y be a homogeneous vector field on N of the same parity as 1. Finally let fi : W ---'>- N be two smooth maps that coincide on {to} x M such that T fi Oat = Yo J;. Then II = h Proof Define gi : W x N ---'>- M x N by gi(t, m, n) = (m, fi(t, m)). It follows that T gi Oat = (Q. x Y) ° gi and that the gi coincide on {to} x M x N. The result now follows IQEDI from uniqueness of (local) flows with initial condition [4.8] and [4.17].

5.3 Corollary. Let mo E BM be a point with real coordinates and X a homogeneous vector field on M with flow 1> X such that X (mo) = O. Then for all t : 1> X (t, mo) = mo· Proof Let Wx be the domain of definition of 1>x and fmo = Wx n Aa x {m o }. If II : fmo ---'>- M is defined as lI(t) = 1>x(t, m o), then by definition of the flow we have Tho at = X ° II. But for h : fmo ---'>- M defined by h(t) = mo we also have Tho at = 0 = X (m o ) = X ° f2. We conclude by [5.2]. IQEDI

5.4 Corollary. Let f : M ---'>- N be a smooth map and let X be an integrable vector field of parity a on M with flow 1> x· Then T foX = 0 if and only iff ° 1> X = f ° 7r2 on W x, where 7r2 : Aa x l\!f ---'>- M denotes the canonical projection. Proof If f ° 1> X = f ° 7r2 then, by applying the tangent map, we obtain the equation 0= TfoT7r2 Oat = TfoT1>xoat = TfoXo1>x. Restriction of this map to {O} xM gives T foX = O. If on the other hand T foX = 0, we deduce that T(f ° 1>x) Oat = Q. and T(f ° 7r2) Oat = Q.. Since the two maps f ° 1> X and f ° 7r2 coincide on {O} x M, the result follows from [5.2]. IQEDI

§5. Commuting flows

237

5.5 Lemma. Let 1 : M ----+ N be smooth, let X be an integrable vector field of parity a on M, and let Y be an integrable vector field of the same parity a on N. Ifwe have ¢y 0 (id x 1) = 1 0 ¢x on an open set containing {O} x M then X and Yare related by f. On the other hand, if X and Yare related by 1, then ¢y 0 (id x 1) = 1 0 ¢ X on W x· Proof Suppose ¢y 0 (id x 1) = 1 0 ¢ x. Applying the tangent map of this relation to the on Ao x M (more precisely, on the open set containing {O} x M) gives us vector field

at

T 1 0 T ¢X

T10 T10

~ ~

x x

0 0

0

at = T ¢y

0

T( id x 1) 0

at

(at

x Q) 0 (id x 1) ¢x = Yo ¢y 0 (id x 1) = Yo 1 0 ¢ X

¢ X = T ¢y 0

Restricting this identity to {O} x M proves that Y 0 1 = T 1 0 X. To prove the second part we start with a local statement. Since 1 is smooth, for any point m E M there exist open sets m E U c M and l(m) EVe N, and an open interval I containing 0 such that 1(u) c V, I x U c Wx, and I x V c W y . We compute T(J 0 ¢ x) 0 = Yo (J 0 ¢ x) and T( ¢y 0 (id x 1)) 0 = Yo (¢y 0 (id x 1)). Since the maps 1 0 ¢ X and ¢y 0 (id x 1) coincide on {O} x U, we conclude from [5.2] that the maps coincide on I x U. By gluing these local subsets I x U together, we conclude that 1 0 ¢x and ¢y 0 (id x 1) are defined and coincide on a wave W in Au x M around O. Let Wo be the biggest wave with these properties. We claim that Wo = W x , a fact which is obvious in the odd case. In case X and Y are even, suppose Wo is strictly included in W x. There thus exists a border point (tl' ml) E Wx \ Wo for WOo Since Wo is open and ¢x continuous, there exist open tIE I and ml E U such that (I - I) x ¢ X (I X U) c W o , where I - I = {a - b I a, bEl} is an open interval containing O. Moreover, since W X is open, we may assume I x U c W x. By definition of (it, mI) there exists a t2 E I such that (t2, mI) E WOo Since Wo is open, we may assume {t2} x U C Wo (shrinking U if necessary). For an arbitrary tEl we compute

at

1(¢x(t, m))

at

= 1(¢x(t, ¢x( -t2, ¢X(t2, m)))) = 1(¢x (t - t2, ¢X(t2, m))) = ¢y(t - t2, 1(¢x(t2, m)))

by [4.12] and [4.13]

= ¢y(t - t 2, ¢y(t2' l(m)))

because {t2} xU

= ¢y(t, l(m))

by [4.12] and [4.13].

We conclude that I x U proves that W X = WOo

because (I - I) x ¢x(I x U)

c Wo bymaximality, contradicting (tl, mI)

c Wo

c Wo

E I x U\ Woo This

IQEDI

5.6 Corollary. If 1 : M ----+ N is a diffeomorphism and X an integrable homogeneous vector field on X with flow ¢ x, then Y = T 1 0 X 0 1-1 is an integrable vector field on N with flow ¢y = 1 0 ¢x 0 (id xI-I).

238

Chapter V. The tangent space

5.7 Lemma. Let X be a homogeneous integrable vector field on M and let P be any open subset ofN x M. If ¢x : Wx ----+ M is the flow of X on M, then the flow of Q x X on P is given by

(5.8)

¢QxX : (t, n, m)

f-*

(n, ¢x(t, m)) .

Its domain of definition WQxx is Al x P for odd X; if X is even it is given by WQxx

=

{(t, n, m) E AD x P

I (t, m)

E Wx & :JIo,t :

{n} x ¢x(Io,t, m)

c P} ,

where Io,t denotes an open interval containing both 0 and t. Proof We apply [5.5] to the two canonical projections 7rN : P ----+ Nand 7rM : P ----+ M, which satisfy T7rM 0 (Q X X) = X 07rM and T7rN 0 (Q X X) = Q0 7rN. This gives us as result that 7rM 0 ¢Qxx = ¢x 0 (id X 7rM) and 7rN 0 ¢Qxx = ¢Q 0 (id x 7rN)' This proves that (5.8) is valid on the domain of definition of ¢Qx x. This finishes the proof for odd X. For even X it remains to show that the given W Qxx coincides with the domain of definition of ¢Qx x, which we temporarily denote by WOo Since ¢ X is continuous, P open and WX a wave, it follows that WQx X is a wave in AD x P around 0 on which the map 'IjJ : (t, n, m) f-* (n, ¢x(t, m)) makes sense. An elementary calculation shows that T'IjJ 0 at = Q x X 0 'IjJ. By maximality ofWo this implies that WQxx CWo. On the other hand, if I, U C NI, and V C N are open such that I x V x U cWo, it follows from our first observation that I x U c W x. Moreover, since W X is a wave, we may assume that o E I. It follows that I x V x U C WQxx, proving the other inclusion. IQEDI

5.9 Corollary. Let X and Y be integrable homogeneous vector fields on M. Their flows commute if and only if Y and Q x Yare related by ¢x (ifand only if X and Q x X are related by ¢y). Proof Consider the vector field Q x Y on W X . It follows from the explicit expression (5.8) that the flows of X and Y commute if and only if ¢y 0 (id x ¢x) = ¢x 0 ¢QxY. According to [5.5] this is true if and only if T ¢ X 0 (Q x Y) = Yo ¢ x. Interchanging the IQEDI roles of X and Y proves the second part.

5.10 Discussion. We continue with the preparations for our characterization of commuting flows and we take a closer look at the tangent map of the flow ¢ X of a homogeneous vector field X on an A-manifold M. We will use the symbol t for the time parameter, even if the vector field X is odd. In order to simplify this discussion, we will drop (in this discussion) the subscript X in ¢ X . We start by choosing a point (s, t, m) E W X (i.e., such that ¢(s, ¢(t, m)) makes sense) and we choose local coordinate systems x~ around m, x~ around ¢(t, m), and x~ around ¢(s, ¢(t, m)) == ¢(s + t, m). We then note that a T¢: -a i ICt,m) Xa . a T¢. atICt,m)

~ a¢b

f-*

f-*

L j

a~ a -ai (t,m). a j I¢Ct,m) Xa Xb a

_

i

a

~ at (t,m)· a jIW,m) -Xb(¢(t,m))a jl¢Ct,m.), j

Xb

Xb

§5. Commuting flows

239

where the last equality follows by definition of a flow. Since IP3e = id, the same is true for the tangent map: (TIP X)2 = id; applied to the vectors ax~ I(t,m) and at! (t,m)' this identity gives the relations .

k

~ a¢~ a¢b ~ - . (-t,¢(t,m))· -a. (t,m) i

aXb

x~

k

= OJ ,

LX~(m). ~!~(t,m) = Xi(¢(t,m)). i

a

Applying likewise the identity (TIP X)3 = id to the vectors ax~ I(s,t,m) gives the additional relation (5.11)

a¢b a¢~ ~ -a. (t,m). - . (s,¢(t,m)) ~ x' ax) j a b

=

a¢~

-a. (s+t,m). x'a

Now if sand t are close to zero, we may assume that the coordinate systems are the same, i.e., a = b = c. We then obtain in particular the relation

a¢~ (0 m) = Oi

(5.12)

ax~'

)

.

Applying [111.3.13] then gives the useful identity a2¢~

-~. (O,m) =

(5.13)

ax~ ax~

o.

5.14 Discussion. Let us now return to the actual characterization of commuting flows. According to [5.9], the flows of X and Y commute if and only if Y and Q x Y are related by ¢x, i.e., Yo ¢x = T¢x 0 (Q x Y). Composing on the right with the diffeomorphism IP X shows that this is the case if and only if Yo 7r2 = T¢ X 0 (Q x Y) 0 IP x, where 7r2 : W X ---+ M denotes the canonical projection (t, m) f---+ m. Since the left hand side is rather easy to understand, we have to study the right hand side in more detail. We thus define the function 'IjJ : W X ---+ T M by

'IjJ

= IPx 0 (Q x Y) oT¢x .

Now IPx maps (t, m) to (-t, ¢x(t, m)), then Q x Y maps this point to a tangent vector at this point, and finally T¢x, which maps the base point (-t, ¢x (t, m)) back to m, sends this tangent vector to a tangent vector at m. In short, 'IjJ(t, m) E TmM. If'IjJ equals 7r2 0 Y, this implies that 'IjJ(t, m) should be equal to Ym E T mM, independent oft. It now follows easily from the relation (5.12) that 'IjJ(0, m) = Ym , so it remains to show that 'IjJ(t, m) is independent of t. Applying [5.4] to the vector field at on W X shows that this is the case if and only if at 0 T'IjJ = O. Since the vector field at and the map 'IjJ satisfy the conditions of [3.13] and since at is homogeneous, we conclude that at 0 T'IjJ = 0 if and only if at'IjJ = O. To summarize this discussion, we have shown that the flows of X and Y commute if and only if at'IjJ = O. Hence our interest in the quantity at'IjJ, where we recall that t denotes the time parameter whose parity equals the parity of X.

240

Chapter V. The tangent space

5.15 Proposition. Let X be a homogeneous integrable vector field on M and let Y be an arbitrary vector field on M. If 'IjJ = X 0 (Q x Y) 0 T¢ X : W X ---+ T M, then 8t 'IjJ : Wx ---+ TM is given by 8t 'IjJ = x 0 (Q x [X, Y]) oT¢x. Using the notation of the generalized tangent map [3.19J and abbreviating ¢x to ¢, this can be written as (5.16) Proof To simplify the notation, we will, as in [5.10], write ¢ for ¢ X throughout this proof. Let us first consider the last statement. Writing the definition of'IjJ(t, m) explicitly gives

'IjJ(t,m) = <(Q-t'Y¢t(m»)IIT¢> =
xi

Xes, t, m)

~ k 8¢~ =~ Yb (¢(s + t, m)) . - k (-s -

k ,t

8Xb

t, ¢(s

+ t, m)) . -88i 1m Xa

where we introduced in = ¢(t, m) and where we used (5.11). In the same trivialization we now compute (8 s X)(O, t, m), which, according to (3.12), means that we have to differentiate the fiber function (meaning the coefficients of 8x~ 1m) with respect to sand not touching the base point (meaning m, which indeed does not depend upon s).

§5. Commuting flows

241

To obtain the second equality we have applied in particular (5.12) and (5.13). The additional sign appears because we have to change the order of the partial derivatives as and axk. We now note that c:(xt) is the parity of the k-th basis vector in E, and thus that b

+ c:(xt) = c:(Y). We also note that c:(s) = c:(X). Now comparing the expression in parentheses with (1.22), we see that it represents the coefficients of [X, Y] (m) with respect to the basis ax] I~. We conclude that (asX) (0, t, m) == (at 7/!) (t, m) gives the local m C:(Ybk )

b

expression of x 0 (Q x [X, Y]) have finished the proof.

0

T¢x. Since the point (t, m) is chosen arbitrarily, we IQEDI

5.17 Remark. An analogous result for k-forms is given in [7.27]; a generalization to arbitrary A-Lie group actions is given in [VI.S.3].

5.18 Theorem. Let X and Y be two integrable homogeneous vector fields on M. Then their flows commute if and only if [X, y] = O.

a

Proof If [X, y] = 0, then t 7/! = 0 by [5.15] and thus by the conclusion of [5.14] the two flows commute. On the other hand, if the two flows commute, we have by the same reasoning that at 7/! = O. In particular we have 0 = (at 7/!) (0, m) = [X, Y]lm. IQEDI

5.19 Remark. If X is an odd integrable vector field, the proof of [5.1S] can be given by a direct computation as follows. If X is odd and integrable, its flow is given by the expression ¢X(7, m) = m + 7' Xc(m), where we have used the shorthand introduced and thus we have in [4.1S]. Using the same shorthand, we write Y(m) = Yc(m) . (Q x Y) (7, m) = 0 . + Yc ( m) . Computing the explicit expression for the equality Yo x = T¢x 0 (Q x Y) at (7, m) gives us

aT

am.

am

5.20 Remark. With [5.1S] we have another way to view the integrability condition of an odd vector field X. The property of its flow ¢x(t, ¢x(s, m)) = ¢x(s + t, m) implies that ¢x (t, ¢x(s, m)) = ¢x(s, ¢x(t, m)), i.e., the flow commutes with itself. According to the quoted result, this is the case if and only if [X, X] = 0, i.e., exactly the necessary and sufficient condition for X to be integrable.

242

6.

Chapter V. The tangent space

FROBENIUS' THEOREM

In this section we first define the notion of an involutive subbundle of the tangent bundle TM. We then show Frobenius' theorem which states that a subbundle is involutive ifand only if it is locally generated by axl, ... , axk with k the rank of the subbundle. Using this we show that through all points with real coordinates there passes a maximal integral manifold (a leaf) of such a subbundle. For points that do not have real coordinates, there is no guarantee that a leaf passes through them.

6.1 Proposition. Let Xi, 1 ::::: i ::::: k be k homogeneous vector fields on M satisfying [Xi,Xj] = 0,1::::: i,j::::: k. Ifm E Mis apoint at which the k vectorsXi(m) are independent, then there exists a local chart U :1 m with coordinates (Xl, ... , xn) such that Xilu = axi, 1 ::::: i ::::: k. Proof Note first that we have in particular that [Xi, Xi] = 0, i.e., all vector fields are integrable. If we define ma = Bm E BM, it follows from [11.6.6] and [111.2.7] that the vectors BXi(m) = Xi(m a) are independent. If (yl, ... , yn) are local coordinates on a chart Ua containing ma (and thus m), the vectors Xi(m a) are represented as

By a constant affine transformation with real coefficients (which is smooth!) we may assume that the coordinates of ma are all zero and that Xi (ma) = ayi. We now define a map 'ljJ by the expression

i.e., starting from (0, ... ,0, xk+l, ... , xn) we first follow the flow of X k for a time xk, then we follow the flow of X k- l for a time xk-l, ... , and we finish by following the flow of X I for a time xl. If the (Xl, ... , xn) are taken in a sufficiently small neighborhood UI C Ua of zero (representing ma), this expression makes sense and takes its values in the chart Ua . In order to compute the Jacobian (a'ljJj / ax i ) (0, ... , 0) of'ljJ at zero, it is obvious that it equals 8{ for j > k, using that a flow is the identity at time zero. With the same argument and the flow property we obtain (a'ljJ/ax j ) (0, ... ,0) = Xj(O, ... , 0) = ayj for 1 ::::: j ::::: k. We conclude that Jac( 'ljJ) (ma) is the identity, and thus, by the inverse function theorem, there exists a neighborhood ma E U such that 'ljJ-I : U --4 'ljJ-I(U) is a diffeomorphism. We claim that, in the coordinates (Xl, . .. , xn) = 'ljJ-I (yl , ... , yn), the vector fields Xi are represented as Xi = axi. For i = 1 this is fairly easy to show. Using the group property of a flow it follows that

§6. Frobenius' theorem

243

ax',

Since this is the flow of the vector field it follows from [5.6] that Xl is represented by in the coordinates x. One might think that it is more difficult for i > 1, but it is as easy. Since the vector fields Xi commute, their flows commute. For i = 2 we thus may write

ax,

'ljJ(XI, ... ,xn) = ¢x 2 (x 2, ¢x, (Xl, ¢x 3 (x 3, ... ¢X k (xk, 0, 0, ... ,0, xk+l, ... ,xn) ... ) . We thus can apply exactly the same argument to prove that Xi is represented by axi in the k. IQEDI coordinates x for 1 < i

s:

6.2 Definitions. Let M be an A-manifold and V a subbundle of TM. V is said to be involutive if for any two smooth vector fields X and Y that take their values in V, their commutator also takes values in V, i.e., X, Y E r(V) ===} [X, Y] E r(V). A subbundle V is said to be integrable iffor each m E M there exists a chart m E U with coordinates x such that Vlu is generated by ax', ... ,axk, where k is the rank of V. An integrable subbundle is also called afoliation.

6.3 Theorem (Frobenius). A subbundle VeT M is involutive if and only if it is integrable. Proof If V is generated on U by ax" ... , axk, it follows immediately from [1.21] that X, Y E ru(V) implies [X, Y] E ru(V). Since this is true all over M, V is involutive by [1.20]. To prove the implication in the other direction, suppose V is involutive and let m E M be arbitrary. Since V is a subbundle of rank k, there exists a chart m E U with coordinates (yl, ... , yn) and homogeneous Xi E ru(TM), 1 i n generating ru(TM) such that the Xi, 1 i k generate VI u [Iv'3.14]. If we define ma = Bm, the coefficients of Xi (ma) with respect to the basis ayj are real; by a constant linear transformation with k. Using these coordinates, real coefficients we may assume that Xi (ma) = ayi, 1 i we define a projection 7rk : U --4 V, (yl, ... ,yn) 1----* (yl, ... ,yk), from U to an open set V in an A-vector space of total dimension k. By construction the map

s: s:

s: s:

s: s:

Tmo7rklv",o : Vmo is bijective, simply because Xi(m o ) =

xl

ayi.

--4

T7rd mo)V

Writing Xi(m) =

s: s:

Lj Xl(m) . ayj,

our

s: s:

assumption implies that (mo) = 5{, 1 i k, 1 j n. It follows that the (m), 1 i, j k is invertible in a neighborhood UI of mo. And square matrix thus Tm 7rk Iv", : Vm --4 T7rdm) V is bijective for all m E UI . Hence there exist vector fields Yi E r u, (V) generating Vlu, such that T7rk 0 Yi = ayi 07rk. (The Yi( m) are given i,j k.) explicitly by multiplying the Xi(m) by the inverse of the matrix X{ (m), 1 We deduce that

xl

s:

s:

s:

s:

a a

T7rkO[Yi,Yj] = ["il',~]07rk =Q. uy' uyJ

Since V is involutive and T7rk bijective on Vlu" it follows that [Yi, Yj] are independent, we can apply [6.1] to conclude.

= 0.

Since the Yi IQEDI

244

Chapter V. The tangent space

6.4 Definition. Let VeT M be a sub bundle of rank k. A smooth map f : N ----t M is said to be tangent to V if for all n E N we have Tf(TnN) C Vlfen). An integral m11nifold of V is a pair (i, N) such that (i) dim( N) = k and (ii) i : N ----t M is an injective immersion tangent to V. If V is a foliation, one defines a leaf ofV to be a connected integral manifold i : L ----t M such that i( L) is maximal with respect to inclusion.

6.5 Corollary. Let V C TM be afoliation of rank k and let ma E BM be a point with real coordinates. Then there exists an integral manifold (i, N) of V such that i : N ----t M is an embedding and such that ma E i(N). Proof Let U be a chart around ma as in the definition of an integrable subbundle and let N CUbe the subset N = {m E U I 'Vi > k : xi(m) = xi(ma)}. Then N is a submanifold of M of dimension k (the xi(ma) are real!) and thus the canonical injection i : N ----t M is an embedding. Moreover, since V is spanned by the axi, i :::; k, (i, N) is an integral manifold for V. IQEDI

6.6 (Counter) Examples. One usually says that a subbundle V C TM is integrable if through every point passes an integral manifold. Defined that way, Frobenius' theorem states that V is involuti ve if and only if through every point passes an integral manifold. However, in the context of A-manifolds problems arise due to the fact that the image of an immersion has to pass through points with real coordinates. The following examples show what can happen, justifying our definition of integrability . • Consider first the A-manifold M = of dimension 210 with coordinates (Xl, x 2 ), on which we define V as the subbundle of rank: 1 generated by the vector field X I = axl. This subbundle is involutive: (Xl, x 2 ) is a coordinate system satisfying [6.2]. If i : N ----t M is an integral manifold, it is fairly obvious that i(N) should be contained in a slice x 2 constant (we will show it explicitly in the proof of [6.9]). But a point with real coordinates in N is mapped to a point with real coordinates in M, implying that x 2 should be real. It follows that no integral manifold passes through points (Xl, x 2 ) with x 2 non-real. • With this example in mind, one might think that it should be sufficient to demand that through every point with real coordinates passes an integral manifold. As the next example will show, this condition is too weak to ensure that a subbundle V is involutive. Consider the A-vector space E of dimension 112 and define M = Eo with coordinates (x, e). The subbundle V of rank 2 is generated by the global vector fields Xl = ax and X 2 (x, = aEl + aE2. Now consider the A-vector space F of dimension 111 and define N = Fo with coordinates (Y,7]). It is elementary to show that fue smooth map i : N ----t M, (y, 7]) ~ (x, = (y, 7], 0) is an integral manifold ofV. Moreover, it passes through every point with real coordinates of M. However, V is not involutive because [X2' X 2] = eae rf- V.

A6

e,

e, e)

ee e, e)

6.7 Lemma. Let V be an involutive subbundle of T M of rank k. Let furthermore U be a chart with coordinates Xi such that VI u is generated by axi, 1 :::; i :::; k. Finally define

§6. Frobenius' theorem

245

the slices Sm C U as Sm = {m' E U I 'Vi > k: xi(m') = xi(m)}. If f: N --4 M is tangent to D, then each connected component off (N) n U is contained in a slice Sm withm E BM. Proof The main problem of this proof is to show that f(N) cannot "fill up" parts of U; the crucial ingredient is that N is second countable. Let F be the A-vector space of the appropriate dimension such that X k +1 ) ... ) xn (n = dim M) are coordinates on F o , and define s : U --4 Fo by s(xl) ... ; xn) = (xk+l) ... ) xn). By definition s is constant on slices and Sm = s-1 (s( m)). Since f is tangent to D, it follows that T( so f) = O. By [3.21] we conclude that so f is constant on connected components of f- 1 (U). In particular if B is a connected component of f- 1(U) and b E B, then f(B) is contained in the slice Sf(b) (because (sof)(B) is constant equal to s(f(b»). Since N is locally homeomorphic to the even part of an A-vector space which is locally connected [III. 1.3], a connected component B of the open set f- 1(U) is open. Hence BB C B and B contains a point b EBB c B with real coordinates. And thus s(f(B)) E BF and f(B) is contained in a slice Sm with mE BM, namely m = f(b). Now f(N) n U = f(f-l(U)) and thus the image f(B) of a connected component B of f-l(U) is contained in a connected component C of f(N) n U. Since each f(B) is contained in a slice, C is the union of (parts of) slices. It is true that the union of two slices is no longer connected, but an arbitrary union of slices could be connected. For instance, U itself (if it were connected) is the union of all its slices. In order to prove that C is contained in a single slice, we invoke the fact that N is second countable and thus that there are (at most) countably many connected components (open!) B of f-l(U). Since for each B the image s(f(B)) is a single point in BF, s(C) is a countable subset of BF. Now C is connected, BF is homeomorphic to some Rd and the only countable connected subsets ofRd are points (the only connected subsets ofR are intervals). Hence s(C) is a IQEDI single point and thus C is contained in a single slice.

6.8 Proposition. Let D be an involutive subbundle of TM, let i : L --4 M be an integral manifold of D, and let f : N --4 M be a smooth map. If f(N) c i(L), then there exists a unique smooth map 9 : N --4 L such that f = i 0 g. Proof Since i is injective, existence and uniqueness of a set theoretic map 9 is guaranteed. The only difficulty is in proving that this 9 is smooth. So let no E BN be arbitrary, mo = f(no) E M and Po = g(no) E L. Let furthermore U, No, and io be as in [6.5]. Finally let U L be the connected component of i-I (U) containing Po and let UN be the connected component of f-l(U) containing no. These sets are open because i and fare smooth. Since i is tangent to D, it follows from [6.7] that i(UL) C io(No). Since io is an embedding, there exists a unique smooth j : UL --4 No such that i = io 0 j [2.18]. Since Ti is injective, Tj is injective; since No and L have the same dimension, j is a diffeomorphism onto its image [2.14]. And now: f(U N ) is connected and contained in i(L) n U, and the connected components ofi(L) n U are contained in the slices Sm [6.7]. Hence f (UN) is contained in the slice Sm o' because f (no) = mo E Sm o' Once again because io is an embedding, there exists a unique smooth map fa : UN --4 No such that

246

I =

Chapter V. The tangent space

io 0 10' It follows that the set theoretic map 9 is given on UN by glUN which is smooth. Since no is arbitrary, we conclude that 9 is smooth.

=

j-I 010'

IQEDI

6.9 Proposition. Let'D be an involutive subbundle of TM of rank k. Through every point mo E M with real coordinates passes a unique (up to diffeomorphism) leaf(i, L). Moreover, ifI : N ----t M is tangent to 'D, if N is connected, and if I (N) n i (L) =I- ¢, then there exists a unique smooth map 9 : N ----t L such that I = i 0 g, and thus in particular I(N) C i(L). Proof Let m E M be arbitrary and choose a chart U as in [6.2]. Shrinking U if necessary, we may assume that all slices Sm' with m' E U are connected. Since M is second countable, there exists a countable set of such charts U = { Ui liE N } covering M. Let Uo E U be such that mo E Uo, and let So = Sma be the slice in Uo containing mo [6.7].

If S is a slice in U E U and S' a slice in U' E U, we will say that Sand S' are related if there exists a sequence Ui E U, 1 :::; i :::; C, and slices Si in Ui such that S = S I (and thus U = UI ), Si n Si+ I =I- ¢, and S e = S'. We now define S as the set of all slices in any U E U that are related to So = Sma' Then we define the topological space X = ilSES S and the continuous map j : X ----t M such that j IS is just the canonical injection of the slice S in M. We finally define an equivalence relation", on X by x '" y {=:} j(x) = j(y), and the topological space L = XI'" with the canonically induced injective continuous map i : L ----t M. We claim that this (i, L) is the sought for leaf passing through mo. The proof of this claim breaks down into several steps . • The first step is to prove that j is an immersion. We will say that a slice S in U E U is a real slice if it is of the form S = Sm with m E BM. It follows that if S is a real slice, the canonical injection i : S ----t M is an integral manifold and an embedding. Now let Sa be a real slice in Ua E U and Ub E U arbitrary. It follows that the connected components of Sa n Ub are contained in the slices of Ub. Hence, if Sb is a slice in Ub, the intersection Sa n Sb is a union of connected components, and thus open in Sa. In particular, Sa n Sb being open in Sa, there is a point with real coordinates in this intersection, i.e., Sb is a real slice. Since So is a real slice, we conclude that all S E S are real slices, and that j : X ----t M is an immersion . • The next step is to prove that L is a proto A-manifold and that i is an injective immersion. If Sa, Sb E S intersect, we have seen that Sab = Sa n Sb is open in Sa. We thus can define 'Pba : Sab ----t Sba by 'Pba = (jISb)-I 0 (jlsJ (use [2.18] with the embedding j ISb : Sb ----t M). It follows immediately that x '" y if and only if x E Sab, y E Sba and y = 'Pba(X) for some indices a, b. We thus have the complete set of ingredients to form a proto A-manifold [111.4.9] (recall that all slices are essentially open sets in a k dimensional A-vector space). We conclude that L = XI'" is a proto Amanifold. The induced map i verifies j = i 0 7r, where 7r denotes the canonical projection 7r : X ----t L. Since 7rISa is a diffeomorphism, we have il7r(Sa) = (jlsJ 0 (7rlsJ. It follows that i : L ----t M is smooth, injective, an immersion, and tangent to 'D, i.e., an integral manifold of'D, except that we do not know that L is an A-manifold.

§7. The exterior derivative

247

• Since i : L ----t M is injective and smooth, BL is Hausdorffbecause BM is. To prove that L is second countable is harder. We will show that S is countable, which implies that X, and thus L is second countable. First fix a sequence Ui E U, 1 ::; i ::; f!.. If Si is a slice in Ui , the connected components of the intersection Si n Ui+l are contained in slices of Ui+l. Since Si is second countable, there are only countably many slices Si+l in Ui+l that intersect Si. It follows that there are only countably many slices related to So = Smo by a sequence of slices contained in the given sequence of Ui E U. Since U is countable, there are only countably many such sequences, proving that S is countable. We conclude that L is a genuine A-manifold, and that i : L ----t M is an integral manifold passing through mo. • Two items remain to be proved: that L is connected and that it is maximal. The connectedness follows from the fact that the slices S in the charts U E U are all connected. The actual argument is a bit tedious and left to the reader. Maximality will be proved at the end. • For the second part, let f : N ----t M be tangent to 'D. For any Ui E U we define the open sets Vij C N as the connected components of f-1(Ui ). It follows from [6.7] that f(Vij) is contained in a slice Si of Ui . By construction of L, if a slice Si in Ui intersects i (L ), it must be contained in i (L ). Thus iff (Vij) n i (L) =I- ¢, it must be that f(Vij) C i(L). If we define NL = {n E N [ f(n) E i(L)}, this implies that Vij is contained either in N L or in its complement. Hence N L is open and closed. By hypothesis N is connected and NL is not empty, so NL = N and f(N) C i(L). The last conclusion follows from [6.8]. • To finish the proof, suppose i' : L' ----t M is an integral manifold passing through mo. By the previous result, i'(L') C i(L), proving that L is maximal. If we have equality i'(L') = i(L), we have induced smooth maps L ----t L' and L' ----t L by [6.8]. Standard arguments using uniqueness of these factorizations then proves that Land L' are diffeomorphic, proving that (i, L) is unique up to diffeomorphism. [QED[

6.10 Remark. The natural idea of proving this proposition using Zorn's lemma does not work. It is true that one can construct an upper bound to any chain of integral manifolds (chain with respect to inclusion of their images in M). This upper bound has a canonical structure of a proto A-manifold, and even its body is Hausdorff. However, in this approach it is very hard to prove that it is second countable.

7.

THE EXTERIOR DERIVATIVE

In this section we define differential forms and the exterior derivative as well as some of its standard properties.' it is a derivation of square zero and commutes with pull-backs. For the last property we define the notion of the pull-back of a differential form as well as a generalization using the generalized tangent map. Defining the Lie derivative of

248

Chapter V. The tangent space

differential forms by the formula of H. Cartan [H.Ca]: £(X) = do L(X) + L(X) 0 d, we also show that £(X) a equals Ot¢;a at t = 0, where ¢t denotes the flow of the homogeneous integrable vector field X.

7.1 DefinitionIDiscussion. Let M be an A-manifold. A k-form on M is a section of the bundle I\k *TM, and thus in particular a smooth k-form is an element of r(l\k *TM). In accordance with standard notation, we denote the set of all smooth k-forms on M by nk (M), i.e., A k-form at m E M is any point in the fiber of I\k *T M above m. It follows that if a is a k-form on M, a(m) == am is a k-form at m E M. Since 1\0 *TM is the trivial bundle M x A, it follows that a O-form is just a function on M, and thus in particular nO(M) = COO(M). For k = 1 we find 0. 1 (M) = r(*TM), i.e., a I-form is a section of the bundle *TM, the left dual bundle of the tangent bundle TM. This left dual bundle *TM is usually called the cotangent bundle of M. According to [IV.5.14] and [1.5.5] we have the identifications (7.2)

nk(M) ~ Hom~k(r(TM)k; COO(M)) ~ HomL(N r(TM); COO(M))

== *(N r(TM)) ,

which tells us that we may interpret a smooth k-form as a (left) k-linear graded skewsymmetric map of smooth vector fields on M with values in the smooth functions on M. Note that the k-linearity is over COO(M) and not over A (which does not make sense because r(T M) is not an A-module). For k = 1 the identifications (7.2) reduce to r(*TM) ~ *r(TM). Even for k = 0 (7.2) makes sense: nO(M) == r(M x A) and *(Nr(TM)) == *COO(M) because N gives the basic ring [1.5.3], which here is COO(M). Since the trivial bundle M x A comes with its canonical trivialization, we have a canonical identification r(M x A) ~ COO(M). And thus for k = 0 (7.2) reduces to the obvious identification COO(M) ~ *COO(M). For future reference we define n(M) as the direct sum over all k : 00

n(M) =

EB

nk(M) .

k=O

Obviously n(M) is a Z x Z2-graded COO (M)-module, where the Z-grading is given by the k from k-form. Using the wedge product of such sections as defined in [IV.6.l],n(M) becomes a Z x Z2-graded commutative COO (M)-algebra. This becomes even more explicit when we also use the identification *(N r(T M)) ~ N *r(T M) [Iy'5.20], which tells us that n(M) is (isomorphic to) the exterior algebra 1\ *r(TM). Now if f E COO(M) ~ nO(M) is a O-form and a E nk(M) a k-form, we can form the wedge product f 1\ a as well as f . a, which uses the COO ( m)-module structure ofnk(M). Since the wedge product is pointwise, [1.5.8] tells us that these two are equal: f 1\ a = f . a (and similarly a· f = a 1\ f).

§7. The exterior derivative

249

7.3 Definition. Let 0: be a k-form on M. The exterior derivative of 0: is the (k do: on M defined by (7.4)

+ I)-form

(_I)k. L(Xo, ... , X k) do: = =

L

i+ L(E(Xp)iE(X i »

(-1)

p
Xi(L(XO, ... , Xi-I, Xi+I, ... , Xk) 0:)

O~i~k

j+ L

(-1)

+

i
(E(Xp)IE(Xj»

L(Xo, ... , Xi-I, [Xi, Xj], Xi+I, ...

O~i<j~k

Of course this formula needs a lot of explanation before it can be used as a correct definition. In the first place, the Xi are homogeneous vector fields on M. As such, (7.4) defines a map of k + 1 homogeneous vector fields to functions on M. Taking the sum over homogeneous parts, it defines a (k + 1)-additive function of vector fields on M to functions on M.

7.5 Lemma. The map do: is (k + I)-linear overCOO(M) and graded skew-symmetric. It thus defines a (k + 1 )-form on M. Proof(sketched). The definition of do: breaks up into two parts: a single summation and a double summation. Graded skew-symmetry means that interchanging two neighboring vector fields Xr and X r + I changes the result by a global sign _(_I)(E(X r )IE(X r +Il). U sing that 0: itself is graded skew-symmetric, the only terms in the single summation that do not produce this global sign are the terms with i = r, r + 1. But these two terms transform into each other, up to this global sign. In the double summation the terms with i, j ¢:. {r, r + I} produce the correct sign change. The summations i < r with j = r, r + 1 transform into each other up to the needed sign, as do the summations r + 1 < j with i = r, r + 1. The remaining term i = r, j = r + 1 produces the correct sign because of the graded skew-symmetry of the commutator of two vector fields. We thus conclude that do: is graded skew-symmetric. Knowing that do: is graded skew-symmetric, it suffices to verify that do: is linear over COO(M) in the first vector field X o, i.e., that

The single summation produces, apart from the terms one wants, some terms involving Xd. Using [1.19], it is easy to verify that these unwanted terms are canceled by the unwanted terms appearing in the double summation. IQEDI

7.6 Discussion. Once we know that (7.4) gives a correct definition ofthe exterior derivative of a k-form, it is useful to write this explicitly in simple cases. So let X be a vector field,

250

Chapter V. The tangent space

and let Yo, Yi , Y2 be homogeneous vector fields. Furthermore, let f be a O-form, i.e., a smooth function, let be a I-form, and let w be a 2-form. Their exterior derivatives are given by

e

L(X) df = Xf - L(YO, Yi ) de

= Yo (L(Yd e) -

(_1)(E(Yo )IE(Y2 )) L(YO,Yi

, Y2) dw

(-1) (E(Yo)IE(Yt))

Yi (L(YO) B) - L([YO, Yi ]) e

=

(_1)(E(Yo)IE(Y2 ))

+ (_l)(E(Y )IE(Yo)) + (_1)(E(Y2)IE(Y l

l

))

+ L(YO, [Yi , Y2]) w) (Yi (L(Y2, Yo)w) + L(Yi' [Y2, Yo])w) (Y2(L(YO, Yd w) + L(Y2' [Yo, Yi ]) w)

(YO(L(Yi , Y2) w)

.

In deriving the last (rather symmetric looking) formula, one has to use the graded skewsymmetry of wand of the commutator. Notice also the minus sign in front of L(YO, Yd de (see also [7.14]). However nice this definition of the exterior derivative might be, it is not one that allows for easy calculation. In order to compute da in terms of a local expression on a chart, we need a preliminary lemma. If U c M is open, it is in particular an A-manifold in its own right. Restricting a and da to U gives us a k-form al u and a (k + 1 )-form (da) u on U. Denoting (temporarily) the exterior derivative on the A-manifold U by d u , we are thus faced with the question whetherdu(alu) equals (da)lu. 1

7.7 Lemma. For any open U eM: du(alu)

= (da)lu.

Proof Let m E U be arbitrary, let p be a plateau function around m in U, and let X o, ... ,Xk be k + 1 smooth homogeneous vector fields on U. Multiplying the Xi by p if necessary, we may assume that the Xi are (smooth!) vector fields on M, zero outside U. According to [Iy'6.2] we have (L(X il , ... , X ik ) a)lu = L(Xill u , ... , X ik lu) alu. Using [1.5], it follows immediately that we have

Since the values Xi (m) can be chosen "arbitrarily" (see the proof of [Iy'5.9]), it follows that (dualu )(m) = (da)lu(m). IQEDI

7.8 Discussion. If U c M is a chart with coordinates Xi (even and odd together), we know from [1.8] that r(TU) = ru(TM) is a free graded COO (U)-module with the Oi as basis. Since the Oi are finite in number, it follows in particular that ru (TM) is f.g.p. If we denote by dXi the left dual basis in *r(TU) ~ r(*TU) = ru(*T M), i.e., L(Oi) dx j = 8{, we can apply [1.8.16], [1.8.14], and [1.8.15] to conclude that the space of k-forms on U nk (U) = ru (N *T M) ~ N *r(TU) is a free graded Coo (U)-module generated by the

dxil

A ... A

dX ik .

§7. The exterior derivative

251

7.9 Proposition. Let 0: be a k-form on M, (3 an £-form, and let f be a smooth function. Furthermore, let Xl, ... , xn be coordinates (even and odd) on a chart U c M. Then: (i) dflu

=

E:

l

dxi.ad;

(ii) on U : d u (dx i1 A ... A dXik . fl u) = (_I)k dX i1 A ... A dXik A dfl u; (iii) d (0: A (3) = (do:) A (3 + (-I)ko: A d(3; (iv) d(do:) = O.

Proof • Using the vector fields ai on U, the equality in (i) is immediate from the definition of dj. • Denote by'Y the k-form dXil A ... A dX ik , and denote by Xp = ajp one of the basis vectors of the tangent space. Since L(XO, ... , Xi-I, X i+ l , ... , X k ) 'Y is either 0 or ±1, it follows that

Xi(L(Xo, .. . , Xi-I, Xi+l, ... , Xk)'Y· f) = (E(XdIE(-Y)+ EE(Xp» (-1) p#i (L(Xo, ... ,Xi-l,Xi+l, ... ,Xk)'Y)·Xd. From this and the fact that the Xi have commutator zero, it follows that

(_I)k . L(Xo, ... , X k ) db· f)

=

k i+(E(X,)IE(-y)+ EE(Xp» 2:)-1) p>i L(XO, ... ,Xi-l,Xi+l, ... ,Xk)'Y·Xd. i=O

Using that L(XO, ... , X k ) equals L(XO) 0 • • • 0 L(Xk) [1.7.16], and that L(Xi ) is a right derivation of degree ( -1, c:(Xi )) [1.6.16], one can show by induction that the right hand side of this formula equals L(XO, ... , X k ) ('Y A df). This proves (ii) because the (Xo , ... , X k ) generate (TU)k+l. • The direct way to prove (iii) is to use induction as above and the derivation property of vector fields. An easier way is to note that the exterior derivative is additive and commutes with restrictions. It follows that it suffices to prove it in a chart with 0: = dX i1 A· .. A dxik-f and(3 = dx j1 A· .. Adxjl.g for some homogeneous functions f and g. Using the definition of d on 0-forms, it is immediate that d(f . g) = (df). 9 + f . dg. The result then follows from (ii) and the graded skew-symmetry of the wedge product. • As before, it suffices to prove (iv) on a chart with 0: = dX i1 A· .. A dX ik ·f. Applying (ii) and (i) twice we obtain n

d(do:)

= (_I)k+(k+l)

L

dX i1 A··· A dX ik A dx i A dx j . aja./ .

i,j=l

From [ai,aj] = 0 we deduceajad = (_I)(E(xi)le(xj»aiajf. On the other hand, graded J skew-symmetry of the wedge product gives us dx i A dx j = -( _1)(E(x')IE(x »dx j A dXi. It follows that d( do:) equals its opposite, and thus is zero. IQEDI

252

Chapter V. The tangent space

7.10 Discussion. If Xi is a coordinate on a local chart U of an A-manifold M, it is in particular an element of Coo (U). As such we can calculate its exterior derivative dU(Xi), which is a I-form on U. On the other hand, we have defined the I-form dXi on U as being an element of the basis dual to the basis OJ ofr(TU). It follows immediately from [7.9-i] that these two I-forms coincide:

This observation justifies the name dXi for these I-forms. In particular it follows that the I-form dx i does not depend upon the choice of the other coordinates on U. This in contrast to the vector fields Oi, where anyone of them depends in general upon the whole set of coordinates.

7.11 Lemma. Let M be a connected A-manifold and Then f is constant if and only if df = O.

f :M

--4

A a smooth function.

Proof This is a direct consequence of [3.21] Gust use all vector fields) and the definition of df [7.3], [7.6]. IQEDI

7.12 Discussion. By definition the exterior derivative is a map d : Dk(M) --4 Dk+l(M), so officially it should be indexed by a k. Taking the direct sum over all k gives us a map (still denoted by d): d: D(M) --4 D(M) .

Property [7.9-iii] shows that the map d is a right derivation of degree (1,0). However, there is a pitfall to be avoided: the exterior derivative is not linear over Coo (M)! Luckily R is a subring ofCOO(M), and d is linear over R. It follows that the exterior derivative is a right derivation of the Z x Z2-graded commutative R-algebra D(M). Since commutators of derivations are again derivations, we can compute the commutator [d, d]. Since the degree ofd is (1,0), itfollows that [d, d] = 2d 0 d == 2d 2 . Property [7.9-iv] then can be stated equivalently as [d, d] = O.

7.13 Summary. The exterior derivative on differential forms d : D(M) --4 D(M) is a right derivation of the Z x Z2-graded R-algebra D(M) of bidegree (1,0) and of auto commutator zero.

7.14 Nota Bene. Some readers might wonder about the global factor (_l)k in the definition of the exterior derivative of a k-form. We will give two explanations. In the first place, this sign can be attributed to our way to identify I\k *E with *(I\k E), which allowed us to write

§7. The exterior derivative

253

without an additional sign (_1)k(k-l)/2 [1.7.22]. And indeed, introducing this extra sign will give us on the left hand side of (7.4) the factor (_1)k(k+l)/2 and on the right hand side the factor (_1)k(k-l)/2, and then our global factor (_l)k disappears. In the second place, the given d is a right derivation of degree (1,0), so applying the inverse transpose 'I-I gives us a left derivation 'r-1d. Since linearity is over R for which left and right do not make a difference, the only difference comes from the degree. Now if 0: is a k-form of parity (Z2-degree) a, we have €( 0:) = (k, a) and thus

It follows that we could have defined the left derivation 'r-1d by (7.4) without the additional sign. Looking at [7.9], the conclusions (i) and (iv) remain unchanged when we replace d by 'r-1d; in (ii) the sign (_l)k disappears, and the conclusion in (iii) has to be replaced by [(0: A (3)'r- 1d = 0: A ([((3)'r- 1d) + (_1)l([(o:)'r-ld) A (3, i.e., by the standard property of a left derivation.

7.15 Definition. If X is a smooth vector field on M, we have defined in [1.6.16] (see also [IV.6.1])a right derivation [(X) of the Z x Z2-graded algebra D(M); if X is homogeneous, [( X) has degree ( -1, €( X)). It is a right derivation of the Coo (M)-algebra structure, and hence a right derivation of the R-algebra structure. Since the exterior derivative d is also a right derivation of the R-algebra structure of D(M), we can define a new right derivation £(X) by taking the commutator:

£(X) = [d, [(X)] == do [(X)

+ [(X)

0

d.

The right derivation £ (X) of D( M) is called the Lie derivative in the direction X; if X is homogeneous, £(X) has bidegree (O,€(X)). For O-forms, i.e., for functions, the Lie derivative reduces to £(X) f = [(X)df = Xf because the contraction operator [(X) acts as the zero operator on O-forms. The definition of the Lie derivative in the direction of a smooth vector field X is extended to include an action on smooth vector fields Y E r(T M) by

£(X) Y = [X, Y] , i.e., the action of the Lie derivative of Y in the direction of X is just the commutator [X,Y].

7.16 Proposition. Let X and Y be vector fields on M, then (i) [d, £(X)]

== do £(X) - £(X) d = 0; = [([X, Y]) == [(£(X)Y) = [[(X), £(Y)]. 0

(ii) [£(X), [(Y)]

Proof. The proof of (i) is immediate when using that dod = o. • To prove the equality [£(X), [(Y)] = [([X, Y]), we first note that any operator [(Z) commutes with restrictions, simply because it is defined pointwise. Since the same is true

254

Chapter V. The tangent space

for d, it follows that it also is true for both sides of the equality [.c (X), L(Y)] = L( [X, Y]). It thus suffices to verify this equality on a local chart U c M. But on a local chart U the R-algebra D(U) is generated by the functions Coo(U) and the (local) I-forms dXi. Since both sides of the equality [.c(X) , L(Y)] = L([X, Y]) are right derivations of this algebra, it suffices to verify it on generators. Now on functions both sides act as the zero operator, hence they are equal. To see what happens on a generator dXi, write Xlu = Ei Xi. Oi and YI u = Ei yi . Oi with Xi, yi E Coo (U). Since the equality is linear in X and Y, we may assume that X and Yare homogeneous. We then compute

[.c(X), L(Y)] dx i

.c(X)(L(Y) dx i ) - (_I)(E(X)IE(Y» L(Y)(.c(X) dx i ) = .c(X) yi _ (-I)(E(X)IE(Y»L(Y)(d(L(X) dx i )) =

= X(yi) _ (_I)(E(X)IE(Y»Y(X i ) = L([X, Y]) dx i .

The last equality of (ii) follows by interchanging X and Y and noting that for homogeneous vector fields X and Y we have not only [X, Y] = -(-1) (E(X)IE(Y» [Y, X] but also

[.c(X), L(Y)]

= -( _1)(E(X)IE(Y» [L(Y), .c(X)].

7.17 Corollary. Let X, Y 1 , have the operator equality

... ,

IQEDI

Y k be smooth homogeneous vector fields on M, then we

[.c(X) , L(Yd ° ... ° L(Yk)] = k E (E(X)IE(Yj» 2)-I)l S j
7.18 Remarks . • Using the definition of the Lie derivative as well as [7.I6-ii] and omitting the k-form a, the defining equations for the exterior derivative can be written as the operator equality

(_I)k. L(Xo) ° . ··L(Xk)od = i+ E(E(Xp)IE(X;) = (-1) p
L

J+

+

(-1)

E i
(E(Xp)IE(Xj» L(XO) ° ... ° L(Xi-d ° L([Xi' Xj]) ° L(Xi+d ° ...

0:5i<j:5k

On the other hand, this operator equality could have been derived directly from [7.l6-ii]. But then we use a circular argument because the Lie derivative is defined in terms of the exterior derivative.

255

§7. The exterior derivative

• For homogeneous vector fields X and Y, and a k-form a, property [7.16-ii] can be written as

.c(X) (L(Y)a)

= L(.c(X)Y)a + (_l)(e(X)le(Y» L(Y).c(X) a

.

If we interpret the operation L(Y) as a kind of multiplication, this equation tells us that .c(X) remains a derivation of bidegree (0, c:(X)).

7.19 Definition. Let M and N be two A-manifolds and let f : M ----t N be a smooth map. According to [2.1] we have an induced even vector bundle map Tf : TM ----t TN. We thus can form the even bundle morphism I\k T f : I\k T M ----t I\k TN and its associated pull-back map [IV.6.3] *(NTf) : r-I(*(NTN)) ----t r-I(*(NTM)). Using the identification * (N T M) ~ I\k *T M and [1.7.25], we can interpret this pull-back as going from k-forms on N to k-forms on M; restricting attention to smooth k-forms, it maps nk(N) to nk(M). In this interpretation (see [IY.6.9] and [IY.6.10]) it is customary to denote the pull-back map * (I\k T f) between k-forms by f*. (Note that, although convention dictates that the * is at the right of f, f* is a left linear pull-back; this does not really pose a problem because f itself does not admit any kind of dual.) Although f* is not a vector bundle morphism, its definition is pointwise: it maps a k-form af(m) at f(m) E N to a k-form (f*a)m at mE M. If Xl, ... , Xk E TmM are tangent vectors at m E M, the definition of the pull-back gives us the formula (7.20) In the particular case k = 0, this formula tells us that the pull-back of a smooth function (M) is given by f* ¢ = f <> ¢ = ¢ 0 f. This is the special case alluded to at the end of [IV.6.7], with the map h = * T f) : M x A ----t N x A, (m, a) 1----* (f(m), a). For future use we recall that f* is an algebra morphism [IV.6.10], i.e.,

¢ E Coo (M) ==

n°

(N

1* (a A (3) = (1* a) A (1* (3) , and that 1* commutes (more or less) with restrictions [IV.6.6].

7.21 Lemma. Let f : M ----t N be a smooth map and X a vector field on M. Thenfor any point mE M we have (with some abuse of notation) the equality

In particular,

if Y is a vector field on N related to X by f, then L(X) (I*a)

= 1* (L(Y) a) .

Proof The second part follows from the first by noting that (( Xm II Tn = Yf(m). The abuse of notation is that the f* in I*(L(((XmIITn)af(m») should be seen as the dual

256

Chapter V. The tangent space

map I\k *TJ from I\k *Tf(m)N to I\k *TmM, i.e., as the pointwise expression of the pull-back map, rather then the official definition of the pull-back map on sections. With this explanation, let Zl, ... ,Zk-l E T mM be k - 1 arbitrary tangent vectors at a point m E M. With these we obtain the algebraic equalities L(Zl, ... , Zk_l)(L(X) (f*a))m

7.22 Proposition. Let d(f*a) = f*(da).

J :M

= L(Zl, ... , Zk-l, Xm)(f*a)rn = L(((ZlIITn,···, «Zk-lIITn, ((XmIITn) af(m) = L(((ZlIITn,···, «Zk-lIITn)(L(((XmIITn) af(m)) = L(Zl, ... , Zk-d(f*(L( «Xm IITn) af(m))). IQEDI

--4

N be a smooth map, and let a be a k-form on N. Then

Proof To prove this global equality, it suffices to prove it on charts U that cover M. We may even assume that J(U) C V for some chart V because such charts U still cover M. Using that d commutes with restrictions to open subsets [7.7], and using the way the pull-back commutes with restrictions [IV.6.6], we thus have to show the equality du((flu )*a) = (flu)* (dvalv). Now let Xi be coordinates on U and yj be coordinates on V, and let us not note the various restrictions. Since J* is an algebra morphism [IV.6.l0] and since d is a right derivation, it suffices to prove the equality on the generators ¢ E Coo(V) and dyj. Using the chain rule [III.3.8] we have the equalities

On the other hand we can compute L( oi)J*d¢ at any point m E U by using [7.21]:

Since this is the same as L(Oi)(d(f*¢))m, we conclude that J*(d¢) = d(f*¢). Since we know that dyj is the exterior derivative of the coordinate function yj E Coo (U), we immediately obtain d(f*(dyj)) = d2 (f*yj) = 0, as well as f*(ddyj) = J*O = 0 (both IQEDI because d 2 = 0).

7.23 Definition/Construction. Let M, N, and Q be A-manifolds, 0 C M x N an open subset, J : 0 --4 Q a smooth map, and a a (not necessarily smooth) k-form

§7. The exterior derivative

257

on Q. With these ingredients we want to define a pull-back of a to N but in an Mdependent way. More precisely, for each mE M we define the open subset Om eN by On {m} X N = {m} X Om and the map 1m : Om --4 Q by Im(n) = I(m, n). Using the definition of the generalized tangent map TIm [3.19] we now define for each m E M the k-form l:na on Om by its action on tangent vectors:

In order to discuss smoothness properties of the k-forms l:n a, we need another way to describe them. We thus define a map 'l/Jcx : 0 --4 I\k *TN such that for each (m, n) E 0 the image 'l/Jcx(m, n) is a k-form at n E N, i.e., 'l/Jcx(m, n) E *TN)n. We will call 'l/Jcx an M-dependent k-form on N (even though it is only defined for (m, n) EO C M X N). In terms of its action on tangent vectors at a point n E N, 'l/Jcx is defined by

(N

L(Y1I n , ... , Ykln)'l/Jcx(m, n) = L((Qm, Y1I n ), ... , (Qm, Ykln))(f*a)(m,n)

= L( (( (Qm, Y1In)llT(m,n)n,···, «(Qm' Ykln)llT(m,n)n) af(m,n) . For a fixed mE M, the map'l/Jcx(m, Jis a k-form onO m eN; comparing the definitions immediately shows that 'l/Jcx(m, n) = U:na)n.

7.24 Lemma. If the k-form a is smooth, the map 'l/Jcx is also smooth.

Proof Being smooth is a local property, so we may restrict attention to (trivializing) charts U for M and V C Eo for N (we assume that N is modeled on the A-vector space E). This gives us a local trivialization V x I\k *E for I\k *T N. In this local trivialization the map 'l/Jcx is given as (m, n) ~ (n, ;j(m, n)) for some map ;j : U x V --4 I\bE. The map ;j is smooth if and only if its coordinates with respect to some basis for I\k *E are smooth. Those coordinates (the right version) are given by L( ei l , . . . , eik );j( m, n), where (ei) is a basis for E. According to [1.16] those coordinates are also given by L( Oil' ... ,Oik )'l/Jcx (m, n). This means that we have to prove that the functions

are smooth. But that is obvious because a, I, TI, the zero section Q, and the vector fields OJ are all smooth maps. IQEDI

7.25 DiscussionlDefinition. Let the ingredients M, N, 0, Q, and I be as in [7.23], let a be a k-form on Q and let f3 be an C-form on Q. We then can form the M-dependent k-form'l/Jcx and the M-dependent C-form 'l/Jf3. More precisely, for each (m, n) E 0 we *TN)n and 'l/Jf3(m, n) = U:n(3)n E *TN)n. We have 'l/Jcx(m, n) = U:na)n E thus can form the wedge product U:na)n A U:n(3)n E (I\k+f *TN)n. Generalizing the symbol A, we define the M -dependent (k + C)-form 'l/Jcx A'l/Jf3 : 0 --4 I\kH *T N by

(N

(N

258

Chapter V. The tangent space

Looking at this formula in local charts shows immediately that 'l/J", A'l/J(3 is smooth if 'l/J", and'l/J(3 are smooth. On the other hand, we can form the (k + C)-form a A (3 on Q and the M-dependent(k + C)-form 'l/J",/\(3' Nobody will be surprised that this is the same as 'l/J", A'l/J(3.

7.26 Lemma. 'l/J", A'l/J(3 =

'l/J",/\(3

or, equivalently, f:n a A f:n(3 = f:n(a A (3).

Proof This is an immediate consequence of the definition of'l/J", in terms of f*a and the fact that we have f* (a A (3) = f* a A f* (3 [7.19], [IV.6.10]. IQEDI

7.27 Proposition. Let X be a smooth homogeneous integrable vector field on an Amanifold M, let ¢ : W X ----t M be its flow, and let a be a smooth k-form on M. We then have a smooth map 'l/J", : Wx ----t I\b TM, (t, m) ~ (¢;a)m. The vector field at and this map satisfy the condition of[3.10] and we have at'l/J", = 'l/JC(X)"" i.e.,

Proof The property we have to prove is local (equality of maps), so let us choose a local chart V around ¢(t, m) with coordinates yi, and let J c AE(x) be an interval around t and U a coordinate chart around m with coordinates x j such that J x U c W X and ¢(J x U) c V (continuity of ¢ assures the existence of such sets). If M is modeled on the A-vector space E, then, as in the proof of [7.24] the map 'l/J", is given on I x U as (t, m) ~ (m, ;;';",(t, m)) for some smooth map ;;.;'" : J x U ----t I\k *E. From this it is obvious that indeed the vector field at and 'l/J", verify the conditions of [3.10], so our statement makes sense. Moreover, the map at'l/J", is, in the same local trivialization, given as (t, m) ~ (m, at;;';",(t, m)) (3.12). We thus have to show that at;;';", = ;;';C(X)",' On the local chart V there exist smooth functions ai 1,... ,ik : V ----t A such that the k-form a has the form a = 2::.'tl, ... ,'tk . ai 1... ik· dyi1 A ... A dyik. Using [7.26] we thus have (¢; a) = 2::i 1 , ... ,ik (¢;ai 1... ik) (x) . (¢;dyi1 ) A ... A (¢;dyik). Taking a closer look at the separate factors in this expression, we first note that (¢; ail ... ik) (x) is the function ai1 ... ik (¢t( x)) = ai1 ... ik (¢(t, x)). And then we note thatthe definition of ¢; implies that we have L(axi)¢;dyi = L(((axil(t,x)IIT¢)))dyi = (axi¢i)(t,x). Hence all terms of the form ¢; dyi are of the form

We conclude that the local expression 'l/J", of ¢;a is a sum, with signs, of terms of the form ai1 ... ik (¢(t, x)) . (axil ¢i1) (t, x) ... (axik ¢ik) (t, x) . dx j1 A ... A dx jk . Since the derivative of an A-vector space-valued function can be computed by deriving each of its components (111.3.9), we thus have to compute the action of at on the coefficients of dx lt A ... A dx jk . Since at is a derivation, we thus have to derive each of the factors ai 1... ik(¢(t,X)) and (axi¢i)(t,x) separately.

259

§7. The exterior derivative

If on the chart V the vector field X has the local expression XI (y) = Ei Xi(y). 0y' I(y), then the definition of the flow gives the explicit expression (Ot¢i)(t,X) = Xi(¢(t,X)). Instead of computing each OtOxi ¢i separately, we compute the time derivative for a whole group at a time:

Ot(¢*dyi) = " ( _l)(E(t)IE(x i »dx j ~

t

)

= L

.

02¢i. (t x) = "dx j otoxJ' ~ )

02¢i (t x) oxJot'

. o(Xio¢) . O¢k OXi dxJ . ox) (t,x) = LdxJ . ox) (t,x)· 8k(¢(t, x))

)

y

j,k

= ""'*d ~ 'f't Yk

"'* 8k OXi = . 'f't Y

k

Since Xi

.

"'*(" OXi) = ¢t*(dXi) ~ dYk . 8k

'f't

k

.

Y

= L(X)dyi, it follows

immediately from the definition of the Lie derivative that dXi = £(X)dyi. By definition of the flow we have Ot((ail ... ik o¢) = (Xail ... ik)o¢. Since, again by the definition of the Lie derivative, we have Xai, ... ik = £(X)ai, ... ik' we conclude that and It now only remains to knit these results correctly together:

ot(¢;a) = . L

Ot (¢;a i1... ik . ¢;dyil A··· A ¢;dyik)

=.L (Ot(¢;ail ... ik)·¢;dyil A"'A¢;dyi k+ 1.1,··· ,tk

k (E(t)IE(ai , .. ik)+ E E(yiq» L(-l) lSq
¢;ai, ... ik . ¢;dyil A···

p=l

.. , A

¢;dyi p- 1 A Ot(¢;dyi p) A ¢;dyip+l A··· A ¢;d yik )

(¢;(£(X)ai1 ... ik)' ¢;dyil A··· A ¢;dyi k+

.L 'l..1 , ... ,tk

"'*a· .. "'*dyil A··· 'f't .,···.k 'f't p=l

... A ¢;dyi p- 1 A ¢;(£(X)dyip) A ¢;dyip+l A··· A

= ¢; . L. (£(X)ai1 ... ik· dyil

A··· A dyik+

'l.b ... ,1,k

k (E(t)IE(ai "'k)+ E E(yiq» " L(-l) lSq

... A dyip-l A £(X)dyip A dyip+l A ... A dyik )

¢;d yik )

260

Chapter V. The tangent space =

¢; (£(X) . L

O!i1 ... ik·

dyil A··· A dyik) .

~b···,tk

The last equality is a direct consequence of the fact that the Lie derivative £(X) is aright derivation of parity (O,c(X)) = (O,c(t)). IQEDI

7.28 Remark. An analogous result for vector fields is given in [5.15]; a generalization to arbitrary A-Lie group actions is given in [VI.S.I9]

8.

DE RHAM COHOMOLOGY

Using Batchelor's theorem we show that the de Rham cohomology of an A-manifold M (closed forms modulo exact forms) equals the de Rham cohomology of the underlying ordinary R-manifold BM.

8.1 Discussion. When we apply the body map to an A-manifoldM, we get the R-manifold BM. If we apply B to a smooth function f E COO(M), we get a smooth function Bf on BM. Things change a little when we apply B to vector fields: the vector bundle BTM splits into its two homogeneous parts BTM = B(TMo) EB B(TMI). On a local chart of M with coordinates x and ~, taking the body map means ignoring the coordinates~. Since the tangent space is generated by the and the it follows easily that B(T Mo) is generated by the and that B(TMl) is generated by the One thus immediately sees that B(TMo) can be identified with the tangent bundle of BM. In other words, applying the body map to any vector field on M does in general not yield a vector field on BM, but applying it to an even vector field yields a vector field on BM. If we now turn our attention to k-forms, it becomes even more delicate. Applying B to a O-form on M (i.e., a function) yields a function on BM, and applying B to an even I-form (i.e., a section of *TM) yields a I-form on BM. But already for 2-forms it becomes more complicated. If E is an A-vector space of dimension plq, the dimension of N E is ~p(p - 1) + pq + ~q( q + 1), of which pq are odd. In local coordinates this means that the body part of an even 2-form still might contain terms involving d~ A d~. We thus are naturally led to ask the question: does there exist a (canonical) way to obtain a k-form on BM from a k-form on M?

ax

ax

aE,

aE'

8.2 Construction. Let M be an A-manifold of dimension plq, and let U be a chart with coordinates (xi, ~j). On U we define an even vector fieldXE by q

(8.3)

XE =

.

a

L e . a~j j=l

.

§S. de Rham cohomology

261

The subscript E stands for Euler because X E is essentially the Euler vector field, although only in the odd coordinates. An elementary calculation shows that

Moreover, C(XE)d~j = d~j and C(XE) dx i = O. Since C(XE) is aright derivation of degree (0,0), it follows that any k-form a on U can be written as a sum

It is easy to describe this decomposition in words: the operator C(XE) counts the number of fs in a k-form (whether or not preceded by a d) and each term a(f) has exactly C fs. A simple count shows that there will be no non-zero a(P) for C > k + q.

To show that this local decomposition a = L;~ci a( P) is unique, we suppose that a = 0 and we apply C(XE) s times to get the equations 0 = L;~ci CSa(f) (writing formally 0 0 = 1). Taking s = 0, ... , k + q we can see these k + q + 1 equations as linear equations in the k + q + 1 "unknowns" a(O), ... ,a(k+q). The entries of the matrix A of these linear equations are given by Asp = CS. Since the determinant of this matrix is n~!i n!, the matrix A is invertible. We conclude that all a(f) are zero, proving uniqueness of the decomposition. The obvious candidate for a k-form on BM associated to a k-form a on M is the form described locally as the zero-th order term a(O) in this decomposition. But we have to be careful to see what happens in another chart. So let V be another chart with coordinates (yi, r/), on which we define the Euler vector field YE = Lj 7]j . In these terms, a k-form a on V can be decomposed according to the number of 7]'s. On the overlap Un V we thus have two decompositions: a = Lp a~) = Lp a~), according to X E and YE respectively. In particular the homogeneous terms a~o) decompose according to

a.,.,i.

X E as a~o) = Lp a~uf). Since in the coordinate change (x,~) ~ (y, 7]) the 7]'s are at least linear in the ';'s, it follows easily that a~o) = Lf>e a~uP), i.e., the number of odd coordinates does never diminish. Uniqueness of the decomposition on U then implies (00) (0). (0) (0)" (of) . that a vu = au ' I.e., a v = au + L.JP>o a vu ' ThIS formula tells us that the k-form a~), which does not contain any 7] nor d7], is given in the (x,O coordinates as a~), which does not contain any ~ nor d~, plus terms involving ~s and/or d~s. We now recall that in the coordinate change (x, 0 ~ (Y,7]) the y contain only even powers of~. This implies that all terms a~V with C > 0 must contain ~s and not only d~s, simply because d(~i~j) = ~id~j - ~j d~i. We conclude that Ba~) = Ba~). This proves that, if a is a _

0

k-form on M, then we can decompose it in a local chart as alu =

Lp a~), and then the

local forms Ba~) coincide on overlaps, i.e., they define a global k-form on BM. But the reader has to be careful: Ba is in general not a k-form on BM (the d~s do not necessarily disappear), and the local k-forms a~) do in general not glue together to form a global k-form on M. The latter can be achieved by making a particular choice for the charts U to be used (see the proof of [8.9]).

262

Chapter V. The tangent space

8.4 Definition. Let M be an A-manifold, and a a k-form on M. We define the k-form BMa on BM as being the k-form B(a(O)) constructed in [8.2]. If a is a O-form (a function), this definition coincides with the standard body map: BM f = Bf. If a is a I-form, this definition coincides with the standard body map applied to the even part of a : B M a = B ( ao). In the same spirit we define the map B M on vector fields by

BMX = B(Xo).

8.5 Discussion. Using the Lie derivative£(XE) on vector fields (£(XE)(X) = [XE, Xl) we could have defined an analogous (local) decomposition of a vector field X = X(f) according to the number of times an odd coordinate ~ appears in a (local) description. As for k-forms, one then could associate to any vector field X on M a vector field B(X(O)) on BM. However, as for I-forms, this coincides with the given definition of BM : B(X(O)) = B(Xo) = BMX. On the other hand, this approach to BM on vector fields can be easily extended to other kinds of tensor fields on M (if the need arises).

Le

8.6 Remark. Our construction of B M on k-forms can be stated in a more fancy language. The fact that the number of odd coordinates appearing in a local expression never decreases, implies that we can define the subsets F;(M) c [lk(M) as consisting of those k-forms that do contain at least C odd coordinates in any local coordinate system. We thus obtain a filtration [lk(M) = F~(M) ::l Ff(M) ::l F~(M) ::l .... The existence of our map BM : [lk(M) ----t [lk(BM) then can be rephrased as the existence of a (canonical) map F~(M)/Ff(M)

----t

[lk(BM).

8.7 Lemma. Let M be an A-manifold, a a k-form, (3 an C-form, and X a vector field on M. Then (i) BM : [lk(M) ----t [lk(BM) is R-linear and surjective; (ii) BM(a A (3) = (BMa) A (BM{3); (iii) BM(L(X) a) = L(BMX) BMa; (iv) BM(da) = d(BMa).

Proof The properties (ii), (iii), and (iv) are rather obvious in a local chart, and thus globally when one realizes that the given constructions all commute with restrictions. The IQEDI only non-trivial part is the surjectivity, which will be proven in [8.9].

8.8 Definition. A k-form a on M is said to be closed if da = 0; it is said to be exact if there exists a (k - I)-form (3 on M such that a = d{3. Since d2 = 0 it follows that any exact k-form is closed. An equivalent way to state these definitions is that a is closed if a E ker(d : [lk(M) ----t [lk+l(M)), and it is exact if a E im(d : [lk-l(M) ----t [lk(M)). The implication exact ===} closed gets translated into the inclusion

263

§S. de Rham cohomology

Since d is R-linear and homogeneous, it follows from [1.3.9] and the fact that R is a field that these subsets of n,k (M) are graded R-vector subspaces of the R-vector space n,k(M). (Of course n,k(M) is also a graded COO(M)-module, but that structure is not preserved by these subsets.) We thus can define the (quotient) R-vector space

The R-vector space Hk(M) is called the kth de Rham cohomology group of M. If a is a closed k-form on M, its cohomology class in Hk(M) will be denoted by [a]. Note that Hk(M) is an R-vector space for A-manifolds M as well as for R-manifolds M.

8.9 Theorem. IfM is anA-manifold, then the map [a] ~ [BMa], Hk(M) is an isomorphism ofR-vector spaces.

--4

Hk(BM)

Proof From [S.7] we know that this map is a well defined R-linear map, so only injectivity and surjectivity remain to be proven. To do so, let U be an atlas as in Batchelor's theorem [IV.S.2]. On each chart U E U we can define the Euler vector field X E (S.3), but now the special form of the transition functions for this atlas guarantee that these vector fields coincide on overlaps, i.e., there exists a globally defined Euler vector field X E (Nota Bene. This global vector field X E depends upon the choice of the special atlas U!). The same argument (the special form of the transition functions for our atlas) shows that the decomposition of a k-form a in homogeneous parts a(e) is invariant under a change of coordinates, i.e., each global k-form a decomposes (uniquely) as a = Le a(e) with ate) a global k-form satisfying .c(XE) a(e) = C· a(e). If a is closed, the uniqueness of the decomposition implies that all a(e) are closed, and thusthatC·a(e) =.c(XE)a(e) =dL(XE)a(f). We thus can write (8.10)

da

=0

a = a(O)

+ d (L i

L(XE)a(e») .

/2: 1

Since a(O) does not contain any ~ nor d~, we may identify a(O) with its body part Ba(O). In other words, we have a bijection between k-forms on BM and k-forms a on M satisfying .c(XE)a = O. This proves enpassant the surjectivity of BMclaimed in [S.7-i]. We now are prepared to prove the isomorphism between Hk(M) and Hk(BM). If a is a closed k-form on BM, it can be seen as a closed k-form on M satisfying .c(XE)a = 0, proving that the given map Hk(M) --4 Hk(BM) is surjective. Now [BMa] = 0 is equivalent to saying that a(O) is exact. But then according to (S.lO) the whole form a is IQEDI exact, proving injectivity of the map Hk(M) --4 Hk(BM).

8.11 Remark. The essential ingredient in the proof of [S.9] (and thus in the proof that BM is surjective) is the existence of a special atlas with transition functions as in Batchelor's theorem [IV.S.2]. Since that theorem depends in an essential way on the existence of partitions of unity, it follows that [S.9] depends in an essential way on the existence of partitions of unity.

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Chapter VI

A-Lie groups

Just as differential geometry can be seen as the interplay between analysis and geometry, so can the theory of Lie groups be seen as the interplay between differential geometry and group theory/algebra. An A-Lie group is an A-manifold which happens to be also a group in such a way that the group multiplication is compatible with the A-manifold structure. We have already introduced A-Lie groups in chapter IV, but there they did not playa very important role. On the other hand, this chapter is entirely devoted to the study of A-Lie groups and subjects directly related. Even so, we can only scratch the surface of the theory of A-Lie groups. Any A-Lie group determines an A-Lie algebra, which is the linearized version of the group in the sense that it is the tangent space at the identity equipped with the linearized version of the group multiplication. Once we have the A-Lie algebra associated to an A-Lie group, we can also go backfrom the A-Lie algebra to the A-Lie group with the exponential map. This map is defined in terms of the flow of a vector field and generalizes the usual exponential (series) of matrices. The most important aspect of A-Lie groups is that an A-Lie group is completely determined by its A-Lie algebra up to coverings. More precisely, there is, up to isomorphisms, a unique simply connected A-Lie group G associated to each (finite dimensional) A-Lie algebra 9 and any A-Lie group with the same A-Lie algebra is a quotient of G by a countable discrete subgroup. This fact means that a lot of global geometric properties of an A-Lie group can be translated into algebraic properties of its associated A-Lie algebra. For instance, ifwe define an A-Lie subgroup of an A-Lie group G as an A-Lie group H that admits a smooth injective homomorphism into G, then connected A-Lie subgroups of G are in bijection with the A-Lie subalgebras of its associated A-Lie algebra and normal connected A-Lie subgroups correspond to ideals. Once we have groups and subgroups, it is natural to consider homogeneous spaces, i. e., the quotient of a group by a subgroup. A homogeneous A-manifold is an A-manifold M with a transitive action of an A-Lie group G. Homogeneous A-manifoldsfor a given 265

266

Chapter VI. A-Lie groups

A-Lie group G are completely determined by the proper A-Lie subgroups ofG, where proper A-Lie subgroup means an A-Lie subgroup which is also a submanifold. More precisely, any homogeneous A-manifold is the quotient of G by a proper A-Lie subgroup. Another example where properties related to an A-Lie group can be translated into properties of its A-Lie algebra is the case of invariant vector fields and invariant differential forms. A vector field on a connected A-manifold is invariant under the group action if and only if it commutes with allfundamental vector fields associated to the A-Lie algebra and a differentialform is invariant ifand only ifthe Lie derivative ofthisform in the direction of the fundamental vector fields is zero. In a separate section we prove that any action ofan A-Lie group on an A-manifold can be transformed into a pseudo effective action of a quotient group. This shows that the restriction to pseudo effective actions for structure groups of a fiber bundle is in reality not a restriction at all.

1.

A-LIE GROUPS AND THEIR A-LIE ALGEBRAS

In this section we (re)define an A-Lie group G and we show that topologically it is the direct product ofGw.o.d and the even part of anA-vector space with only odd dimension. We then define the associated A-Lie algebra 9 as the set of left-invariant vector fields on G with the usual commutator of vector fields as bracket; as an A-vector space 9 is isomorphic to the tangent space at the identity of G. We give a formula for the structure constants of9 in terms of the multiplication function in a neighborhood of the identity. We finish with a detailed discussion on how to interpret the A-Lie algebra associated to the A-Lie group Aut(E). Of course this is EnciR(E), but the identification in terms of matrices is subtle and is not what one would think first.

1.1 Definitions. Some of the definitions we will give here were already given in §IV.l. We recall them because they belong rightfully in this chapter. • An A-Lie group is an A-manifold G that admits at the same time a group structure for which the multiplication m : G x G --t G is smooth. Depending on context, we will denote the product m(g, h) also by m(g, h) = go h = 9 . h = gh. The identity element will usually be denoted bye, sometimes also by id. We will show in [1.2] that e has necessarily real coordinates, and in [1.6] that for any A-Lie group taking the inverse is automatically a smooth map. The basic example of an A-Lie group is the group Aut(E) of automorphisms of an A-vector space E (see [IV. 1.3]) . • A smooth left action of an A-Lie group G on an A-manifold M is a smooth map 1> : G x M --t M such that for all m EM: 1>( e, m) = m and such that for all m, g,h: 1>(g,1>(h,m)) = 1>(gh,m). For a fixed 9 E G we will denote the map m 1---4 1>(g, m) by 1>g. If no confusion is possible, we will denote a left action also as 1>(g, m) = 1>g(m) = g(m) = g·m = gm. The evaluation map Aut(E) x E --t E is a left

§ 1. A-Lie groups and their A-Lie algebras

267

action when we equipAut(E) C EndR(E) with the usual composition of endomorphisms as group structure (see [IV.l.3]). • A smooth right action of an A-Lie group G on an A-manifold M is a smooth map 1>: M x G --4 M such that for all m EM: 1>(m, e) = m and such that for all mE M, g,h E G: 1>(1)(m,g),h) = 1>(m,gh). For a fixed 9 E G we will denote the map m f--4 1>( m, g) by 1>g. If no confusion is possible, we will denote a right action also as 1>(m, g) = m· 9 = mg. • A left/right action of an A-Lie group G on an A-manifold M is called transitive if 'rim, m' EM 3g E G: 1>g(m) = m'. • A (homo )morphism ofA-Lie groups is a smooth map p : G --4 H between two A-Lie groups G and H that is at the same time a homomorphism of (abstract) groups. • An isomorphism ofA-Lie groups is an A-Lie group morphism p : G --4 H between two A-Lie groups G and H that is at the same time a diffeomorphism of A-manifolds. It follows that p-1 is also an A-Lie group morphism. • A linear representation of G on E, or just a representation of G is an A-Lie group morphism p : G --4 Aut(E), E being an A-vector space.

1.2 Discussion. Associated to any A-manifold M we have two subsets: Mw.o.d, which is a submanifold, and BM, which is an R-manifold. Moreover, restricting a smooth map to one of these subsets maps it to the corresponding subset in the target space. Looking at BG we thus find that m : BG x BG --4 BG. From the commutativity of the diagram in [111.4.22] (i.e., Bom = moB) we deduce B(e· e) = (Be)· (Be) and thus e = Be, i.e., the identity element has real coordinates. From e = B(g . g-l) = (Bg) . B(g-l) we deduce (Bg)-l = B(g-l), i.e., ifg has real coordinates, then so has g-l. Itfollows immediately that BG is an R-Lie group (BG is an R-manifold and restriction of m to BG remains smooth) and that B : G --4 BG is a homomorphism of (abstract) groups. Looking now at Gw.o.d we can deduce from the fact that inversion Inv is smooth [1.6] that Gw.o. d is an A-Lie group (because Inv then also preserves Gw.o.d). (Note that the proof of [1.6] uses that e has real coordinates, so we have to be careful in what order we prove our statements.)

1.3 Nota Bene. If 1> is a smooth left action of an A-Lie group G on an A-manifold = 1>gh (for a right action we obtain 1>g o1>h = 1>hg), and thus in particular all maps 1>g : M --4 M are bijective with inverse 1>g-1 (because 1>e = id(M). Since the action is smooth, it is in particular continuous. It follows that all maps 1>g are homeomorphisms of M. Moreover, it follows from [III.1.23-g] that if 9 E G has real coordinates (i.e., 9 E BG), then 1>g : M --4 M is smooth. Since (BG)-l = BG we conclude that such a 1>g is a diffeomorphism of M. However, if 9 does not have real coordinates, there is no reason to suppose that 1>g is smooth. Consider for example the smooth left action of Aut(E) on E = E~. According to [11.6.22] and [III. 1.27] a map 1>g = 9 E Aut(E) is smooth if and only if the matrix elements of 9 (i.e., its coordinates) are real.

M, it follows immediately that for g, h E G we have 1>g 0 1>h

Chapter VI. A-Lie groups

268

Even though 9 is in general not smooth, we have defined generalized tangent maps Tg in [Y.3.l9] by the formula

The property 9 0 h = gh then easily gives the property T 9 0 T h = T gh, as if the chain rule were still valid. (For right actions the defining formula for Tg would be Tg(Xm) = T(Xm,Qg), and then we get Tg oTh = Thg, in accordance with the equality 9 0 h = hg.)

1.4 DefinitionIDiscussion. We can interpret the multiplication m : G x G --t G as either a left or a right action of the A-Lie group G on the A-manifold G. If we view it as a left action, i.e., m : G gp x G mfd --t G mfd , the maps mg = m(g, J are usually denoted as mg = Lg and are called left translations of Gover g. In case we view m as a right action, i.e., m : G mfd x G gp --t G mfd , the maps mg = m( _, g) are usually denoted as mg = Rg and are called right translations ofG over g. All left and right translations are homeomorphisms of G; they are diffeomorphisms if and only if 9 has real coordinates (e.g., if Lg is adiffeomorphism, Lg(e) = 9 must have real coordinates by property (A2) of smooth functions).

1.5 Lemma. The generalized tangent maps TLg and TRh commute, i.e., 'rig, h E G: TLgoTR h = TRhoTL g. Proof The associativity of the multiplication says m(g, m(k, h)) = m(m(g, k), h). Applying the tangent map to this identity allows us to compute

TLg(TRh(Xk)) =

= Tm(Qg, Tm(Xk,Qh)) = Tm(Tm(Qg, Xk),Qh) = TRh(TLg(Xk)) . TLg(Tm(Xk,~))

1.6 Lemma. IfG is an A-Lie group, then the map Inv : G --t G, 9 ~ Inv(g) = g-l describing the inverse is a smooth map andfor an arbitrary Xg E TgG we have the equality Tlnv(Xg) = -TL g-1(TR g-1(Xg)) = -TR g-1 (TL g-1 (X g )). In particular

Tlnv(Xe)

= -Xe.

Proof Since h = Inv(g) is the unique solution of the equation m(g, h) = e, smoothness is given by the implicit function theorem. The details are as follows. For go E BG we compute the partial derivatives 8ml8h of m with respect to the second variable at the point (go, g;; 1). Using [III.3.l3] this is exactly the map T Lgo. Since this map is invertible with inverse T L go-1, we can apply the implicit function theorem [111.3.27] to conclude that h = Inv(g) is the unique smooth solution in a neighborhood of go E BG. Since these neighborhoods cover G [111.4.12], Inv is globally smooth [111.4.18].

269

§ 1. A-Lie groups and their A-Lie algebras

To compute TInv, we consider the map 1> : G the chain rule we find

0= T1>(Xg)

--4

G, g

~

m(g, Inv(g)) == e. Applying

= Tm(Xg, Tlnv(Xg)) = Tm(Qg, Tlnv(Xg)) + Tm(Xg,Qg-l)

= TLg(Tlnv(X g))

+ TRg-l(Xg) .

Applying TLg-l gives the announced result (also using [1.5]).

1.7 Proposition. Let G be an A-Lie group which is modeled as an A-manifold on the A-vector space E ofdimension plq. Ifwe denote by F the A-vector space of dimension Olq, then there exists a diffeomorphism Gw.o.d x Fo --4 G, i.e., as an A-manifold G is the direct product ofGw.o.d, which contains only even coordinates, and Fo, which contains only odd coordinates. --4 0' c Eo be a chart around the identity element e E G. In the A-vector space E we not only have the canonical graded subspace Ew.o.d spanned by the even basis vectors (in the equivalence class), but also the graded subspace F spanned by the odd basis vectors. Obviously dim F = Olq and E = Ew.o.d EB F. By a translation over a vector with real coordinates (a diffeomorphism) we may assume that
Proof Let

Chapter VI. A-Lie groups

270

1.8 Definition. A (not necessarily smooth) vector field X on an A-Lie group G is called left-invariant if it satisfies the condition Tm 0 (Q x X) = X 0 m, i.e., if we have a commutative diagram

TGxTG ~ TG

G x G

------4

G.

m

Applying this condition to a point (g, h) E G x G and using the generalized tangent maps gives us T Lg(Xh) = X gh . An alternative definition of a left-invariant vector field thus is 'rig E G: TLg 0 X = X 0 L g, but this involves the generalized tangent map TL g. The vector field X is said to be right-invariant if it satisfies Tm 0 (X x Q) = X 0 m or equivalently 'rig E G : TRg 0 X = X 0 R g.

1.9 Lemma. The map X ~ Xefrom left-invariant vector fields on G to points in the tangent space at the identity e EGis a bijection. Moreover, X is smooth ifand only if Xe E BTeG, i.e., ifXe has real coordinates. If X and Yare two smooth left-invariant vector fields on G, then so is their commutator [X, Y}.

Proof Since Xg = T LgXe, the map X ~ Xe is injective. And if X is smooth, Xe has real coordinates because of condition (A2) of smooth functions and the fact that e has real coordinates. On the other hand, we can define a vector field X from Xe by Xg = T LgXe . Since T LhXg = T LhT LgXe = T LhgXe = X hg this vector field is left-invariant. Moreover, if Xe has real coordinates, the map G --t TG, g ~ TLgXe = Tm(Qg,Xe) is smooth (essentially by [III.1.23-g]). The last claim follows immediately from [Y.2.29] IQEDI and the fact that [Q x X,Q x Yj = Qx [X, Yj.

1.10 Lemma. ffX is a left-invariant vector field on G, then T Inv 0 X oInv is a rightinvariant vector field with (T Inv 0 X 0 Inv) (e) = -Xe.

Proof A direct corollary of [1.6].

1.11 Definition. In [1.6.1] we defined a (Z2-) graded A-Lie algebra (g, [_, J) as a graded A-module 9 equipped with an even bilinear graded skew-symmetric map [_, J (called the bracket) satisfying the graded Jacobi identity. In this chapter we will be more restrictive and we define an A-Lie algebra without the adjective graded to be a (Z2-) graded A-Lie algebra 9 in which 9 is an A-vector space and where the bracket is smooth. In the same spirit we define a morphism ofA-Lie algebras to be a morphism ¢ : 9 --t £) of(Z2-) graded A-Lie algebras [1.6.9] which is smooth in the sense of A-vector spaces (and recall that it is already supposed to be even). A morphism ¢ of A-Lie algebras is called an isomorphism ofA-Lie algebras if it is bijective, in which case¢-l is also anA-Lie algebra morphism.

271

§ 1. A-Lie groups and their A-Lie algebras

1.12 Construction. To any A-Lie group we will associate an A-Lie algebra which we will usually denote by the same letter as the corresponding A-Lie group, but in gothic (or Fraktur) font. In order to define the A-Lie algebra 9 associated to an A-Lie group G, we proceed in two steps. As an A-vector space, 9 is the tangent space at the identity e E G : 9 = Te G. Since e has real coordinates, TeG is indeed an A-vector space, not only a free graded A-module (see [IV.3.2]). If G, as an A-manifold, is modeled on an A-vector space E, this means that 9 is, as an A-vector space, the space E. We can turn this argument around and say that an A-Lie group G is modeled (as an A-manifold) on its A-Lie algebra 9 (as an A-vector space). According to [1.9] we can and will identify 9 with the space of all left-invariant vector fields on G. In order to distinguish between these two aspects of the A-Lie algebra g, we will usually denote an element of TeG by a lower case letter (e.g., x), and the associated left-invariant vector field by putting an arrow over this letter (e.g., X). In view of [1.9] this means in particular that for x E 9 we have = x. The usual way to define a bracket on 9 is by means of the commutator ofleft-invariant vector fields. However, in the graded case there are too few smooth left-invariant vector fields to fill 9 = TeG, and for non-smooth vector fields the commutator is not defined. The "obvious" way out (and we will use this trick quite often) is not to consider two fixed left-invariant vector fields at a time, but to consider them all together, i.e., as variables. t We thus introduce the A-manifold 9 x 9 x G, which is modeled on the A-vector space g~ x g~ x g. On this A-manifold we define two vector fields Zl and Z2 by

xe

These two maps are indeed smooth vector fields because the zero sections are smooth, as is the identity map 9 --4 TeG. The vector field Zl (and similarly for Z2) consists of all left-invariant vector fields on G : restricted to {x} x {y} x G it is the (not necessarily smooth) left-invariant vector field whose value at e EGis x. But remember: if either x or y has non-real coordinates, the slice {x} x {y} x G is not a submanifold of 9 x 9 x G. Denoting by 7r3 the projection 9 x 9 x G --4 G, we finally define the commutator on 9 by

[x, y]

=

T7r3([Zl, Z2l<x,y,e)) E TeG = 9 .

By construction, this map 9 x 9 --4 g, (x, y) ~ T7r3([Zl, Z2lcx,y,e)) is smooth. Remains to prove that this map gives 9 the structure of an A-Lie algebra.

1.13 Lemma. The map (x, y) ~ [x, y] == T7r3([Zl, Z2l<x,y,e)) is an even graded skewsymmetric bilinear map satisfying the graded Jacobi identity, thus turning 9 into an A-Lie algebra. Moreover, if the coordinates of x and yare real, then [x, y] represents the commutator of the two smooth left-invariant vector fields x and y, i.e., [x, y] = [x,

YJe.

tThe same trick can be used to describe the unconventional super Lie groups of DeWitt [DW, §4.1). For instance, the unconventional super Lie group G = Al with group structure m-y : G x G --> G, m-y(E, 1]) = E+1] - ~,E1] with 1 E Al a fixed parameter is not a A-Lie group in our sense because the multiplication m-y is not smooth for non-zero I' However, if we consider 1 as a variable, we obtain a family of group structures m : Al x G x G --> G, mb, E, 1]) = m-y(E, 1]) for which the map m is smooth.

Chapter VI. A-Lie groups

272

Proof Let (Vi) be a basis of TeG = g, which has by definition real coordinates. An elementary calculation shows that the vector field Zl is given by

Zl (x, y, g) = (Q", Qy, EiXi . Vilg) , where the Xi denote the left coordinates of x with respect to the basis (Vi) : x = Ei Xi. Vi. A similar formula holds for Z2, and we find for their commutator the expression

To derive this formula we have used that the vector field Vi has parity C(Vi), and that the commutator of vector fields is bi-additive. That the coefficients (smooth functions on 9 x 9 x G!) Xi and yj come out as they do is because the vector fields Zi do not contain any derivatives with respect to these variables. It thus follows that our bracket is given by (1.14) Since the commutator of vector fields is even, this formula shows immediately that our bracket on 9 is even and bilinear. Since the commutator of vector fields is also graded skew-symmetric and verifies the graded Jacobi identity, it follows easily that our bracket on 9 does so as well. The last part follows immediately from the given formula for [x, y] in terms of the left-invariant vector fields Vi. IQEDI

1.15 Remarks . • It is not hard to show that the vector field W given by the formula = T7r3([Zl, Z2](x,y,g)) E TgG is left-invariant; it is the left-invariant vector field whose value at e is We = [x,y] = T7r3([Zl, Z2](x,y,e))' In this way the bracket [x,y] really is the commutator of two vector fields . • We could have avoided using the A-manifold 9 x 9 x G and the vector fields Zl and Z2 by introducing a basis on 9 = E and defining the bracket on 9 directly by (1.14). The advantage of the given way is that it is intrinsic (avoiding the choice of a basis) and that it introduces a technique we will use more often . • Although it is customary to define the bracket using left-invariant vector fields, nothing prohibits the use of right-invariant vector fields to define a bracket on 9 = TeG. This would mean that one defines the bracket of x, y E 9 as being the value at e E G of the commutator of the right-invariant vector fields whose values at e are x and y respectively. It follows immediately from [1.10] and [V.2.29] that the bracket one obtains that way is the opposite of the bracket defined by means ofleft-invariant vector fields.

Wg

1.16 DefinitionIDiscussion. Let 9 be an A-Lie algebra with basis (Vi)' Then there exist such that constants

cfj

[Vi, Vj]

= EkC~jVk .

If the bracket is smooth, these constants must be real; they are called the structure constants ofg with respect to the basis (Vi). Given that the bracket is bilinear, it is immediate that the bracket is completely determined by these structure constants.

273

§ 1. A-Lie groups and their A-Lie algebras

Now if G is an A-Lie group with multiplication map m, it automatically has an associated A-Lie algebra 9 with basis (Oi Ie) relative to some coordinate system (xl, ... ,xn) in a neighborhood U of the identity e. By continuity of m there exists a chart V c U such that m(V x V) c U. Let yi = Xi 0 7rl, yn+i = Xi 0 7r2, 1 ::; i ::; n be the 2n local coordinates on V x V c G x G. Writing the local coordinates of m(g, h) for g, h E V as m k (g, h), 1 ::; k ::; n, we can compute the structure constants of g.

1.17 Lemma. The structure constants of 9 with respect to the basis (Ok Ie) are given by

Proof Since the Vi = Oi Ie have real coordinates, their brackets are the commutators of the associated left-invariant vector fields [1.13]. These are given by the formula

Vilg

om k

0

= TLgVi = (((~, oile)llTm)) = L oyn+Jg, e) ox k Ig . k

Using (V. 1.22) we thus find for the commutator of Vi and

0:

om k 0 ome 0 [ 'Ek oyn+Jx, e) oxk ' 'Ee oyn+j (x, e) oxe lie = om k o2m e 0 'Ek,e oyn+i (e, e) . oykoyn+j ( e, e) oxe Ie k e ( ). o2m _ (_l)(E(xi)jE(xj)) om ( )~I '" +. e, e '" k '" '" e e uyn J uy uyn+t. e, e uX Since m(e, g)

= g,

we have (On+imk)(e, e)

= 8f.

•

The conclusion follows from [III. 3.6]. IQEDI

1.18 Lemma. JfG is an A-Lie group with A-Lie algebra g, then its tangent bundle TG is trivial, a trivialization being given by the map ¢ : G x 9 ---t TG, (g, x) ~ T Lgx = xg. Moreover, if H is another A-Lie group with A-Lie algebra £) and if p : G ---t H is an A-Lie group morphism, then the tangent map Tp is described in this trivialization by (g, x) ~ (p(g), TeP(x))

Proof The map ¢ : G x 9 ---t TG, ¢(g,x) = Tm(Qg,x) is smooth because Tm, the zero section, and the identity 9 ---t Te G are smooth. On the other hand, the map 'ljJ: TG ---t G x 9 given by 'ljJ(Xg) = (7r(Xg),Tm(Q71'(Xg)-l,Xg)) is also smooth, just because Tm, the canonical projection 7r : TG ---t G, and the inverse 9 ~ g-1 are smooth. We finish by showing that they are each others inverse: 'ljJ(¢(g, x)) = 'ljJ(TLgx) = (g, TLg-ITLgx) = (g,x) and ¢('ljJ(Xg)) = ¢(g, TLg-IXg)) = TLgTLg-IXg = X g. The second statement follows from the computation: T p(g, x) = T p(Tma (Qg, x)) = TmH(Tp(Qg), Tp(x)) = TmH(Qp(g), TeP(x)), where we used that p is an A-Lie group IQEDI morphism.

274

Chapter VI. A-Lie groups

1.19 Corollary. Let G be an A-Lie group with A-Lie algebra 9 with basis (Vi). Then r(TG) is a free graded COO (G)-module with basis (Vi). In particular the (vilg) generate the tangent space TgG.

1.20 Discussion. We now attack the question of how to interpret the A-Lie algebra 9 of the A-Lie group G = Aut(E) for an A-vector space E. We have interpreted Aut(E) as consisting of right linear morphisms [IV. 1.3], and we already argued that it is an open subset ofthe even part EndR(E)o [111.2.26]. Since for any U c Fo open in the even part of a finite dimensional A-vector space F we have a natural identification TU ~ U x F [V.1.2], it follows that we have a natural identification T Aut(E) ~ Aut(E) x EndR(E), and in particular In other words, we identify the A-Lie algebra 9 of Aut(E) with the right linear endomorphismEndR(E) on E. But there is more to an A-Lie algebra then just its A-vector space structure: it also has a bracket. Now both 9 and EndR(E) are naturally equipped with a bracket: 9 by the commutator of left-invariant vectorfields, and EndR (E) by the (right) commutator [¢,1/'lR = ¢01/' - (-l)(E(¢)IE(,p))1/'o¢ [1.6.3]. We want to show that our identification 9 ~ EndR(E) respects these brackets. Our first concern is to find a suitable coordinate system for Aut(E) in order to describe the left-invariant vector fields. The obvious choice is the matrix representation MR defined in [11.4.1], which nicely respects the group structure. Unfortunately this choice is not really compatible with more important choices already made: coordinates on (the even part of) an A-vector space are supposed to be left coordinates with respect to a basis. If (ei)i=l is a basis for E, the natural basis for EndR(E) is given by (ei Q9 ej)r,j=l [11.4.1]. Our approach thus imposes the coordinate functions Xij = (eMR)i j defined by

¢= LX i j(¢).eiQge j . i,j

We know that the composition ° corresponds to an elementary (inner) contraction [11.5.2], so we obtain the following composition rule in these coordinates: (1.21) The conjugation \tE(ei)+E(ej) appears because we have to permute the coefficients x j k( 1/') with the ei Q9 ej . Writing for an arbitrary element g E EndR(E) the coordinates as gij = Xij(g), we obtain as multiplication in Aut(E) : (1.22) j

j

The conjugation of h j k gets transformed into the given sign because h is even, and thus the parity of h j k is c( ej Q9 ek ) = c( ej) + c( ek). Now let X be any element of EndR(E)

275

§ 1. A-Lie groups and their A-Lie algebras

with coordinates Xij = Xij(X). In the given coordinates on Aut(E) C EndR(E), the point (h, X) E Aut(E) x EndR(E) corresponds to the tangent vector

Writing z = m(g, h) we now compute, using (1.21) and (1.22), ((Xh I T Lg)) as

=

L

. .

't,),p,q,r

O ( hq r .gP q ) '--1 X'··--. {} 9 h= J {}h'.J ozP r .

L' X'··gP··--1 ° ozP.

·

= "'gP .. \tE(ep)+E(ei)(X i ). ~I h ~ , J {}zp. 9

=

J'

"'(goX)P.

~ J . ),p

. . t,),P

.

J

't,),P

J

9

h

~I h. ozP. 9 J

Similar computations complete the proof of [1.23]. The above computation shows that for a left-invariant vector field X on Aut(E) we have, in the natural identification T Aut(E) ~ Aut(E) x EndR(E), Xg = go Xe. It follows immediately that the vector fields Zi on 9 x 9 x Aut(E) [1.12] are given by

°

Z2(X,Y,g) = "'(goY)pj' ~. ~ ug P . j,p J A simple calculation gives for the commutator

where [X, Y]R denotes the commutator on EndR(E). From this it is clear that the commutator ofleft-invariant vector fields on Au t( E) corresponds to the usual commutator on EndR(E) under our identification 9 ~ EndR(E).

1.23 Lemma. In the natural identification T Aut(E) ~ Aut(E) x EndR(E), the tangent maps T L g , T R g , and TInv are given by

TLg(h,X)

=

(gh,gX)

TRg(h, X)

=

(hg, Xg)

1.24 Remark. Of course, even with 0 as group law, we could have seen Aut(E) as left linearmorphisms. That would have led to a natural identification ofg with End L (E), but an EndL(E) equipped with 0 as composition law, not with o. A careful computation reveals

Chapter VI. A-Lie groups

276

that under this identification the commutator ofleft-invariant vector fields corresponds, in

End£(E), to the bracket defined on homogeneous elements by [X,

Yl = (_l)(E(X)IE(Y)) X

0

Y - Yo

X

= -(X 0

Y -

(_l)(E(X)IE(Y))Y 0

X)

=

-[X, Y1 L

.

It should come as no surprise that this bracket is exactly the bracket obtained from the usual bracket [X, Y1 R on EndR(E) under the isomorphism '1'-1 [1.6.4]. Another variation on the above theme is to consider Aut(E) equipped with the composition law o. In this case the most natural identification for 9 is with End£( E) equipped with 0 as composition. A similar computation as for 0 shows that in this case the commutator ofleft-invariant vector fields corresponds exactly to the usual bracket [X, Y1 L on

End£(E).

1.25 Discussion. We have argued that our approach imposes the left coordinates eMR on Aut(E) instead of the more natural middle ones given by NIR [11.4.1]. A variation upon the given argument is the following. Going from G = Aut(E) to its Lie algebra 9 = Te G = EndR(E) is a kind of derivative (infinitesimal form). Since the map NIR on Aut(E) c EndR(E) is neither left nor right linear, it is not its own derivative (nor its transpose) [111.3.14], and "thus" NIR will not give the coordinates on the Lie algebra. But let us have a look at what would actually happen if we did use the NIR coordinates. The identifications g = Li,j Xi j (g) . ei Q9 ej = Li,j ei Q9 yi j (g) . ej immediately give us the relations

yij(g) =

(_l)(E(e;JIE(e;)+E(e j ))x i j(g)

,

where we have used that the parity of Xij (g) equals c:( ei) +c:( ej) (because g is supposed to be an even endomorphism). These two coordinate systems on Aut(E) thus differ only by some signs. What then could cause such problems that we insisted on the x coordinates? The answer lies in the way one has to identify EndR(E) with tangent vectors at the identity! Let X E EndR(E) be arbitrary, not necessarily even. We have identified it with the tangent vector X c:::: Li,j Xi j (X) . Ox; j' which in the y coordinates gives us

Unfortunately, the combination (-1) (E(e;)IE(e;)+E(ej)) Xij (X) is not the same as yi j( X), unless X is even. In the general case we get

We conclude that there is nothing wrong with using the (more natural) coordinates y on Aut(E), and that the identification of 9 with EndR(E) using the coordinates y also poses no problems, provided we only use even vectors/endomorphisms (see also [2.8]).

§2. The exponential map

277

Yet another way to interpret the problems with the NIR coordinates is the following observation. The yi j (g) coordinates are in between the ei and e j vectors in the tensor product representation 9 = Ei,j ei Q9 yi j (g) . ej . It thus would be natural to put the coordinates of a tangent vector also in between these indices. But the basis vectors ayi j can not be written in a natural way as a product of terms with separate indices i and j. Hence there is no natural way to put the coordinates of a tangent vector between the indices i andj in ayi j •

2.

THE EXPONENTIAL MAP

In this section we start with the definition of the exponential map, which goes from go to G with g being the A-Lie algebra of the A-Lie group G. We then show that the exponential map intertwines an A-Lie group morphism with its associated A-Lie algebra morphism (its tangent map at the identity). We finish with the definition of the Adjoint representation Ad : G --4 Aut(g) and the fact that the derivative of the Adjoint representation is the algebraic adjoint representation: Te Ad = adR : g --4 EndR(g).

2.1 Construction. The usual way to define the exponential map is by following the flows of the left-invariant vector fields. However, as we have seen, for A-Lie groups there are far too few smooth left-invariant vector fields. As before we circumvent this difficulty by looking at all (even) left-invariant vector fields at the same time. We thus consider the A-manifold go x G, on which we define the even smooth vector field ZL by

Since the zero sections and the identity map g --4 TeG are smooth, this defines indeed a smooth vector field on go x G. Obviously this vector field regroups all even left-invariant vector fields on G. Since ZL has no components in the direction of go, its flow has necessarily the form ¢(t, x, g) = (x, 1j;(t, x, g)).

2.2 Proposition. Let W Z L C Ao x go x G be the domain of definition of the map 1j; : W ZL --4 G, then (i) 1j;(t, x, g) = 9 ·1j;(t, x, e) == m(g, 1j;(t, x, e)); (ii) W ZL = Ao x go x G; (iii) 'VA E Ao: 1j;(tA,x,g) = 1j;(t, AX, g).

Proof· Since 1j; is part of the flow of ZL, we have T1j;(atl(to,x,g)) = TL..p(to,x,g)Xe. We then define ¢(t, x, g) = (x, 9 .1j;(t, x, e)) wherever it makes sense. This ¢ obviously

Chapter VI. A-Lie groups

278

satisfies ¢(O,x,g) = (x,g). Moreover,

T¢(Otl(to,x,g») = (Qx, TLgT'l/J(Otl(to,x,e»))) = (Q", TLgTL1f;(t o,x,e)Xe)) =

(Qx,TLg.1f;(to,x,e)Xe) = ZL(¢(ta,x,g)) ,

where in the second equality we used the definition of'l/J. We thus see that ¢ satisfies the conditions of a flow for Z L; by uniqueness of flows it thus must coincide with ¢ on their common domain of definition . • Since W ZL is the maximal domain of definition of ¢, it must contain the domain of definition of ¢. But if (t,x,e) belongs to W ZL , it belongs to hence any (t,x,g) belongs to W, and thus (t,x,g) belongs to W ZL . But then the group law ¢(t, ¢(s, x, g)) = ¢(s + t, x, g), the openness ofWzL , and the fact that x does not move, these all together imply that (for fixed x) all (t, x, g) belong to W ZL' • To prove (iii), consider the A-manifold Ao x go x G on which we define the even smooth vector field Z f by

W

W,

Its flow ¢E has necessarily the form ¢E (t, A, x, g) = (A, x, 'l/JE (t, A, x, g)). We also define ;j;E (t, A, x, g) = 1/;(t, AX, g), and ;fiE (t, A, x, g) = 1/;(tA, x, g). With these we compute

and

T;fiE(Otl(to,g,x,>.»)

A' T1/;(Otl(to>.,x,g») = A . Tm(Q1f;(to>.,X,g) , xe)

by the chain rule

= Tm(Q:i:E(t o,/\,g,x \ )' AXe)

by left linearity of Tm.

=

'P

From these two computations and the uniqueness ofthe flow of Z f, it follows immediately that 1/;E = ;j;E = ;fiE. IQEDI

2.3 Definition. The exponential map exp : go --t G is defined in terms of the flow of the vector field ZL on go x Gas exp(x) = 1/;(1, x, e).

2.4 Proposition. The exponential map exp : go

--t

G has thefollowing properties.

(i) TheflowofZL isgivenby¢(t,x,g) = (x,g·exp(tx)). (ii) 'tis, t E Ao'tlx E go : exp(sx) . exp(tx) = exp((s + t)x).

279

§2. The exponential map

(iii) If X is any even smooth left-invariant vector field on G, its flow is defined on the whole ofAo x G and is given by (t, g) f---> 9 . exp(txe). (iv) To exp = id(fI). (v) Theflow of the even smooth vectorfield ZR(X,g) = (Q", TRgx) on flo x G (the right equivalent of ZLJ is given by (t, x, g) f---> (x, exp(tx) . g). (vi) If X is any even smooth right-invariant vector field on G, its flow is defined on the whole ofAo x G and is given by (t,g) f---> exp(tXe )· g.

Proof • (i) is a direct consequence of [2.2-i,iii], and (ii) follows from the group property of the flow of ZL' • If X is a smooth left-invariant vector field on G, x = xe has real coordinates, and thus the map X: (t, g) f---> g. exp(txe) = 1jJ(t, x, g) is smooth. We then compute, using the left invariance of X, TX(Otl(t,g)) = Tm(Q,p(t,x,g),xe ) = Xx(t,g). Uniqueness of its flow then proves (iii). • To prove (iv), consider the map X: Ao x flo --4 G, (t,x) f---> exp(tx). By the chain rule we find that TX(Otl(o,x)) = Toexp(xlo). Note however that there is a change in interpretation of the x in this formula. The tangent map of the map (t, x) f---> tx transforms the tangent vector Ot at (0, x) into the tangent vector x == xlo E Toflo ~ fI at 0 E flo. Since X(t, x) = 1jJ(t, e, x), we have Tx(otl(o,x)) = TL,p(O,x,e)xe = xe = x (use that 1jJ(0, x, e) = e and e . 9 = g). • According to (i) we have the equality exp(sx) . exp(tx) = exp(tx) . exp(sx). When we see this as maps defined on (s, t, x) E Ao x Ao x flo, we can apply the tangent maps to the vector osl(o,t,x)' Using (iv) we obtain the equality TRexp(tx)Xe = TLexp(tx)xe , We now define Xby X(t,x,g) = (x,exp(tx)· g), and we compute TX(Otl(t,x,g))

= (Q", TRgT1jJ(otl(t,x,e))) = (Q", TRg(TLexp(tx)Xe)) =

(Q", TRg(TRexp(tx)Xe))

=

ZR(X(t, x, g)) .

=

(Q", TRexp(tx).gXe )

Uniqueness of the flow finishes the proof of (v). The proof of (vi) is a variation of that of (iii). IQEDI

2.5 Nota Bene. The restriction to even elements in [2A-ii] and [2A-iii] is essential. One might be tempted to think that for a smooth odd X, i.e., x E Bfll [1.9], its flow is given by (T, g) f---> 9 exp( TX), using that TX E flo (because the time parameter of an odd vector field is odd). In [3.17] we will show that this is the case if x satisfies [x, xl = 0, i.e., the standard condition for integrability. We will also show that this is equivalent to the homomorphism property exp(Tx) . exp(O'x) = exp((T + O')x). In [2.8] we will give an example in which these conditions are not satisfied.

2.6 Lemma. Let G be a connected A-Lie group and U an open neighborhood of e E G. Then G is generated by U, i.e., any element ofG is a finite product of elements of U.

Chapter VI. A-Lie groups

280

Proof Define V = Un U- 1 = {g E U I g-1 E U}, which is an open neighborhood of e becauselnv is adiffeomorphism(~ homeomorphism), and denote by G 1 the (abstract) subgroup generated by V. For any 9 E G 1 it follows that Lg V is an open neighborhood of 9 (because Lg is a homeomorphism) which is contained in G 1 . Hence Gl is open. On the other hand, suppose 9 E G \ G 1 and h E Lg V n G 1 , then 3v E V : 9 = hv- 1 , i.e., 9 E Gl (because V = V-I). Since this contradicts 9 rf. Gl, we conclude that Lg V c G \ Gl, i.e., that G \ Gl is open. We conclude that Gl is open, closed, and non-empty. Since G is connected, we conclude G = G 1 . IQEDI

2.7 Corollary. There exists an open set U C go containing 0 and an open set V C G containing e such that exp : U ---t V is a diffeomorphism. In particular, if G is connected, it is generated by elements ofthe form exp(x) with x Ego.

Proof The first part is a direct consequence of [2.4-iv] and the inverse function theorem [111.3.23]. The second part follows from [2.6]. IQEDI

2.8 Example. Let G be the multiplicative A-Lie group of invertible elements in A, i.e., G = {x + ~ E A I Bx -I O}. It is modeled on an A-vector space of dimension 111 and its multiplication is given by (2.9)

A basis of g = TeG at e = (1,0) is given by the vectors VI = oxle and V2 = 0Ele. The associated left-invariant vector fields are given by

An elementary computation reveals [VI, VI] [ih, V2] = 0 and [V2, V2] = -2Vl. We could also have used [1.17] to obtain these commutators (structure constants): using (2.9) one obtains

0102+1 m = OxOym = (1,0) 0202+1m

= 0Eoym = (0,1)

= oxorym = (0,1) , 0202+2m = 0Eorym = (-1,0). 0102+2m

Inserting the appropriate signs immediately gives

cil

=

ci2

= c Z1 = c~2 = 0 and

C§2 = -2, in accordance with the previously calculated commutators of the Vi. If we denote by (11, P) the left coordinates of an element f E g with respect tot the basis (Vi), we obtain the full bracket in g by (1.14) as

Integrating the vector field ZL(fl, f2, x,~) = fl . VI P is odd), one finds the flow

fl is even and

+ P . V2

(but now f E go, i.e.,

§2. The exponential map

281

which gives for the exponential map (aVI + aV2)

t---+

exp( a, a) :

Combining the even and odd coordinates in a single A-valued "coordinate" on both sides, we can write this expression as exp(a + a) = e a .(1 + a). If we realize that a is odd and thus a 2 = 0, we can see the term 1 + a as the Taylor expansion of e a , i.e., we can write exp(a + a)

= e a +a

.

In this visualization the exponential map of G thus becomes the ordinary exponential map extended to A (see also [3.11]). Once we know the exponential map, it is easy to compute for odd 0-, 7 E Al the product exp(O-V2)' exp(7V2) = exp(O,o-)· exp(0,7) = (1,0-)' (1,7) = (1 +0-7,0- +7). Since this is not equal to exp( (0- + 7 )V2) = (1,0- + 7), we here have an example in which [2A-ii] is not true for odd vector fields (and odd coefficients). We can also consider the map¢: Al x G --4 GdefinedbY¢(7,(x,O) = (x,~)·exp(7v2) = (X+~7,~+X7). A direct calculation gives T¢(OTI(T,X,E)) = XOE I(XHT,UXT) - ~OxI<xHT,E+XT)' which is not equal to v21(xHT,UXT) = (x + ~7)oEI<xHT,UXT) - (~+ x7)oxl(xHT,E+XT)' This shows that [2A-iii] need not be true in the odd case. This example is also perfectly suited to illustrate the truth of [1.25]. The group G can be realized as a group of 2 x 2 matrices equipped with the usual matrix multiplication:

It seems reasonable that the corresponding matrix representation of 9 is given by

And indeed, if I, 9 E 9 are even (meaning 11, gI E A o, p, g2 E AI), a direct computation shows

[/,g]

~ [(j~

in complete agreement with the official bracket. But ... when we take for 1 and 9 the odd element V2, i.e., 1 = 9 ~ (0,1), the official bracket equals -2VI ~ (-2,0), while the bracket of the corresponding (odd!) matrices gives (2,0). As said in [1.25] the origin of this problem lies in the identification between tangent vectors and matrices. Since we use the standard matrix multiplication, this means that we use the coordinates MR. According to the formulas in [1104.1] we thus have

282

Chapter VI. A-Lie groups

i.e., the matrices (~ ~) and ( ~1 ~ the same formulas,

/ = / 1 . VI + / 2 . V2

::::: /1 . (

)

form a basis. For the Lie algebra we thus find, using

° ° + (_° 1

1

)

/2 .

1

A careful calculation reveals that this matrix representation indeed effectively represents the bracket in g; and it corresponds to the previous identification for even elements.

2.10 Definitions. Let Q be an A-manifold, and let G and H be A-Lie groups. A smooth map 1> : Q x G ---) H is called a family of (A-Lie group) homomorphisms from G to H if for all q E Q the map 1>q : G ---) H, 9 ~ 1>( q, g) is a homomorphism of (abstract) groups. If 9 and £) are A-Lie algebras, then a smooth map ¢ : Q x 9 ---) £) is called a family of (A-Lie algebra) morphisms from 9 to £) iffor all q E Q the map ¢q : 9 ---) £), X ~ ¢( q, X) is a morphism of A-Lie algebras [1.6.9] (and thus in particular even). According to [IV.3.17], such a family is equivalent to a smooth map ¢ : Q ---) HomR(g; £))0 such that all ¢(q) == ¢q are A-Lie algebra morphisms.

2.11 Proposition. Let G and H be A-Lie groups, let 9 and £) be their A-Lie algebras, and let 1> : Q x G ---) H be a family of homomorphisms. Then the map T'1> : Q x 9 ---) £) defined by T'1>(q, x) = T1>q(xe) = T1>(~, xe) [V.3.i9] is a family ofmorphisms from 9 to £). in case Q contains a single point, this reduces to the fact that the tangent map at e of a homomorphism between A-Lie groups is a morphism between their A-Lie algebras. Proof First of all note that T'1> is indeed a smooth map, and that it is even and linear in x (because T1> is a smooth even vector bundle map). Since 1> is a family of homomorphisms, the map 1> q sends the identity of G to the identity of H, proving that T'1> has indeed £) = TeH as target space. In order to prove that T'1> preserves brackets, we recall that the bracket on 9 is defined by the commutator of the vector fields zf on 9 x 9 x G (and similarly for £)). We now extend these vector fields to vector fields 2f on Q x 9 x 9 x G by 2f = Q x zf. We also extend the map 1> to a smooth map 1> : Q x 9 x 9 x G ---) £) x £) x H by

1>(q, x, y, g)

= (T'1>(q, x), T'1>(q, y), 1>(q, g)) .

With these ingredients we compute (T1> 0

-

~ (T1> o ZI )(q,x,y,g)

2f) (q, x, y, g) as

-

= T1>(~,Qx,Qy,TLg(xe)) = (QT'(q,x) , QT'(q,y) , T1>q(T Lg(xe))) = (QT'(q,x) , QT'(q,y) , TL(q,g) (T'1>(q, x)) = z[I (T'1>(q, x), T'1>(q, y), 1>(q, g))

§2. The exponential map

283

H

= (Zl

~

oq,)(q,x,y,g),

where, in going from the second to the third line, we have used that q, is a family of homomorphisms from G to H. We thus have proved that the vector fields and are related by the map Cli (for the computations are similar). By [V.2.29] their commutators are also related by Cli, in particular at the point (q, x, y, e) and its image (T'q,(q, x), T'q,(q, y), e) where we have

zfI

2f

2f

,

~

----t

~

~G

~G

(Q, Q, T q,(q, [x, yJ)) = Tq,(Q, Q, Q, [x, yJle) = Tq,([Zl , Z2 J) = [z[i, z£iJ = (Q, Q, [T'q,(q, x), T'q,(q, y)

J) .

IQEDI

2.12 Definition. Let G be an A-Lie group and 9 its A-Lie algebra. We define the map 1 : G x G ----t G by 1 (g, h) = ghg-l. It is elementary to verify that / is a left action of G on itself and at the same time a family of homomorphisms from G to G. We thus can define the associated family T'/ of morphisms from 9 to 9 by T'I (g, x) = T I/ie. Formally the tangent map is left linear, but since it is also even, it is right linear too. We thus can apply [IV. 3.17] to obtain a smooth map Ad : G ----t EndR(g)o, i.e., Ad(g)(x) = T' I(g, x) = T I/ie. According to the definition ofI(g, h), this can also be written as

According to [2.11], each Ad(g) is an A-Lie algebra morphism, i.e., we have the equality Ad(g)([x, yJ) = [Ad(g)(x), Ad(g)(y)J. And even more: the map Ad: G ----t EndR(g) is an A-Lie group homomorphism. To prove this, we compute Ad(h)(Ad(g)(x))

= Th(TIg(xe)) = Thg(xe) =

Ad(hg)(x) .

Since obviously Ad( e) = id(g), this shows that Ad takes its values in Aut(g) and that it is a linear representation of G on g. This representation is called the Adjoint representation ofG.

2.13 Example. Let G be the group Aut(E) of automorphisms of some finite dimensional A-vector space E, and recall that we have identified its A-Lie algebra 9 with EndR(E). We now want to compute the Adjoint representation explicitly. Using [1.23], we find for Ad(g)x = TLg TRg-l xe in thetrivialization T Aut(E) ~ Aut(E) x EndR(E) the expression TLg T Rg-l (e, x) = (e, gxg- 1 ). We thus obtain for the Adjoint representation of Aut(E) on EndR(E) the following formula: I;fg E Aut(E), I;fX E EndR(E) ~ 9

Ad(g)(X)

= go X

0

g-l .

284

Chapter VI. A-Lie groups

2.14 DiscussionIDermition. We can apply [2.11] to the morphism Ad : G ---t Aut(g) of A-Lie groups to obtain a morphism Te Ad : 9 ---t EndR(g) of A-Lie algebras. In [2.1S] we will show that this map is the right adjoint representation of the A-Lie algebra 9 defined by adR(x)(y) = [x, yj, i.e., Te Ad = adR. Once we know this, the fact that it is an A-Lie algebra morphism, i.e.,

'v'x,y E g: ad([x,y])

=

[ad(x),ad(y)jR'

is just a reformulation of the graded Jacobi identity.

2.15 Proposition. Let G be an A-Lie group, and let Ad : G ---t Aut(g) be its Adjoint representation. Then the A-Lie algebra morphism Te Ad: 9 ---t EndR(g) is the algebraic adjoint representation adR : (Te Ad(xe))(Y) = [x, yj.

Proof Since Te Ad is an even linear map, it is sufficient to know its values on homogeneous vectors with real coordinates (these contain at least a basis). Similarly, to know the linear map Te Ad(xe) E EndR(g) it is sufficient to know its values on homogeneous vectors with real coordinates. Since Aut(g) c EndR(g)o, we can see Ad as a smooth function with values in the even part of the A-vector space EndR(g). We know from [Y.3.2] that (( xe I T Ad )) = xe Ad E End R (g). Using [1.9] it thus is sufficient to prove the equality (xe Ad)(y) = [x, Yle for smooth homogeneous left-invariant vector fields. So let X and ybe smooth homogeneous left-invariant vector fields on G and let a = c(x) be the parity of x. We define the maps ¢ : Aa x G ---t G and 1> : Aa x G ---t Aa x G by ¢(t,g) = g. exp(tx) and 1>(t, g) = (-t,¢(t,g)). This looks like the flow of the left-invariant vector field but for a general x this needs not be the case (see [2.S] and [2.8] for more details). What we do have is the property (Ot I(o,g) I T¢)) = Xg (but not necessarily for values of t different from zero). Moreover, since ¢(O, g) = g we also have the equalities (V.S.12) and (Y.S.13). As in [y'S.14] and [V.S.1S] we define 'l/; : Aa x G ---t TG by 'l/; = T¢ 0 (Q x y) 01>. Unlike [V.5.1S] we will not compute Ot'l/; for all values of t, but only at t = 0. The computations are completely similar to those of the proof of [V.S.1S] (without the additional s) and we find (Ot'l/; )(0, g) = [x, Yjg. On the other hand, we can use the explicit form ¢(t, g) = Rexp(tx)g to compute 'l/; directly

x,

'l/;(t,g) = T¢(Q_ t , Ygexp(tx)) = TRexp(-tx)'Yaexp(tx) = TLgTLexp(tx)TRexp(tx)-lYe = TLgT1exp(tx)Ye = TLg (Ad(exp(tx)) (y)) , where for the third equality we used that the vector field Y is left-invariant. We thus find that 'l/;(t, e) = Ad(exp(tX))(Y) and thus Ot'l/;(O, e) = (xe Ad)(y), where we used that To exp = id. But restriction to real values and differentiating other coordinates commute [111.3.13], so we find (xe Ad)(y) = [x, YJe. IQEDI

§3. Convergence and the exponential of matrices

285

2.16 Proposition. Let : Q x G ----t H be afamily of homomorphisms from G to H, and let T' be the associated family of A-Lie algebra morphisms from 9 to f). Thenfor all q E Q and all x E go :

(q,expc(x))

= eXPH(T'(q,x)) .

In case Q contains a single point, this reduces to (expc(x)) = eXPH(Te(x)). Proof The exponential maps are defined in terms of the flows of the vector fields ZL. We on 9 x G to a vector field on Q x 9 x G by the formula extend the vector field = Q x Zr. We also extend the map to a map Cli : Q x 9 x G ----t f) x H by Cli( q, x, g) = (T' (q, x), ( q, g)). With these definition we compute

if

zf

zf

if and Zf

We conclude that the vector fields [Y.S.7] and [Y.S.S], the result follows.

are related by the map

Cli.

Combining

IQEDI

2.17 Corollary. Let ¢i : G ----t H be two A-Lie group homomorphisms. If Te¢l and Te¢2 are the same as linear maps 9 ----t f), then ¢1 and ¢2 coincide on the connected component of G containing the identity e.

Proof According to [2.16] we have ¢l(exp(x)) ¢2(exp(x)). The result now follows from [2.7].

= exp(Te¢l(X)) = exp(Te¢2(X)) = IQEDI

2.18 Example. In [III.3.14] we essentially proved that Te gDet = gtrR when we view Aut(E) as a subset of EndR(E)o. From [2.17] we deduce that gDet is the unique (on the connected component) A-Lie group homomorphism Aut(E) ----t A whose associated algebra morphism is gtrR'

2.19 Corollary. Let G be an A-Lie group and let 9 be its A-Lie algebra. For any 9 E G and any x E go we have

9 . expc(x) . g-l

= expc(Ad(g)x)

and

Ad(expc(x)) = eXPAut(g) (adR(x)) .

In particular ifG = Aut(E) we have go exp(X) 0 g-l

= exp(g

0

X

0

g-l).

Proof This is a direct consequence of [2.15] and [2.16]. The particular case G = Aut(E) follows from [2.13]. IQEDI

286

3.

Chapter VI. A-Lie groups

CONVERGENCE AND THE EXPONENTIAL OF MATRICES

In this section we show that the derivative of the exponential map is given by the formula 1 e-ad(x) Tx exp = T Lexp(x) -ad (x) . In order to prove this we have to introduce a notion of convergence of a sequence of smooth functions. We use this notion also to define the exponential of a matrix, and we show that it corresponds to the exponential map defined previouslyfor abstract A-Lie groups.

3.1 ConstructionIDefinition. Let E be an A-vector space of dimension plq, let F be a finite dimensional A-vector space, and let U be open in Eo. Ifwe want to define a suitable notion of convergence of functions in COO(U; F), the usual pointwise convergence will not do, because the non-Hausdorff topology of F prohibits uniqueness. In order to obtain a satisfactory notion of convergence, we decompose a function IE COO(U; F) according to [111.2.23] as q

I(x,~) =

L

where the li1 ... ik (x) are ordinary smooth functions on BU with values in BF. Since these ordinary vector valued functions 1i1 ... ik are uniquely determined by I, it seems natural to define convergence in terms of these components lil ... i k. We thus will say that a sequence In E COO(U; F) converges (pointwise/uniformly on compacta) to I E COO(U; F) if and only if all components (fn)il ... ik E COO(BU; BF) converge (pointwise/uniformly on compacta) to the corresponding component lil ... ik E COO(BU; BF), using any suitable norm on BF to define these notions of convergence in Coo (B U; BF). Note that with this definition we have uniqueness of convergence: if In converges to I and to g, then I = g. This is an immediate consequence of the bijection between a function I and the set of its components li1 ... ik'

3.2 Discussion. Our main application of the notion of convergence will be in the construction of functions on EndR(E)o, with E an A-vector space of dimension plq. On EndR (E) we will use left coordinates X = Xi j . ei Q9 e j . However, in order to simplify notation, we denote the p2 + q2 even ones among the Xi j by Xi, and the 2pq odd ones by ~i. Finally we define the smooth functions gn : EndR(E)o ---t EndR(E)o by

Li.i

gn(X)=xn=~

.

n times

In terms of coordinates, these functions decompose into components

§3. Convergence and the exponential of matrices

287

Each (gn)i, ... ik takes it values in B EndR(E), i.e., in the space of (p + q) x (p + q) matrices with real entries. It is immediate from matrix multiplication that each matrix entry of (gn)i, ... ik is a homogeneous polynomial of degree n - k in the Xi (provided n ;::: k, else it is zero). Nearly as immediate is the estimate

This estimate will permit us to define functions on EndR(E)o by means of converging power series.

3.4 Lemma. Let I(z) = L~=o anz n be a convergent power series on the whole of C, and let E be a finite dimensional A-vector space. Then the sequence offunctions Ii : EndR(E)o -+ EndR(E)o defined by li(X) = L~=o anxn converges uniformly on compacta to a smooth function I : EndR(E)o -+ EndR(E)o (slight abuse of notation). Proof In terms of the coordinates x and~, the functions Ii obviously decompose as

li(X,~) = L k

L ~il i , <···
i

.. .

~ik. (Lan· (gn)i1 ... ik(X)) . n=O

Using the estimate (3.3) and the fact that I( z) is convergent on the whole of C shows that the series L~=o an· (gn)i, ... ik (x) converges to a function li, ... ik (x). Standard arguments of real analysis show that the convergence is uniform on compacta and that the resulting function is smooth. We thus can define the function I(X) by

00

=L k

i,

L ~il ... ~ik·(Lan·(gn)il ... ik(X)) <···
3.5 Definition. If E is a finite dimensional A-vector space, we define the exponential map EndR(E)o -+ EndR(E)o, X ~ eX as the smooth function associated to the power series e Z = L~=o zn In!.

e :

3.6 Example. Let E be an A-vector space of total dimension 2. If its graded dimension is either 210 or 012, the space EndR(E)o is described by 4 even coordinates and the exponential map is just the G-extension of the ordinary exponential of a 2 x 2 matrix. However, if the dimension of E is 111 it gets more interesting. Using the matrix representation NIR [11.4.1] we describe an even endomorphism X with two even coordinates a and d and two

Chapter VI. A-Lie groups

288

odd coordinates f3 and 'Y by NIR(X)

= (~ ~).

An elementary calculation then gives

= (an ( a (3)n d 0 ~(

As a final we result one finds

ex ((a p

'Y

a

d

d

(3))=(e +ea B'(a-d)f3'Y d e aB(a-d)f3 ) d e B(d - ah e - e B'(d - a)f3'Y '

with B(z) = (e Z -l)/z,and B'(z)itsderivative. We can apply this result to the matrix representation of [2.8], in which case we have a = d = 11 E Ao and f3 = 'Y = 12 E AI. Our exponential of matrices then yields the result (

J

iJ1

e·f

2

el' el

/2 ), in agreement with the exponential found in [2.8].

3.7 Lemma. Let g, I, In E COO(U; F) be smoothfunctions, and let Xi denote any (even or odd) coordinate on U. Ifthe sequence In converges pointwise to I and if the sequence OXi In converges uniformly on compacta to g, then 9 = OXi f. Proof From (111.3.5) we see that for an odd coordinate Xi, the set of components of 0xi In is essentially the same as that of In. The announced result then follows immediately. From (111.3.5) we also see that if Xi is an even coordinate, we have (OXi f)i, ... ik = 0xi (fi , ... ik)' The result now follows from the corresponding result for ordinary vector valued functions in COO(BU; BF). IQEDI

3.8 Lemma. Let I, In E COO(U; F) be smooth functions, let V c Fo be open, and let 9 E COO(V; G) be arbitrary. Suppose that the sequence in converges (pointwise/ uniformly on compacta) to f. If go In and go I are defined, then go In converges (pointwise/uniformly on compacta) to go I Proof Let [ E COO(U; F) be an arbitrary function, but such that h = go [is defined. If we denote the coordinates on U by (x,~) and those on V by (Y,7]), we can write [(x, 0 = (y(x, ~), 7](x, ~)) and thus:

(g 0 [)(x, 0 = g(y(x, ~), 7](x, ~)) =

LL

7]), (x,~)

... 7]j£ (x, 0 . gj, ... j£ (y(x, 0)

.

From this expression, it follows that the components hi, ... ik of the composite function h = go[ are of the form hi, ... ik(X) = ¢Ji, ... ik(fj, ... jt(x)), where each ¢i, ... ik(Z) is a smooth function in z whose structure depends only on g, not on Since all components of In converge (pointwise/uniformly on compacta) to the corresponding components of I, it follows that the functions ¢i, ... ik((fn)), ... j£) converge

1.

289

§3. Convergence and the exponential of matrices

(pointwise/uniformly on compacta) to ¢i, ... ik (fj, ... ji)' For pointwise convergence this is immediate from the continuity of ¢i, ... ik; for uniform convergence on compacta this is a more delicate reasoning using that a continuous function is uniformly continuous on compacta and using that a compact set in BV can always be enlarged a small amount IQEDI while remaining inside BV.

3.9 Corollary. Let I, g,!n, gn E COO(U; EndR(F)) be smooth functions such that In converges to I and gn converges to g.

x F converge to thefunction ¢(m,v) = I(m)(v). (ii) Thefunctions hn(m) = In(m) 0 gn(m) converge to h(m) = I(m) 0 g(m). (i) Thefunctions ¢n(m,v) = In(m)(v) definedon U

Proof If the functions In converge to I, then the functions in : U x F --4 EndR(F) X F defined by !n(m, v) = (fn(m),v) converge to !(m,v) = (f(m),v). This is immediate when we look at the coordinates in EndR(F) x F separately. Since the evaluation map EndR( F) x F --4 F is smooth, the result follows from [3.8]. A similar argument applies in (ii): if the functions In converge to I and the gn to g, then the functions in (m) = (fn (m), gn (m)) converge to (f, g). Since the composition map EndR(F) x EndR(F) --4 EndR(F) is smooth the result follows again from [3.8]. IQEDI

3.10 Discussion. If E is a finite dimensional A-vector space we now have two exponential maps on EndR(E)o : exp(X) defined by means of the flow ofleft-invariant vector fields on Aut( E), and eX defined by means of a converging power series. Let us show that these two are actually the same. Consider the smooth map;j; : Ao x EndR(E)o x Aut(E) --4 EndR(E)o defined by ~

'IjJ(t,X,g) = go e

tX

,

and use it to define the map ¢ : Ao x EndR(E)o x Aut(E) by

--4

EndR(E)o x EndR(E)o

¢(t,X,g) = (X,;j;(t,X,g)). If we can show that this ¢ satisfies the equation T¢ 0 at = Z L 0 ¢, then uniqueness of flows proves our claim. In [1.20] we already proved that we have Tm(Q g, Xe) = go X, giving us the equality

ZL(¢(t, X, g)) = (Qx, ;j;(t, X, g) 0 X) = (Qx, 9 0 etX 0 X). On the other hand we have tX the equality T¢(at!;j;(t,X,g») == at I;j;(t,x,g)(¢) = (QX' at;j;(t, X, g)) = (Qx,goat(e )). But then we are in position to apply our lemmas. The function (t, X) ~ etX is the limit of the partial sums (t, X) ~ L~=o t n . xn In!. The partial derivative with respect to t of these sums is given by (t, X) ~ L~-==~ t n . Xn+l In! = (L~-==lO t n . xn In!) 0 X. Combining [3.7] and [3.9-ii] we obtain that at (e tX ) = etX 0 X. We conclude that ¢ must

be the flow of ZL and thus both definitions of the exponential map coincide. In particular we deduce that the power series takes its values in Aut ( E ).

290

Chapter VI. A-Lie groups

3.11 Remark. Some readers might notice that there is actually nothing that prevents us from defining the exponential in terms of the power series on the whole of EndR (E). The obvious question is whether this gives us something new. The answer is negative. The ring EndR(E) can be identified in a natural way with a subring ofEndR(E~)o, in the same spirit we used to identify the invertible elements of A with 2 x 2 matrices [1.25], [2.8]. Taking the exponential on EndR(E~)o preserves this subring, and thus we find that the map X ~ eX takes its values in the A-Lie group G of invertible elements in EndR(E), i.e., G = {A E EndR(E) I ~B E EndR(E) : A 0 B = id}. Another way to interpret the exponential on the whole ofEndR(E) is in terms of the A-Lie group G introduced above. The space EndR(E) can be identified with the even part of the A-Lie algebra 9 of G. As such, the exponential as power series is exactly the exponential map exp : go --4 G.

3.12 Proposition. Let E be a finite dimensional A-vector space, and let the convergent power series I and In be defined on C as

e Z -1 I(z) = -z- = eZ

f n (z) = Then

-1

8 00

Zi

(i + I)! n-l

00

n-l

.ei

fz n = ""' .1 e / = ""'(""' - ) . Zi n(ez/n -1) ~ n ~ i! ni+l

20

In converges uniformly on compacta to I, seen asfunctions

.

on EndR(E)o.

Proof It is an elementary computation in real analysis to show that the power series In (z) converge uniformly on compacta to f. This plus the estimate (3.3) and the usual arguments on interchanging the order oflimits show that the functions 2::0 (2:~':-~ i! ~;+1) . (gi)il ... ik converge uniformly on compacta to the function 2::0 (i~l)! . (gi)i1 ... ik ·IQEDI

3.13 Discussion. We are now finally in position to give an explicit expression for the tangent map of the exponential map. Let G be an A-Lie group, 9 its A-Lie algebra, and let exp : go --4 G be the associated exponential map. If y is a tangent vector at x E go, the map Tx exp maps it to a tangent vector at exp(x) E G. Since it is easier to deal with the tangent space at e E G than with the tangent space at a general point of G, we left-translate this vector back to e E G. Since Tx(go) is in a natural way isomorphic to g, we thus are led to the definition of the smooth map M : go x 9 --4 9 by (3.14) M(x, y) = TLexp(-x) ((Tx exp)(y)) = Tm(Qexp(_x), (Tx exp)(y)) E TeG = 9 . Since this map is obviously even and linear in y, we can interpret this as a smooth map M : go --4 EndR(E)o [IY.3.l7].

§3. Convergence and the exponential of matrices

291

3.15 Proposition. For all x E go the map M(x) [3.14J is given by 1- e- ad(x) M(x) = ad(x) , i.e., M(x)

=

f( - ad(x)) with f as in [3.12J.

Proof Let us denote by mk the map mk : G k --4 G, (gl, . .. ,gk) ~ gl ... gk of k-fold multiplication. For any n E N* it follows easily from [2.4-ii] that we have the equality exp(x) = mn(exp(~x), ... ,exp(~x)). In the canonical identification Tgo = go x g [v. 1.2], we obtain by the chain rule that for 'ljJ(x) = exp(~x) and y E g we have (Tx'ljJ)(y) = ~(Tx/n exp)(y) = (Tx/n exp)(~y). Using extensively the associativity of the multiplication we then compute:

M(x, y) = Tm2(!4xp(_x) , (Tx exp )(y)) =

Tmn+1 (!4xp(-x) , (Tx/n exp)(~y), ... , (Tx/n exp)(~y)) v

"

.f

n terms

n-1

= L Tmn+1 (!4xp( -x), !4xp(x/n) , ... ,!4xp(x/n), (Tx/n exp )(~y), k=O

' v

'

n-1-k terms !4xp(x/n) , ... ,!4xp(x/n)) v

"

J

k terms

n-l

=

L Tm 4(!4xp(-kx/n), !4xp( -x/n), (Tx/n exp )(~y), !4xp(kx/n)) k=O

n-l

=

L Tm 3(!4xp(-kx/n), M(~x, ~y), !4xp(kx/n)) k=O

n-l

=L

Ad(exp( -~x), M(~x, ~y)) , k=O where we used that Ad(g,x) = Tm3(Qg,Xe,Qg-l). Using [IY.3.17], [2.19], and the linearity of M in its second variable, we thus have established, for all n E N*, the equality

n-1

M(x) =

(L ~ .exp( -~ ad(x)))

0

M(~x) .

k=O But in the limit n --4 00 the sequence M(~x) converges uniformly on compacta to the function M(O), which is the identity by [2.4-iv]. The result then follows from [3.9-ii] and IQEDI [3.12].

3.16 Corollary. For x E go and y E g we have (Tx exp)(y) = TLexp(x) M(x)(y)

=

TRexp(x) M( -x)(y). Proof The first equality follows immediately from the definition (3.14) of M (x). The second one follows from the observation that TRexp(_x)(Tx exp)(y) = Ad(exp(x))M(x, y) and that Ad(exp(x)) 0 M(x) = exp(ad(x)) 0 M(x) = M( -x). IQEDI

292

Chapter VI. A-Lie groups

3.17 Proposition. Let x be an odd smooth left-invariant vector field on an A-Lie group G. Then [x, x] = 0 ifand only ijVt7, TEAl: exp(t7x) . exp(TX) = exp((t7 + T)X). In that case its flow is given by ¢(T, g) = g. exp(Tx).

Proof Suppose first that [x, x] = 0, which means that x is integrable. Since obviously ¢(O, g) = g, it follows from the uniqueness of flows that it is sufficient to prove that T¢(orl(r,g)) = x(r,g)' To do so, we first note that T¢(orl(r,g)) = TLgTexp(xlrx). According to (3.14) and [3.15] we find for Texp(x rx ) the value TLexp(rx)M(TX)(X). Since ad(Tx)(X) = [TX, x] = T[X, x] = 0, we conclude from the explicit expression for M that M(TX)(X) = xe and thus T¢(orl(r,g)) = TLgTLexp(rx)(xe) = xg.exp(rx). That we have exp(t7x) . exp(Tx) = exp((t7 + T)X) follows immediately from the group property of a flow [VA.13]. Now suppose that for all odd 17, TEAl we have exp(t7x), exp(TX) = exp((t7 +T)X). Sincex is smooth, X has real coordinates, and thus the map1/! : (17, T) ~ exp( t7x) ·exp( TX) is smooth. Since ()" and T are odd, B1/! is constant the identity e E G. This implies that 1/!(t7, T) remains in a chart U around e. Working in local coordinates, we can expand 1/! as

+ t70a1/!(0, 0) + TOr 1/!(O, 0) + t7T{}rOa1/! (0, 0) e + (17 + T)Xe + t7TOr Oa1/! (0, 0) .

1/!(t7, T) = 1/!(0, 0) =

Since 17 and T are both odd, we find as in the proof of [1.17] that in the chart U oroa1/!(O, 0) is the vector [x, x]le. On the other hand, we know that 1/!( 17, T) equals exp( (17 + T)X), which develops as e + (t7 + T)X e. We conclude that [x, x]e = O. But the value of the commutator of two left-invariant vector fields is determined by its value at the identity, hence [x, x] = o. IQEDI

3.18 Remark. If in [3.17] we do not have [x, x] ------+

= 0, then the factthat T2 = 0 immediately

gives that M(TX)(X) = xe - ~T[X, x]le, which allows one to compute T¢(orl(r,g)) with ¢( T, g) = 9 exp( TX) for non integrable smooth odd left-invariant vector fields on G.

4.

SUBGROUPS AND SUBALGEBRAS

In this section we define an A-Lie subgroup to be an injective A-Lie group morphism. We then show that A-Lie subgroups of an A-Lie group G correspond to subalgebras of 9 and that normal A-Lie subgroups of G correspond to ideals of g.

4.1 Definition. Let G be an A-Lie group. An A-Lie subgroup of G is a pair (i, H) such that H is an A-Lie group and such that i : H --4 G is an injective A-Lie group homomorphism. If no confusion is possible we will denote an A-Lie subgroup just by H.

§4. Subgroups and sub algebras

4.2 Lemma. Ifi: H immersion.

--4

293

G is an A-Lie subgroup ofG, then the map i is an injective

Proof We know that i is injective, and we want to prove that Ti is injective. According to [1.18] it suffices to prove that Tei is injective. So suppose Tei(x) = Tei(y) for some x -I y E f). We may assume, multiplying them by a small real number if necessary, that both belong to the open neighborhood of 0 E f) on which the exponential is bijective [2.7]. We thus have exp(x) -I exp(y). But by [2.16] we have i(exp(x)) = exp(Tei(x)) = exp(Tei(y)) = i(exp(y)), contradicting injectivity of i. IQEDI

4.3 Definition. Let 9 be an A-Lie algebra. A subalgebra of 9 is a graded subspace (in the sense of A-vector spaces [11.6.23]) f) c 9 that is invariant under the bracket, i.e., 'Vx, y E f) : [x, yJ E f). It follows that f) itself, equipped with the bracket inherited from g, is an A-Lie algebra.

4.4 Proposition. Let G be an A-Lie group, and let i : H --4 G be an A-Lie subgroup. Then fj = Tei(f)) is a subalgebra of 9 isomorphic to f). Moreover, for all 9 E i(H) : Ada(g)(f)) c fj. Proof Since e E H has real coordinates, Tei is smooth; since Ti is even, Tei(f)) is a graded subspace of 9 in the sense of A-vector spaces by [11.6.24]. Since Tei is injective [4.2] and an A-Lie algebra morphism [2.11], the first result follows. For the second part, we compute for h E H and x E f), using that i is a homomorphism, Ti AdH (h)x = Ti(TLh(TRh-1 (x))) = TLi(h) (TRi(h)-' (Ti(x))) = Ada(i(h))Ti(x). IQEDI

4.5 Definition. Let G be an A-Lie group. Two A-Lie subgroups i : H --4 G and j : K --4 G of G are said to be equivalent if there exists an A-Lie group isomorphism p : K --4 H such that for all k E K: j(k) = i(p(k)) (and thus in particular i(H) = j(K».

4.6 Corollary. Let i : H --4 G and j : K --4 G be two equivalent A-Lie subgroups ofan A-Lie group G with A-Lie algebras f) and e respectively. Then f) and e are isomorphic and Tei(f)) = Tej(e).

4.7 Theorem. Let G be an A-Lie group with A-Lie algebra g. For each subalgebra fj of 9 there exists a unique (up to equivalence) connected A-Lie subgroup (i, H) of G with A-Lie algebra f) such that Tei(f)) = fj. Proof If fj is a sub algebra of total dimension k, then in particular there exists a basis (vi)f=l of 9 such that (Vi)~=l is a basis of lj. We denote by :F the subbundle of total dimension k of TG generated by the vector fields (Vi)~=l (see also [IV.3.14]). (In terms of the trivialization [1.18] of TG this subbundle is given as fj x G.) Since fj is a subalgebra,

294

Chapter VI. A-Lie groups

the commutator [Vi, Vj], 1 ::::: i, j ::::: k belongs to fj, in other words, it is a linear combination of the vp, 1 ::::: C ::::: k. From this and [V.l. 19] it follows that F is an involutive subbundle of TG of total dimension k. Using [V.6.9], we define (i, H) as the (unique) leaf passing through e E G. We will denote by eo E H the (unique) element such that i( eo) = e. The proof now proceeds in several steps. The first step is to prove that the abstract subgroup Ho of G generated by exp(fjo) is contained in i(H). The next step is to prove that Ho is the whole of i(H). This proves that i(H) is an abstract group. The third step is to transport this group structure to H such that (i, H) becomes an A-Lie subgroup with A-Lie algebra £) and Tei(£)) = fj. The last step is to prove uniqueness of this A-Lie subgroup up to equivalence. • We start with the first step. We define the subset Ho c G as the set of all finite products of elements of the form exp(x) with x E fjo. Since exp(x)-l = exp( -x), this is an abstract subgroup of G. Next we consider the map ¢ : H x fjo ---) G defined by ¢( h, x) = i( h) . exp( x). In [4.8] we prove that it is tangent to:F. But Hand fjo are connected, and¢(eo,O) = i(e o) E i(H). By [Y.6.9] weconcludethat¢(Hxfjo) C i(H). We now start induction with h = eo E H to prove that Ho C i(H). • The second step is more or less a topological argument to prove that Ho = i(H). In [4.9] we prove that = i-l(Ho) CHis open and closed in H. Now H is connected = H and thus i(H) = i(Ho) = Ho n i(H) = Ho as wanted. and hence • Once we know that i( H) = Ho, life becomes easy. Consider the map ¢ : H x H ---) G defined by ¢(g, h) = i(g)· i(h). Since i(H) = Ho is an abstract subgroup ofG, we have ¢(H x H) C i(H), and thus by [V.6.8] there exists a smooth map mH : H x H ---) H such that ¢ = i 0 mH. It is immediate that this mH makes H into an A-Lie group, i into an injective A-Lie group homomorphism, and that Tei(£)) = fj. • The fourth step is to prove uniqueness up to equivalence. Let (jl, Kd and (j2, K 2) be two connected A-Lie subgroups of G with A-Lie algebras tl and t2 respectively such that Teji(t i ) = fj. In [4.10] we prove that both must be equivalent to the A-Lie subgroup (i, H) constructed above. Hence (jl, K 1) and (j2, K 2) are equivalent. IQEDI

it

it

4.8 Technical lemma. The map ¢: H x (£))0 ---) G defined by ¢(h, x) = i(h) . exp(x) is tangent to :F. Proof We have to show that for any (A, B) E ThH x Tx(£))o ~ ThH x fj we have T(h,x)¢(A, B) E Flq,(h,x)' By definition of H, Thi(A) belongs to Fli(h), and thus there exists y E fj such that Thi(A) = TLi(h)(Ye). We now compute

T(h,x)¢(A, B) = Tm( (Thi(A), Q.,xp(x)) + (Qi(h)' Tx exp(B))) = TRexp(x) (TLi(h) (Ye)) + TLi(h)(Tx exp(B)) = TLh.exp(x) (Ad(exp( -x)(y)) + TLh.exp(x)(M(x)(B)) , where we used (3.14). Since x, y, and B belong to fj which is a subalgebra, the elements ad(x)(y) and ad(x)(B) belong to fj. But now Ad(exp( -x)) = e- ad(x) and M(x) are convergent power series in - ad(x) [3.15], hence by [3.9-i] Ad(exp( -x))(y) and

295

§4. Subgroups and subalgebras

M(x)(B) belong to fj. Substituting this back in the formula for T(h,x)¢(A, B), we see that it belongs to :F. IQEDI 4.9 Technical lemma. The subset ita

Proof Let 0 E U C go be an

= i-I(Ho) cHis open and closed in H. open subset such that exp : U -+ V = exp(U)

eGis a diffeomorphism. By [4.8] the map exp : fj n U -+ V is integral for :F and thus there exists a smooth map exp : fj n U -+ H such that i 0 exp = exp : fj n U -+ G [Y.6.9]. Since T exp is injective on T(fj n U), T exp must be too. Since the graded dimensions of fj n U and H are the same, it follows that exp is a diffeomorphism [Y.2.14]. We denote by Uo C H the open set Uo = exp(fj n U); by construction Uo C ita. Now let ho E BH, for which we define ¢h o : H -+ G by ¢ho(h) = i(ho)-I . i(h). This is a smooth map because ho has real coordinates and because i and m are smooth. Moreover, ¢h o is integral for :F because :F is generated by left-invariant vector fields. Since ¢ho(h o) = i(e o) it follows that there exists a smooth map ¢h o : H -+ H such that ¢h o = i 0 ¢h o [Y.6.9]. Denoting h~ = ¢ho(eo) E BH, we find i(h~) = i(ho)-I (and remember, we do not yet know that Hhas a group structure). Since obviously ¢h o and ¢h~ are each others inverses, we find an open neighborhood Uho

= ¢h~(Uo) of any ho

E BH.

Since B(exp(xI)··· exp(xe)) = exp(BxI)··· exp(Bxe)), it follows that if hI E flo, then also ho = BhI E ilo. We thus have an open neighborhood Uho = ¢h~ (Uo) of ho satisfying

i(UhJ = i(¢h~(Uo)) = i(ho) . i(Uo) . Hence Uho C Ho and thus Ho is open in H, just because any neighborhood ofho is also a neighborhood of hI. On the other hand, suppose hI E H \ Ho, then ho E H \ Ho because Ho is open. But then Uho C H \ Ho. For suppose 9 E Uho n Ho = ¢h~ (Uo) n Ho, then i(g) = i(h o) . i(h) E Ho for some hE Uo. But then i(ho) = i(g) . i(h)-I E Ho, contradicting the hypothesis. It follows that Ho is also closed in H. IQEDI

4.10 Technical lemma. Let (j, K) be a connected A-Lie subgroup of G with A-Lie algebra £ such that Tej(£) = fj. Then (j, K) and (i, H) are equivalent A-Lie subgroups.

Proof Since Tei and Tej are smooth and injective, the equality Tei(£)) = fj = Tej(£) implies the existence of a smooth isomorphism ¢ : £ -+ £) such that Tej = Tei 0 ¢. By [2.16] we obtain for x E £0 : j(exPK(x)) = eXPa(Tej(x)) = eXPa(Tei(¢(x))) = i(exPH(¢(x))) . Since ¢ is an isomorphism, it follows thatj(K) = i(H) (use [2.7]). By [V.6.8] there thus exists a smooth map] : K -+ H such that j = i 0]' Since Tj is injective, the same holds for T], but then the fact that Hand K both have total dimension k implies that] is a diffeomorphism onto its image [V.2.14]. From j(K) = i(H) we deduce that this image equals H and thus] : K -+ H is a diffeomorphism. It is elementary to show that it is an A-Lie group homomorphism, proving that both A-Lie subgroups are equivalent. IQEDI

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Chapter VI. A-Lie groups

4.11 Lemma. Let G be an A-Lie group, (i, H) an A-Lie subgroup of G, and f a smooth map. If f(M) C i(H), then i-I 0 f : M -4 H is smooth.

:M

-4

G

Proof Using the trivialization of [1.18], the map Ti : T H -4 TG is described as Ti(h, x) = (i(h), Tei(x)) E G x Tei(£)). Hence Ti(TH) is contained in the involutive subbundle :F as defined in the proof of [4.7]. It follows that (i, H) is an integral manifold

IQEDI

of:F. The conclusion now follows from [Y.6.8].

4.12 Definitions. Let G be an A-Lie group and (i, H) an A-Lie subgroup. H is called a normal A-Lie subgroup if Vg E G

Vh E H

:3h E H : g. i(h) . g-1

= i(h) .

If 9 is an A-Lie algebra and £) a sub algebra, £) is said to be an ideal in 9 if Vx E 9

Vy E £) :

[x, yj E£).

4.13 Lemma. Let G be an A-Lie group, 9 its A-Lie algebra, (i, H) an A-Lie subgroup with A-Lie algebra £), and fj = Tei(£)). (i) If H is a normal A-Lie subgroup of G, then ~ is an ideal in 9 and Vg E G : Ad(g)(£)) C fj.

(ii) If G and H are connected and

if

fj is an ideal in g, then H is a normal A-Lie

subgroup of G. Proof To prove (i), consider the family of homomorphism

<1> : G x H -4 G from H to G defined by <1>(g, h) = g. i(h) . g-l. Since H is normal, <1>(g, h) lies in i(H) and thus there exists a smooth map 1> : G x H -4 H such that <1> = i 0 1> [4.11]. Uniqueness of 1> proves that it is a family of homomorphisms from H to H. Applying [2.16] we find eXPH(T'1>(g,y)) = 1>(g,exPH(y)), valid for all y E £)0. Composing with i, using [2.16] and [2.19], gives us

Since T ei(T'1>(g,y)) and Ad(g)Tei(y) are smooth in (g,y), it follows that ifyissufficiently close to 0 E £)0, both arguments of expo lie in the neighborhood on which expo is bijective. We deduce (by linearity) that Ad(g)Tei(y) = T ei(T'1>(g, y)) for all y E £)0, i.e., Ad(g) (y) E fj for all y E fjo. In order to prove that this is also true for all y E fj, we denote by 7r the canonical projection 7r : fj C 9 -4 g/fj and we define the smooth map Ad' : G -4 HomR(fj, g/fj) by Ad' (g) = 7r 0 Ad(g). We thus have shown that Vg E G: Ad/(g)I~.o = O. By [111.2.31] we conclude that Vg E G: Ad'(g) = 0, i.e., Ad(g)(£)) C fj as wanted. To finish the proof of (i), consider the smooth map ¢ : G x fj

-4

fj C 9 defined

by¢(g,y) = Ad(g)y. ComputingthederivativeT¢(xe,Qy) for x E g, Y E fj, we find ad( x)y. Since ¢ takes values in fj, this belongs to fj C g, i.e., fj is an ideal of g.

297

§4. Subgroups and sub algebras

To prove (ii), let X E go and YE£)o be arbitrary. Using [2.16], [2.19], and [2.15] we find exp(x). i(exp(y))· exp(-x)

= exp(Ad(exp(x))Tei(y)) = exp(ead(x) Tei(y))

.

Since Tei(£)) is an ideal, ad(x)Tei(y) = [x, Tei(y)] E Tei(£)). From this and [3.9-i] we deduce that ead(x) Tei(y) belongs to Tei(£)), say ead(x) Tei(y) = Tei(fj). It follows that exp( x) . i( exp(y)) . exp( -x)

= exp(Tei(fj)) = i (exp(fj)) .

Since elements of the form exp(x) generate G and elements of the form exp(y) generate H [2.7], it follows immediately that H is a normal A-Lie subgroup of G. IQEDI

A6

4.14 Example. Consider the set G = { (x, y,~, 7]) E x Ai I B(x 2 + y2) =I- o} with its obvious structure of anA-manifold of dimension 212. We endow G with the structure of an A-Lie group by giving its multiplication law:

(Xl, yl,

e,

(x 2, y2, e, 7]2) = (x 1X2 - y1y2 + ee 7]1) .

x 1y2 + y 1x 2 + e7]2 + 7] l e , x 1e - yl7]2 + ex2 - 7]ly2 , Xl 7]2 + y 1e + ey2 + 7] l X2) . 7]17]2,

Attentive readers will recognize this A-Lie group as A C *, the multiplicative group of invertible elements (x +~) + i(y + 7]) E A EB iA. An elementary calculation shows that a basis for the left-invariant vector fields is given by

Vx

= xOx + YOy +

vE = -~ox

~oE

+ 7]0.,., - 7]Oy + xOE + yo.,.,

Vy

= -yox + xOy - 7]0E + ~o.,.,

v.,.,

=

7]Ox - ~Oy - YOE

+ xo.,.,

.

Another elementary calculation shows that the only non-zero commutators among these basis vectors are

Of course these commutators could also have calculated by means of [1.17]. For instance, the structure constants eE.,., are given by 0El 0.,.,2 m+O.,.,l 0E2 m = (0, -1,0,0) +(0, -1,0,0), giving as above [vE' v.,.,] = -2vy. Yet another elementary calculation, but this time a rather tedious one, shows that the exponential map exp : go --4 G is given by exp( a, b, a, (3)

= (e a cos(b), ea sin(b), e a cos(b )a-ea sin(b )(3, e a cos(b )(3+ea sin(b)a)

,

where (a, b, a, (3) denote the coordinates in go of dimension 212 with respect to the basis v x , vy, vE, v.,.,. Given this formula, the reader can ascertain that the expression [2A-i]

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Chapter VI. A-Lie groups

verifies the conditions of the flow of ZL' If we identify G with the multiplicative group of A EB iA and flo with A EB iA, this exponential gets the much nicer form

Looking at the commutators (4.15) we see that the graded subspace Qgenerated by the two vectors Wo = Vy and WI = vE + Vry is a subalgebra of dimension 111. In order to find the A-Lie subgroup (i, H) that corresponds to this sub algebra, we compute the exponentials exp(x) with x belonging to Qo. Our formula gives us

exp(a·

Wo

+ a· WI) =

e ia ·(1

+ (1 + i) . a)

.

The product of two of such elements is given by

(e ia .(1

+ (1 + i) . a)) . (e ib ·(1 + (1 + i) . (3))

= ei (a+b+2aj3) ·(1

From this we "deduce" that H is the A-manifold (GS I )

X

+ (1 + i) . (a + (3))

.

Al with multiplication

and with embedding i : H --4 G given by i(e ia , a) = eia (l + (1 + i)a). It is elementary to verify that this is indeed an embedding and that i( H) eGis the submanifold defined by the equations x 2 + y2 = 1 and y. (~ + 'f}) + x· (~- 'f}) = O. The most general subalgebra Q C fI of dimension 111 is generated by the vectors 2 2 Wo = aVE + bvry and WI = (a - b )v x + 2abvy, (a, b) E R2 \ {(O, On. Except for the cases a = ±b, all the corresponding A-Lie subgroups are isomorphic to A as A-manifolds. The case a = -b is the complex conjugate of the example treated above; the corresponding A-Lie subgroup thus is the complex conjugate of H, i.e., as A-Lie group it is H, but with embedding (a, a) ~ e- ia (l + (1 - i)a).

5.

HOMOGENEOUS A-MANIFOLDS

In this section we are concerned with actions of A-Lie groups on A-manifolds. We therefore introduce the notion of fundamental vector field on an A-manifold associated to an element of the A-Lie algebra. We also introduce the notion of a proper A-Lie subgroup which replaces the notion ofclosed subgroup in the non-graded situation. With these definitions we prove that ifH is a proper A-Lie subgroup ofG, then G/H is an A-manifold. Moreover, if mo EM has real coordinates and if H is the isotropy group at m o , then H is a proper A-Lie subgroup and the map G/ H --4 M is an injective immersion.

299

§S. Homogeneous A-manifolds

5.1 Definition. Let 1> : G x M -4 M be a smooth left action of an A-Lie group G on an A-manifold M. For each x E 9 we define a vector field x M on M by the formula

where T1>m denotes the generalized tangent map [V.3.19]. This vector field is called the fundamental vector field on M associated to x E g. If x has real coordinates, i.e., x E Bg, the vector field x M is smooth (because T1> is smooth and e has real coordinates). A particular case of this definition is when G acts on itselfby multiplication, i.e., 1> = mc : G x G -4 G. Comparing definitions shows that xC is the right-invariant vector field on G whose value at e EGis -x (note the minus sign!). Comparing this with [1.10] shows that xC is exactly the vector field T Inv 0 0 Inv.

x

5.2 Lemma. The notion offundamental vector field enjoys the following properties. (i) T1> gx M lm = (Ad(g)x)MI(g,m) ({=} T1>g 0 x M = (Ad(g)x)M 0 1>g ). Cii) Vx E 9 : T1> 0 (xc x Q) = x M 0 1>, or roughly in words, the right-invariant vector field xC on G and the fundamental vector field x M on M are related by 1>. (iii) For x, y E Bg we have [xM, yM] = [x, y]M. (iv) The flow ¢ of the even vector field ZM on go x M defined by the formula ZM(X, m) = (Qx, xMlm) is given by ¢(t, x, m) = (x, 1>(exp(tx), m)); it is defined on the whole of Ao x go x M.

Proof For (i) we use that 1> is a left action of G on M, i.e., 1>g o1>m = 1>m 0 Lg and 1>m 0 Rg = 1>(g,m)' This allows us to make the following computation: T1> 9 (T1>mxe) = T1>m(TLgxe) = T1>m(TRg (Ad(g)x)) = T1>(g,m) (Ad(g)x). For (ii) the computations are similar. Property (iii) is a direct consequence of (ii), [1.10], and [Y.2.29]. For (iv) we note that, by (ii), the vector field ZM on go x M is related to the vector field Zc on go x G x M defined as Zc(x,g,m) = (~,xClg,Qm) by the map id(go) x 1>. Since the flow of Zc is given by (x, g, m) ~ (x, exp( -tx)g, m) [2A-v], the result follows from IQEDI [Y.S.S], taking g = e. 5.3 Corollary. For x E Bgo the flow ofx M on M is given by (t,m) ~ 1>(exp(tx),m).

5.4 Discussion. If 1> : M x G vector field becomes

-4

M is a right action, the definition of a fundamental

without the minus sign. For the natural right action m : G x G -4 G of G on itself by multiplication, the fundamental vector field xC is exactly the left-invariant vector field X. With this notion of fundamental vector field, the results [S.2-ii] and [S.2-iii] remain unchanged; [S.2-i] changes to T1> gx M lm = (Ad(g-l)x)MI(m,g) and [S.2-iv] changes to ¢(t, x, m) = (x, 1>(m, exp(tx))).

Chapter VI. A-Lie groups

300

The reason to introduce the minus sign for fundamental vector field associated to left actions is twofold. In the first place, it makes that [5.2-iii] comes out without a minus sign, making the map x ~ x M from the A-Lie algebra 9 to the set of vector fields on M a morphism of A-Lie algebras. In the second place, if 1> : G x M --t M is a left action of G on M, the map III : M x G --t M defined by III (m, g) = 1>(g-l, m) is a right action of G on M. The use of the minus sign for fundamental vector fields associated to left actions (and its absence for right actions) makes that the two fundamental vector fields corresponding to these left and right actions are the same (this follows from [1.10]).

5.5 DefinitionIDiscussion. Let G be an A-Lie group and (i, H) an A-Lie subgroup. H will be called a proper A-Lie subgroup if i : H --t G is an embedding, i.e., if i( H) is a submanifold of G. According to [V.2ol2], an A-Lie subgroup (i, H) is proper if and only ifi : BH --t BG is an embedding ofR-manifolds. Moreover, for R-Lie groups an R-Lie subgroup is embedded if and only if its image is closed [Wa, Thm 3.21]. We conclude that (i, H) is a proper A-Lie subgroup of G if and only if i(BH) is closed in BG. On the other hand, for R-Lie groups any closed abstract subgroup is a Lie subgroup [Wa, Thm 3.42]. A similar statement does not hold for A-Lie groups because the topology ignores the odd coordinates completely. [5.8] gives a characterization of proper A-Lie subgroups in the context of A-Lie groups.

5.6 Example. Let G = (A1)2 be the abelian additive group of dimension 012. The A-Lie subgroup H defined as {(~, 0) I ~ E Ad is a proper A-Lie subgroup, but it is neither closed nor open in G.

5.7 Discussion. In the sequel of this section we will often introduce a variant of the exponential function associated to a graded subspace of the A-Lie algebra. The idea is the following. Let G be an A-Lie group, 9 its A-Lie algebra, and let £) C 9 be a graded subspace. Since £) is a graded subspace, there exists a supplement 5 C 9 for £) (see [11.6.23]). Using the decomposition 9 = 5 EB £) we define the map exp : 90 --t G by exp( s, z) = exp( s) exp( z) for s E 50, Z E £)0. By [2.4-iv] T(o,o) exp = id and hence exp is a diffeomorphism from a neighborhood of (0, 0) E 50 EB £)0 = 90 to a neighborhood of e E G. Each time we need this variant of the exponential map, we will give the definition adapted to the circumstances, but we will no longerjustify the existence ofthe supplement, nor will we justify the fact that it is a diffeomorphism in a neighborhood of (0, 0). And obviously we will never say explicitly that exp Iso = exp Iso or that exp I~o = exp I~o

5.8 Lemma. Let G be an A-Lie group. (i) Let H C G be an abstract subgroup of G, let £) C 9 be a graded subspace,

let 5 be a supplement to £), and let exp be as in [5.7J. Suppose there exists an open neighborhood U of 0 E 90 such that exp : U --t V = exp(U) is a diffeomorphism and such that exp(U n £)0) = V n H. If in addition BH C H,

§S. Homogeneous A-manifolds

301

then H is a sub manifold of G and the canonical embedding i : H -4 G turns (i, H) into a proper A-Lie subgroup ofG with A-Lie algebra (isomorphic to) £). (ii) Let (i, H) be a proper A-Lie subgroup with A-Lie algebra £), let 5 be a supplement to 6 = Tei(£)) c g, and let the map exp : 60 EB 50 -4 G be defined as in [5.7J: exp(s, z) = exp(s) exp(z). Then there exists an open neighborhood U of 0 E go such that exp : U -4 V = exp(U) is a diffeomorphism and such that exp(U n 60)) = V n i(H). Proof • Let hI E H be arbitrary, and denote ho = Bhl E H. Since ho has real coordinates, Lh o is a diffeomorphism, and thus W = Lh o (V) is an open neighborhood of ho = Bh 1, and thus of hI. We claim that ¢ : W -4 U C go defined by ¢ = exp-l 0 L-;:1 is a chart satisfying ¢(WnH) = Un£)o. Suppose x E ¢(WnH),thenexp(x) = h;;l.'h for some hEW n H. But then exp(x) E V n H = exp(U n £)0) and thus x E £)0. This proves ¢(W n H) c Un £)0; the other inclusion being obvious, it follows from [III.S.1] that H is a submanifold of G. Hence the restriction of the multiplication ma to H is smooth, making H into an A-Lie group . • Since i : H -4 G is an embedding, i(H) is a submanifold of G. In particular there exists a chart ¢ : V' -4 We go around e E G (G is modeled on g) and a graded subspace F of g such that ¢(V' n i(H)) = W n Fo [111.5.1]. According to [Y.2.16] the graded subspace F must be isomorphic to £). By taking a smaller V' and W if necessary we may assume that there exists an open neighborhood U ' of 0 E go such that exp : U ' -4 V'is a diffeomorphism. Since i is a homomorphism, we deduce that exp(U' n 60) c v' n i(H) [2.16], and thus (¢ 0 exp)(U' n 60) c W n Fo. Since ¢ 0 exp is a smooth injective map and since F and 6 are isomorphic, it follows from [V.2.14] that ¢(exp(U' n 60)) is open in W n Fo. Since W n Fo has the induced topology from Wand since ¢ is a diffeomorphism, there exists an open V C V' such that ¢(exp(U' n 60)) = ¢(V) n Fo = ¢(V n i(H)). Taking U = exp-l(V) C U' we find ¢(exp(U' n 60)) = ¢(exp(U) n i(H)) and in particular U' n 60 c U, i.e., U' n 60 = Un 60. Hence exp(U n 60) = V n i(H). IQEDI

5.9 Theorem. Let G be an A-Lie group and (i, H) a proper A-Lie subgroup. Then: (i) the coset space G/H admits the structure ofan A-manifold modeled on that the canonical left action : G x G / H -4 G / H is smooth;

g/6 such

7r: G -4 0/ H with the natural right action of H on G becomes a principal fiber bundle with structure group H; (iii) ifH is also normal, then G/H is an A-Lie group with A-Lie algebra g/6 and 7r : G -4 G / H is a morphism of A-Lie groups.

(ii)

Proof • As a topological space we equip G/H with the quotient topology. By surjectivity of 7r, any subset 0 of G/H is of the form 7r(V) for some subset V of G. According to the quotient topology, a set 7r(V) is open if and only if 7r- 1 (7r(V)) = V . H is open in G. Since right translations are homeomorphisms [1.3], V . H = UhEH Rh(V) is open whenever V is open in G, i.e., we have proven that the canonical projection 7r : G -4 G / H is an open map.

Chapter VI. A-Lie groups

302

Let S be a supplement to 6 and let exp : So EB 60 -4 G, exp(x, y) = exp(x) exp(y) be as in [5.7]. Now let U and V be as in [5.8-ii] and denote by 7rs : 9 -4 S the associated projection. Our first goal is to find an open neighborhood 0 E Ua C So such that the map X : Ua x H -4 G, X(s, h) = exp(s) . i(h) is a diffeomorphism onto its (open) image. Therefore we consider the smooth map ms : So x So -4 G defined by ms(s, z) = exp( -s) exp(z). Let Wbe an open neighborhood of(O, 0) E So XSo such that ms(W) C V, and define w : W -4 So by w = 7rs 0 exp-l 0 ms. Since w(s = 0, z) = z, (ow / oy)( s = 0, z = 0) = id and hence by the implicit function theorem [111.3.27] there exists a (local) function / : So -4 So such that w(s, z) = 0 = w(O,O) is equivalent to z = /(s). But since we know that z = s is such a local function, we conclude that there exists an open neighborhood Ua of 0 E So with Ua x Ua C W such that 'Vs, z E Ua C So : w(s, z) = 0 { = } s = z. Since ms(W) C V, exp-l(ms(W)) C U. Hence on Ua x Ua the equationw(s, z) = 0 is equivalentto exp-l(ms(s, z)) E Un 60' By definition of exp this is equivalentto exp( -s) exp(z) E exp(U n 60) = V n i(H). Hence we have proven that (5.10)

'Vs, s'

E

Ua

C So, 'Vh,

h'

E

H : exp(s) . i(h) = exp(s') . i(h')

{=}

s = s' .

By taking a smaller Ua if necessary, we may assume that Ua C Un So. It is this Ua that we will use. In order to prove that X is a diffeomorphism from Ua x H onto its (open) image, we will use [V.2.14]. We first compute T(s,x) exp for (s, x) E U C So EB 60 :

T(s,x) exp(y, z) = TRexpxTs exp y + TLexpsTx exp z = TRexpx(Ts expy + TLexps Ad(expx)M(x, z)) . Since (i, H) is a normal A-Lie subgroup and since x E 60' z E 6, it follows from [4.13] and [3.15] that Ad(expx)M(x, z) remains in 6. Bijectivity of T(s,x) exp then shows that (Ts exp)(s) is a supplement for (TLexp(s))(bJ. We next compute TCs,h)X for (s,h) E Ua x H:

Since Ad(i(h))Tei(TLh-1Z) is in 6, we conclude from the bijectivity ofTCs,x) exp that T(s,h)X is bijective. Since we have proven injectivity of X in (5.10), we conclude by [V.2.14] that X is a diffeomorphism from Ua x H onto its open image. To finish our preparations, we define for an arbitrary ga E BG the smooth map Xg o : Ua x H -4 G by Xg o = Lgo 0 X (and thus X == Xe). Since for such ga the map Lgo is a diffeomorphism of G, it follows that Xg o is a diffeomorphism from Ua x H onto Vgo = Xgo(Ua x H), which is open in G. We have now all ingredients needed to start the construction ofthe A-manifold structure on G/H. Since 7r is an open map, Ugo = 7r(Vgo) C G / H is open. Moreover, the map Xgo : Ua -4 Ugo defined as Xgo(x) = 7r(Xgo(x,e)) is a homeomorphism. Bijectivity is immediate, U' C Ua is open if and only if U' x H is open in Ua x H, and (to finish) Xgo(U' x H) = 7r- 1 (XgJU')). We conclude that the map CPgo = X;} : Ugo -4 Ua is a

303

§S. Homogeneous A-manifolds

chart for G/H. Moreover, denoting by verifies immediately that

7r1 :

Ua x H

-4

Ua the canonical projection, one

-1

X90

Ua x H

+---~

X90

l~

~11

(5.11)

Ua

Vao

X90

~

+----

Ugo

<1>90 is a commutative diagram, proving that 7r is a smooth map on these charts. To compute a change of charts, let (Ugo ' cPgJ and (Ugl> CP91) be two charts. Since CPgo is a homeomorphism, there exists an open Ug091 C Ugo such that cPgJUgogJ = Ugo n Ug1 . Since (CP91 a cP ;01)(X) = (7r1 0 X;/ 0 XgJ (X, e) is clearly a smooth map, we find that the charts (Ugo ' CPgo) and (U91 , CP91) are compatible. Since they cover G/H we conclude that G/H is a proto A-manifold. Since Ua is an open subset of .50, which is isomorphic to g/fj as an A-vector space, this proto A-manifold is modeled on gjfj. Remains to prove that B ( G / H) is a second countable Hausdorff space. Since all maps are smooth, they commute with the body map, hence B(G/H) = (BG)/(BH). Since G and H are A-manifolds, BG and BH are second countable Hausdorff spaces. By definition of the quotient topology (BG/BH) is also second countable. Consider 1 BG x BG with the subset R = (BH) where in : Be x Be -4 Be is the smooth map (g, h) f--+ g-l . h. Since BH is closed in BG [5.5], R is closed in G x G. Moreover, by definition of cosets, 7r(g) -I 7r( h) if and only if (g, h) rJ. R. Hence if 7r(g) and 7r( h) are distinct points in BG/BH, there exist open sets U, V in BG such that (g, h) E U x V c (e x e) \ R. It follows that 7r(U) n 7r(V) = ¢. Since 7r is an open map, it follows that BG/BH is Hausdorff. The last item of (i) that remains to be proven is that the canonical left action of G on G/H is smooth. Set theoretically this action 1> : e x e / H -4 e / H is defined as 1>(g, m) = 7r(g. 7r- 1 (m)). It follows that on the local chart Ugo we can write 1>(g,m) = 7r(g. Xgo(cpgo(m),e)). Since the right hand side is composed of smooth functions, we deduce that 1> is smooth . • To prove (ii), we first note that 7r- 1 (UgJ = Vgo by definition of Xg o. We then define the map 1/Jgo : 7r- 1 (UgJ = Vgo -4 Ugo x H by the equation 1/Jgo (g) = (m, h) {=} 9 = Xgo(cpgo(m), h). It follows from (5.11) that 1/Jgo is a trivializing chart for 7r. Moreover, an elementary computation shows that it is compatible with the right actions of H on G and Ugo x H, i.e., 1/Jgo(g . h) = 1/JgJg) . h, with (m, h') . h = (m, h' . h). Finally, for two trivializing charts one easily finds, using (5.11), that we have the equality (1/J91 0 1/J;;,1 )(m, h) = (m, ;j91go (m) . h), where the smooth map ;j91go : Ug1 n Ugo -4 H

m-

is defined by the equation (1/J91 0 1/J;;,1 ) (m, e) = (m, ;J91go (m)). We conclude that the map 7r : -4 H is a principal fiber bundle with structure group H acting in the natural way on the right on G. • To prove that G/H is an A-Lie group it suffices to show that the multiplication is smooth. To that end, let Si : Ui -4 e, i = 1,2 be two local smooth sections of the

e

e/

304

Chapter VI. A-Lie groups

principal bundle 7r : G -4 G / H. The definition of the group structure on G/H shows that for Xi E Ui the multiplication is defined as

It follows immediately that the multiplication mal H is smooth on UI x U2 . Since the domains of such local sections cover G/H, the multiplication is globally smooth. To prove that the A-Lie algebra of G/H is isomorphic to g/rj, we first note that 7r is a morphism of A-Lie groups and hence that T e 7r is a morphism of A-Lie algebras. Since 7r: G -4 G/ His a fiber bundle, Te 7ris surjective. Since7r(expa(x)) = eXPoIH(Te7r(x)), it follows that rj c ker Te 7r. A simple dimension argument then shows that rj = ker Te 7r. It then follows from [11.3.11] and [11.6.24] that the A-Lie algebra of G/H is isomorphic to g/rj as an A-vector space (which we already knew), but because T e 7r is a morphism of A-Lie algebras, this isomorphism is also an isomorphism of A-Lie algebras. IQEDI

5.12 Definition. Let : G x M -4 M be a smooth left action of an A-Lie group G on an A-manifold M, and let ma E M be arbitrary. Then H = {g E G I (g, ma) = ma} is called the isotropy subgroup at ma.

5.13 Proposition. Let : G x M -4 M be a smooth left action of an A-Lie group G on an A-manifold M, and let ma E BM be arbitrary. Then the isotropy subgroup Hat ma is a submanifold of G and the canonical embedding i : H -4 G is a proper A-Lie subgroup ofG with A-Lie algebra £) = {x E 9 I xMlm o = O}. Proof Note first that H is an abstract subgroup of G (because is a left action) and that £) = ker(Te1/'), where 1/' : G -4 M is defined as 1/'(g) = (g, ma)' Since 1/' and Te1/' are smooth it follows that £) is a graded subspace of g. We thus can choose a supplement 5 C 9 for £). The proof now proceeds in two steps. We first show that exp(£)o) C H, and then that there exists an open neighborhood U of 0 E 50 EB £)0 = go such that exp: U -4 V = exp(U), exp(s, x) = exp(s) exp(x) [5.7] is a diffeomorphism and such that exp(U n £)0) = V n H. Applying [5.S-i] finishes the proof. For the first step we define the vector fieldZMon £)0 x Mby ZM(X, m) = (Q", xMlm), i.e., ZM is the collection of all fundamental vector fields on M with x E £)0' We then consider the two smooth maps Ii: Ao x £)0 -4 £)0 x M defined by JI(t,x) = (x,ma) and Jz(t,x) = (x,1/'(exp(-tx))). Obviously TJIoat = 0 = ZM(x,ma) = ZMoJI, but also T Jz ° at = Z M ° Jz because fz is essentially the flow of Z M [5.2-iv]. Since we also have JI(O,x) = (x,m a) = Jz(O,x) we conclude by [V.5.2] that JI = Jz, i.e., 1/'(exp( -tx)) = ma. We thus have proven that exp(£)o) c H. To prove that there exists an open neighborhood U with the desired properties, we first choose any U such that exp : U -4 V = exp(U) is a diffeomorphism. We now focus our attention on the map III : 50 -4 M defined by llI(y) = 1/'(exp(y)). By definition of £) the map To III is injective. It then follows from [111.3.30] that there exists a neighborhood 0 E U ' C 50 such that IlIlul is injective. By taking a smaller U C 50 EB £)0

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305

we may suppose that III is injective on Un 50. We claim that such an U satisfies our conditions. Obviously exp(U n £)0) c exp(U) n H because exp(£)o) c H. So suppose h E exp(U) n H, i.e., h = exp(s) exp(x) E H for some s E 50 and x E £)0. But then exp(s) = h .exp( -x) E H and thus llI(s) = IlI(O). Injectivity of III on U n50 then proves s = O. And thus h = exp(x) E exp(U n £)0). IQEDI

5.14 Proposition. Let 1> : G x M be a smooth left action of an A-Lie group G on an A-manifold M, let mo E BM, and let H c G be the isotropy subgroup at mo. Then the canonical injection III : G / H --4 M defined by llI(gH) = 1>(g, mo) is an injective immersion, equivariant with respect to the G-actions. In particular if the action of G is transitive, then III : G / H --4 M is a diffeomorphism.

Proof First note that H is a proper A-Lie subgroup of G [5.13], and thus G/H is an A-manifold [5.9]. Since III is injective by definition of H, it remains to show that III is smooth and that Till is injective at all points. Let ¢go : Ugo --4 Uo C 50, 90 E BG be a local chart as in the proof of [5.9] with 5 a supplement to £) in g. In terms of this chart, we have (Ill 0 ¢~ol )(x) = 1>(go . exp(x), mo) = 1>(go, 1>( exp(x), mo)). This shows that III is smooth. Moreover, since £) = ker(T(e,m o )1», since 5 is a supplement to £), and since 1>(go, J is a diffeomorphism, it follows that Tx III is injective. Being equivariant with respect to the G-actions means that llI(g· (gH)) = 1>(g, llI(gH)), which is an immediate consequence of the definition of a left action. The last part of the statement follows from [V.2.14]. IQEDI

5.15 Corollary. The structure of G/H given in [5.9J is uniquely determined (up to diffeomorphisms) by the stated properties.

Proof Denote by M the set G/H equipped with some structure of an A-manifold such that the canonical left action 1> : G x M --4 I'll is smooth. Then by [5.14] we obtain a diffeomorphism between G/H with the structure given in [5.9] and M. IQEDI

5.16 Corollary. Let p : G --4 H be a morphism of A-Lie groups, then ker(p) is a proper A-Lie sub group ofG, and the induced morphism p : G / ker(p) --4 H is an A-Lie subgroup of H.

Proof If we consider the left action of G on H defined as 1> (g, h) = p(g) . h, then ker (p) is the isotropy subgroup at e E H. Hence it is a proper A-Lie subgroup by [5.13]. Since it is also a normal A-Lie subgroup, we conclude by [5.9] that G / ker(p) is an A-Lie group. To prove that the induced map p : G / ker(p) --4 H is an A-Lie subgroup, it suffices to prove that p is smooth, because it is an injective homomorphism by construction. On a neighborhood U C G/ker(p) with a smooth section s : U --4 G the map p is defined as P( x) = p( s( x)). Hence pis smooth on U. Since such U cover G / ker(p), the conclusion follows. IQEDI

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6.

PSEUDO EFFECTIVE ACTIONS

In this section we prove that every action can be transformed into a pseudo effective action: ifG acts on M, there exists a proper A-Lie subgroup G1/1o ofG such that G1/1o acts as the identity on M and such that the induced action of G I G1/1o on M is pseudo effective.

6.1 Discussion. In chapter IV we defined fiber bundles with a structure group and we required the action of the structure group G on the typical fiber F to be pseudo effective. In the remaining part of this section we will show that we can transform any smooth (left) action into a pseudo effective action. More precisely, we will show that if : G x M ---> M is a smooth left action of an A-Lie group G on an A-manifold M, then there exists a proper normal A-Lie subgroup G1/1o C G such that (i) all elements of G1/1O act as the identity on M, and (ii) the induced action of G IG1/1O on M is smooth and pseudo effective. Forgetting for the moment the smoothness conditions, the natural approach to obtain an effective action would be the following. One would first define GO c G as the set of all elements of G that act as the identity on M, i.e., g E GO if and only if "1m EM: (g, m) = m. Obviously GO is a normal abstract subgroup of G and GIGo acts effectively on M. When one tries to prove that GO is an A-Lie subgroup, it is natural to think that its A-Lie algebra gO consists of those x E g whose associated fundamental vector field x M is identically zero. The next logical step would be to apply [5.8] to really prove what one wants. The problem with this approach is that it is hard (if possible at all) to prove that gO is a graded subspace of g. The definition of gO is in terms of equations, but these equations depend upon even and odd parameters (the local coordinates of M), and we do not have much control over them. Our approach will be to define subsets G1/1O C GO and g1/1 o C gO described by smooth families in GO and gO respectively. Then our equations are "parameterized" by smooth functions over which we have complete control.

6.2 Definitions. • Let K be an A-manifold and X C K a subset. We will say that k E K is part of a smooth family in X if there exists an A-manifold N and a smooth map 1jJ : N -> K such that k E 1jJ(N) c X . • Let : G x M ---> M be a smooth (left) action of an A-Lie group G on an A-manifold M and let g be the A-Lie algebra of G. The subsets G1/1O C GO c G and g1/1 O C gO C g are defined as :

GO = {g

E

Gig acts as the identity on M }

G1/1o = {g

E

Gig is part of a smooth family in GO }

gO

I x M is identically zero } {x E g I x is part of a smooth family in gO} .

= {x

g1/1 o =

E g

6.3 Remark. With hindsight we now can say that the action of G on M is pseudo effective if and only if G1/1O reduces to the identity element of G.

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307

6.4 Discussion. If g EGis part of a smooth family 1/J : N ----; G in GO, then obviously each element g' E 1/J(N) is part of a (the same!) smooth family in GO. In other words, 1/J(N) C G1/1O. It follows that G1/1O is the union of images of smooth maps into GO. Similarly, if x Egis part of a smooth family 1/J : N ----; 9 in gO, then 1/J(N) C g1/1 O. Since is a left action, it is immediate that GO is a normal (abstract) subgroup of G. Moreover, since multiplication and inverse are smooth operations in G, G1/1O is also a normal abstract subgroup. On the other hand, it is not clear whether GO or G1/1O are A-Lie subgroups of G. Similarly, linearity of the tangent map proves that gO is a graded submodule of g. Continuity of the module operations then shows that g1/1 Oalso is a graded submodule. But again, it is not obvious whether they are subspaces, nor is it completely obvious that they are stable under the bracket operations.

6.5 Lemma. Let E be an A-vector space and let (PI, ... , rPk E B*E be afinite number of smooth homogeneous left linear maps. Then £,1> = {e EEl VI :::; i :::; k : (( e I rPi)) = 0 } is a graded subspace of E in the sense of [Il.6.23]. Proof Let G = {gl, ... , gk} be a set of k elements with parity map c : G ----; Z2 given by C(gi) = C(rPi), and let F = F(G,c) be the free A-module (i.e., an A-vector space) on these generators. With these we define the even (left) linear map : E ----; F as (( e I is smooth. Moreover, E

. The conclusion then follows from [11.6.24]. IQEDI

6.6 Lemma. g1/1 O is a subalgebra of g. Proof Let M be modeled on the A-vector space of dimension plq and let tp : W ----; 0 ,;q). Let el,"" ed be a be a local chart for M with coordinates (xI, ... , x P , homogeneous basis of 9 and let 1/J : N ----; 9 be a smooth family in gO. This implies in particular that 1/J(N) C g1/1 O c gO. On the local chart W c M, the fundamental vector fields ef'I can be written as ef'Ilm = L;!iE;(m)oj with smooth functions E; : W ----; A. Since 1/J is smooth, there exist smooth functions 1/Ji : N ----; A such that d . M 1/J(n) = Li=I1/J'(n)ei. In the local chart W, the fundamental vectorfield1/J(n) thus can be written as

e, ... ,

i,j

The fact that 1/J( n) belongs to gO is equivalent to the fact that for all j = 1, ... ,p + q we must have L i 1/J i (n)E;(m) = 0 as smooth functions of(m,n) E W x N. Using local coordinates (y, "I) on N and using the notational shorthand of [III.3.l7], these equations can be written as Li,I,J 1/J}(y )"11 Ef,J(x)e = O. Since we require A to satisfy (C [00]), we can apply [111.2.21] or [111.2.22] to conclude that these equations are equivalent to the system of equations with x and y taking real values.

Chapter VI. A-Lie groups

308

This suggests that we introduce the functions cP~(x) = Li ie · El,J(x) E B*g, where (ie) denotes the left dual basis. Since the parity of the fundamental vector field ef:4 is the parity of ei, it follows that the parity of the function El,J equals c( ei) + c( OJ) + c(e). Hence cP~ is a homogeneous function of parity c(Oj) + c(e). Moreover, if (ei) is another basis of 9 related to (ei) by ei = Lj a{ ej, then the left dual basis is given by

Li ie a{ = je. It follows that thefunctions cP~(x) are independent of the choice of a basis forgo Our analysis shows that if'!jJ(N) C g,pO, then (( '!jJ(n) I rjJ~(x)>> = 0 for all (j, J, x) (with of course the coordinates x only taking real values). On the other hand, if '!jJ : N --; 9 is a smooth map such that for all (j, J, x) we have (( '!jJ( n) I cP~( x))) = 0, then the same analysis shows thatthe fundamental vector field '!jJ( n) Mis identically zero on the chart W. It follows that g,pO is described by the system of equations (( z I cP~ (x))) = 0, where we also have to vary the chart W in order to cover the whole of M. The set of all cP~ (x) forms a subset of the d-dimensional vector space B *g over R. We thus can find k :::; d independent elements cP~~ (x 1), . . . , cP~: (x k ), wi th x r the (real) coordinates of a point mr E B Wr for local charts Wr of M, such that all other elements cP~(x) are linear combinations with real coefficients of these k elements. It follows that g,pO is described by the k equations (( z I (x r = 0, 1 :::; r :::; k. We conclude by [6.5] that g,pO is a graded subspace of g. To show that g,pO is a subalgebra, we argue as follows. Since g,pO is a subspace, it has a basis iI, ... , ik E Bg. Applying [5.2-iii] we conclude that [1;, fj]M E gO. We now consider the map 1jJ : g,pO x g,pO --; 9 defined by 1jJ(Zl' Z2) = [Zl' Z2]' Since the bracket is bilinear, 1jJ takes values in gO. Since the bracket is smooth, 1jJ is smooth. Hence 1jJ takes IQEDI values in g,pO, i.e., g,pO is a sub algebra.

cPt n

6.7 Proposition. G,po is a proper normal A-Lie subgroup ofG with A-Lie algebra g,po. Furthermore, the induced left action of G / G,po on M is smooth and pseudo effective.

Proof • Let us start with the hardest part: the proof that G,po is a proper A-Lie subgroup with A-Lie algebra g,p°,for which we will use [5.8]. We first introduce the vector fieldsZM and Zc on ggo x M and ggo x G x M respectively, defined as Z M(x, m) = (Q", x M 1m) and Zc(x, g, m) = (Qx, xClg,Qm) (see the proof of [5.2-iv]). These two vector fields are related by the map id(ggO) x If>. But the flow of Zc is given by the expression (t, x, g, m) r--+ (x, exp( -tx )g, m) and the flow of Z M by (t, x, m) r--+ (x, m) because for x E ggo we have x M == O. We conclude by [V.5.5] that exp(ggO) is contained in GO. Since exp is smooth, this means that exp(ggO) is contained in G,po. If g E G,po, there exists a smooth 1jJ : N --; G,po with g E 1jJ( N). Applying the body map we findBg E 1jJ(BN), i.e., Bg also belongs to G,po. In other words, BG,po c G,po. N ow let 5 be a supplement to g,po. Using the decomposition 9 = 5 ED g,pO, we define as usual the map exp : go --; Gbyexp(s,z) = exp(s)exp(z) fors E 50 and z E ggo [5.7]. To find a neighborhood V of e E G and a neighborhood U of 0 E go as required by [5.8], we proceed as follows. In the proof of [6.6] we found k independent elements cP7Jx r ),1 :::; r :::; k in B*g associated to coordinate charts Wr of M, points mr E BWr with (real) coordinates x r , and indices jr (for a generic coordinate on W r ) and Jr (a

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309

multi -index for a product of odd coordinates on W r ). Continuity of implies that there exists neighborhoods W; c Wr of mr and a coordinate neighborhood V of e E G such that Vr = 1, ... , k : (V, W;) c Wr . We thus can use the coordinates on Wr also for points in the image (V, W;). Let (gi)f=l denote coordinates on V and let (mi)f'!i denote coordinates on W r . If we wish to distinguish even and odd coordinates of a point m E W r , we will denote them as (Xi, ~j). It follows that the fundamental vector field associated to Ogi Ie Egis given on W; as

E1

This means that the function (m) of the proof of [6.6] is the function - ~;~ (g = e, m). Using that (dgil e) is the left dual basis of (Ogi Ie) and using [111.3.7] to express the function Ef,J in terms of partial derivatives of Ef, we can write the functions rP~(x) (defined in the proof of [6.6]) as

.

"". 0 J.) (g = e, (dg'le)' (0 oe ogi

rPj(x) = - L

i

X,

~ = 0) .

We then note that the functions j are defined on the whole of V x W; and hence that we can consider the functions V ---> A, g f---+ - ~~;: (g, X = x r , ~ = 0), 1 ::; r ::; k, whose parity is given as c( mjr) + c(er ). In order to combine these functions in a single even map, we introduce an A-vector space F of total dimension k whose basis vectors fr have parity cUr) = c(m jr ) + c(er ). With this A-vector space we define an even map X : V ---> Fo by k

X(g)

=-

L

ojr oer (g,x

= xr,~ = 0)·

fr .

r=l

The fact that the functions rP% (x r ) form k independent elements of *g is equivalent to saying that the Jacobian of X at e has maximal rank k (note that 0e Ogi and Ogi 0e only differ by a sign (_1)(€(gi)I€(e»). Said in yet another way, using thatg>P o is the null space of the equations rP7,. (x r ) = 0, this is equivalent to the fact that g>p o is the kernel of TeX. It follows that the Jacobian at 0 E .50 of the composite map.5o ---> G ---> Fo, s f---+ exp(s, 0) = exp(s) f---+ x(exp(s)) is invertible. Hence if we take the neighborhood V of e E G sufficiently small, we may assume that exp is a diffeomorphism between U = exp-l (V) C go and V and at the same time that X 0 exp is a diffeomorphism between Un.5o and its image in Fo. Now let g E G>P o n V be arbitrary. By definition of G>P o there exists a smooth map 'IjJ : N ---> G>P o such that g E im('IjJ). By taking a smaller N we also may assume that im( 'IjJ) C V, i.e., im( 'IjJ) C G>Po n V. Composing 'IjJ with X, using that taking derivatives with respect to the ~ variables does not interfere with what happens with the g coordinates [111.3.13], and using the definition of G>Po as acting as the identity on M, we deduce that X( 'IjJ( n)) is constant equal to X( e). Since 'IjJ( n) lies in V, there exists (s, z) E U C .50 x gt o such that 'IjJ(n) = exp(s, z). Since is a left action and since exp(g>P O ) C G>Po acts as

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the identity on M, it follows as above that x(1,b(n)) = x(exp(s) exp(z)) = x(exp(s)). Hencex(exp(s)) = x(e), sos = 0 becausexo exp is a diffeomorphism betweenUn.5o and its image in Fo. This proves that G,po n V is included in exp(gt o n U). Since we already know that exp(ggo n U) is contained in G,po n V, we can apply [5.8] to conclude that G,po is a proper A-Lie subgroup of Gwith A-Lie algebra g,po. • To show that G,po is normal, we first note that GO is normal. We then consider the smooth map 1,b : G x G,po ---> GO defined by 1,b(g, h) = ghg-l. Since this is smooth, we conclude by definition of G,po that the image of 1,b lies in G,po, i.e., G,po is normal. • The induced action ind of GjG,po on M is defined by iPind(gG,po, m) = (g, m). This is well defined because the elements of G,po all act as the identity on M. Since 7r : G ---> GjG,po is a locally trivial (principal) fiber bundle [5.9], we choose a local smooth section s : V ---> G for some tri vializing chart V c G j G,po. It follows that the restriction ind : V x M ---> M is given by ind(Z, m) = (s(z), m), which is smooth. Since being smooth is a local property, we conclude that ind is globally smooth. • To prove that iPind acts pseudo effectively, we consider an arbitrary smooth map 1,b : N ---> G j G,po such that all 1,b( n) act as the identity on M. As above we choose a local trivializing chart V and a smooth section s : V ---> G. We then consider the smooth map X : 1,b-I(V) ---> G defined by x(n) = s(1,b(n)). By definition of 1,b, all x(n) act as the identity on M. By definition ofG,po this means that all x(n) lie in G,po. But that means that all1,b(n) are the identity element in GjG,po for n E 1,b-I(V). Since the local trivializing charts cover G jG,po, we conclude that 1,b is constant the identity element. This IQEDI means that ind acts pseudo effectively.

6.8 Lemma. Let G be an A-Lie group and (i, H) an A-Lie subgroup. If there exists a neighborhood V of the identitye E G such that V n i(H) = {e}, then i(H) is a closed discrete subgroup of BG C G. Ifin addition G is connected and H is normal, then i(H)

is contained in the center ofG (and in particular H is abelian). Proof • Since H is an A-Lie subgroup we have g E i(H) =? Bg E i(H). Hence for any g E i(H) we have (Bg)-l . g E V, and thus g = Bg, i.e., i(H) C BG. Now if g E i(H), then Lg is a diffeomorphism and Vg = Lg V is a neighborhood of g such that Vg n i(H) = {g}, i.e., i(H) is a discrete subgroup. Since inversion and multiplication are smooth operations, there exists a neighborhood

W C V of the identity such that W- l . W C V. If g E BG n i(H) \ i(H), then Lg W n i(H) -# ~, i.e., :JXI E BW :Jh l E i(H) : gXI = hI. Since g rJ. i(H), Xl -# e and thus there exists a neighborhood U C W of the identity such that Xl rJ. U (because BG is Hausdorff!). But then again LgU n i(H) -# ~, and thus :JX2 E BU :Jh 2 E i(H) : gX2 = h 2. Butthen hllh2 = x 1l x2 E W-I . Un i(H) c V n i(H) = {e}. Butthis contradicts Xl -# X2 (because X2 E U and Xl rJ. U). This proves that i( H) is closed in BG. • If H is normal, we fix h E H and we consider the smooth map f : go ---> G defined by f(x) = exp(x)· i(h). exp( -x). Since H is normal, f takes values in i(H). Since f is smooth (i(h) E BG), there exists a neighborhood U of 0 E go such that f(U) C Li(h) V. But i(H) n Li(h) V = {i(h)} (i(H) is discrete), hence f is constant i(h) on U. If G

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311

is connected, exp(U) generates G [2.7], and thus we have proven that for all g E G necessarily g. i(h)· g-1 = i(h), i.e., thati(H) is contained in the center ofG IQEDI

6.9 Remark. Unless A does not contain nilpotent elements, a closed subgroup ofBG C G is never closed in G, simply because, by definition of the DeWitt topology, each closed set of G containing e also contains B- 1 { e }.

6.10 Corollary. Let If> : G x M --; M be a smooth left action of an A-Lie group G on an A-manifold M and let G1/1O and 91/1 0 be as in [6.2J.

(i) The action If> is pseudo effective {==} G1/1O = {e} ===} 91/1 0 = {O}. (ii) If 91/1 0 = {O}, then G1/1o is a closed discrete subgroup of BG c G contained in the center ofG. 0

Proof • If 91/1 0 were not {O}, exp(9g ) would contain elements different from the identity acting as the identity on M [6.7], which contradicts that the action is pseudo effective . • Assume 91/1 0 = {O}. According to [6.7] G1/1O is a proper normal A-Lie subgroup, and thus by [5.8] there exists a neighborhood V of e E G such that V n G1/1O = {exp(O)}. The conclusion then follows from [6.8]. IQEDI

7.

COVERING SPACES AND SIMPLY CONNECTED

A-LIE

GROUPS

In this short section we prove that a morphism of A-Lie algebras determines a unique morphism ofthe associated A-Lie groups, provided the source group is simply connected. We thus start with a brief review of covering spaces, universal coverings, and simply connected spaces, and we prove that the simply connected cover ofan A-Lie group has a unique structure of an A-Lie group such that the projection is a morphism of A-Lie groups.

7.1 Definitions. Let X and Ybe a topological spaces. A continuous surjectionp : Y --; X is called a (topological) covering ofX if every x E X admits an open neighborhood U such that p-1(U) = UiE1Ui such that

(i) each Ui is open in Y, (ii) the (Ui)iEI are pairwise disjoint, and (iii) p: Ui --; U is a homeomorphism. It is easy to show that if p : Y --; X is a covering of X and if q : Z --; Y is a covering of

Y, then po q : Z --; X is a covering of X. A covering of X is called connected if Y is connected (and thus X has to be connected too). A connected covering p : Y --; X is called universal if Y is connected and iffor any

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connected covering f : Z ----; X there exists a covering 9 : Y ----; Z such that p = fog. A connected space is called simply connected if the identity map is a universal covering. It follows immediately from the definition of a universal covering that it is unique up to homeomorphisms and that it is simply connected. Theorem [7.2] collects the main results concerning covering spaces that we will need; its proof can be found in most textbooks on algebraic topology (e.g., [Spa]).

7.2 Theorem. Let X be a connected, locally path connected, second countable space admitting an open cover consisting of contractible sets. (i) If X and X' are simply connected, thq; X x X' is simply connected. (ii) There exists a universal covering p : X ----; X.

(iii) Ifp: Y ----; X is a connected covering of X, then Y is second countable. (iv) Let p : Y ----; X be a connected covering of X, and let gi : Z ----; Y be two continuous maps such that po gl = po g2 : Z ----; X. If Z is connected and if there exists a point z E Z such that gl (z) = g2 (z), then gl = g2. (v) Letp: Y ----; X beacovering, f: Z ----; X a continuous map, and(zo,Yo) E ZxY such that f(zo) = p(Yo). IfZ is simply connected, then there exists a (unique) continuous lift 9 : Z ----; Y such that f = po 9 and 9 (zo) = Xo.

7.3 Definitions. In the context of A-manifolds we will call a map p : N ----; M a covering if it is a topological covering [7.1] such that p is smooth and such that the restrictions p : Ui ----; U in condition [7.1-iii] are diffeomorphisms. These additional conditions on a covering exclude maps such as p : Ao ----; A o, x r--+ x3, which is a smooth bijection, but not a diffeomorphism in any neighborhood of O. In view of the inverse function theorem [111.3.23], the additional conditions on a topological covering can be rephrased as saying that p should be smooth and TnP should be a bijection for every n E N, i.e., p is everywhere a local diffeomorphism. The definitions of a universal covering and of simply connected remain the same (but with the changed notion of covering).

7.4 Lemma. Let M be a connected A-manifold. (i) If p : N ----; M is a connected topological covering, then N has a unique structure

of an A-manifold such that p becomes an A-manifold covering. (ii) There exists a universal A-manifold covering p : M ----; M. (iii) Let p : N ----; M be a covering, f: L ----; M a smooth map, and (£0' no) E Lx N such that f (£0) = p( no). If L is simply connected, then there exists a unique smooth lift 9 : L ----; N such that f = po 9 and g(£o) = no. Proof First note that if M is a connected A-manifold, then it is in particular a connected, locally path connected, second countable topological space with an open cover consisting of contractible sets, just by choosing contractible charts. We thus can apply [7.2].

§7. Covering spaces and simply connected A-Lie groups

313

• To prove (i), let p : N ----> M be a topological covering. For any n E N we choose a chart tp : U ----> 0 around m = p(n) in M satisfying condition [7.1-ii] (this can always be done by shrinking U if necessary). Since n lies in a unique Ui , we define tp 0 p : Ui ----> 0 to be a chart for N. We leave it as an exercise for the reader to prove that two charts for N defined in this way are compatible. We thus have constructed an atlas for N, i.e., we have made N into a proto A-manifold for which p satisfies the conditions of an A-manifold covering. With this structure it is elementary to show that Bp : BN ----> BM is a connected covering of BM. Since BM is second countable and Hausdorff, it follows easily from the definition that any covering must also be Hausdorff. We conclude from [7.2-ii] that N is an A-manifold. To prove uniqueness of this structure, it suffices to note that, if p is a smooth local diffeomorphism, then tp 0 p : Ui ----> 0 is a diffeomorphism, and hence a chart for N [IIIA.20]. • Let p : M ----> M be the topological universal covering of M. According to (i) we may assume that it is an A-manifold covering, so it remains to prove that it has the right properties. Therefore, let I : N ----> M be a connected A-manifold covering. It then is in particular a topological covering, and thus there exists a topological covering g : M ----> N such that p = log. It now suffices to prove that this g is smooth. Since p and I are local diffeomorphisms and g a local homeomorphism, we can locally say that g = 1-1 0 p. But this proves that g is locally a diffeomorphism, and in particular globally smooth. • In (ii) we have seen that the universal A-manifold covering is the same as the universal topological covering. This means in particular that the notion of simply connected does not depend upon whether we use A-manifold coverings or topological coverings. Applying [7.2-v] we thus find a continuous lift g : L ----> N with the desired properties. To prove that g is smooth, we choose f!. ELand a chart U :;) I (f!.) of M satisfying the conditions of a covering. Since the Ui are disjoint, there is a unique Ui such that g(f!.) E Ui • Since p : Ui ----> U is a diffeomorphism, it follows that g restricted to l-l(U) coincides with p-l 0 I (use [7.2-iii] if needed). Hence g is smooth in an open neighborhood of f!.. IQEDI

7.5 Discussion. It is elementary to show that if p : N ----> M is a covering (in the sense of A-manifolds), then Bp : BN ----> BM is a covering (in the sense of R-manifolds). On the other hand, if q : Z ----> BM is a covering in the sense of R-manifolds, then one can show (it is elementary but not immediate) that there exists a unique covering p : N ----> M such that Z = BN and q = Bp. In this way one obtains a bijection between coverings of M and coverings of BM. This implies in particular that M is simply connected if and only if BM is simply connected.

7.6 Lemma. Let G be a connected A-Lie group, p : G ----> G its universal covering, and let e E Gbe such that p(e) = e E G. Then Gadmits a unique structure of anA-Lie group such that e is the identity element and p :

G ----> G a morphism

ofA-Lie groups.

----> G is the smooth map defined by I(x, y) = m(p(x),p(y)), then [7.4] there exists a unique smooth map in : G x G ----> G such that p 0 in = I and such by

Proof If I : G x G

Chapter VI. A-Lie groups

314

that m(e, e) = e. By construction p(m(x, y)) = m(p(x),p(y)),i.e., pis amorphism of defines a group structure. To construct the inverse, A-Lie groups once we know that we consider the map Inv 0 p : 0 --; G. By [7.4] there exists a unique smooth map Inv : 0 --; 0 such thatpolnv = Invop. The map h : 0 --; 0, h(x) = m(x,Inv(x)) is such that h(e) = e and po h is constant e E G. Since the constant map e is a lift of this map, by [7.2] we have h(x) = e, i.e., Inv is indeed the inverse and e the identity in (0, m). To prove associativity, we consider the smooth map a: 0 x 0 x 0 --; G defined by a(x, y, z) = p(x) . p(y) . p(z). We have two lifts of this map to 0 : m(x, m(y, z)) and m(m(x, y), z). Since both send (e, e, e) to e, they must be the same by [7.2-iii]. This is associative. The other properties of an abstract group are proved in the proves that same way. IQEDI

m

m

7.7 Discussion. If p : 0 --; G is the universal covering of a connected A-Lie group G, then 71'1(G) == ker(p) is a normal (abstract) subgroup of O. By definition of a covering, there exists a neighborhood V ofe E 0 such that 71'1 (G) n V = {e}. By [6.8] it follows that 71'1 (G) is a discrete central subgroup of O. Since 71'1 (G) is the first homotopy group of G, this proves that the first homotopy group of an A-Lie group is abelian.

7.8 Lemma. Let p : G --; H be an A-Lie group homomorphism and let H be connected. Then p is a covering ifand only ifTeP : 9 --; f) is a bijection. Proof If p is a covering, there exist neighborhoods U of eH E Hand Ui of ec E G such that p : Ui --; U is a diffeomorphism. Hence by the inverse function theorem [III.3.23] TeP is a bijection. If we assume that TeP is a bijection, it follows from [1.18] and [Y.2.l4] that P is everywhere a local diffeomorphism; it follows from [111.3.23] that there exist neighborhoods D :3 eH and V :3 ec such that P : V --; D is a diffeomorphism. Since P is a homomorphism and H connected, it follows from [2.6] that P must be sUljective. We now consider the (smooth) map f : G x G --; G, (9, h) r-+ 9h-1. By continuity of f there exists an open neighborhood V c V of ec such that f(V x V) c V. We finally define D = ker(p) and U = p(V) c D. With these ingredients we can prove that p is a covering. Let h E H and 9 E p-1(h) be arbitrary, then the set Lg(V) is an open neighborhood of 9 and L h (U) is an open neighborhood of h. Moreover, since p is a homomorphism, we have the equality p = Lh 0 po L g-1. Since p is a homeomorphism from V to U and because L g-1 and Lh are (global) homeomorphisms, p = Lh 0 P 0 L g-1 is also a homeomorphism from Lg(V) to Lh(U), Since p is everywhere a local diffeomorphism, it is a diffeomorphism from Lg(V) to Lh(U). Fixing 90 E p-1(h) we claim that the decomposition p-1(L hU) = UdEDLgodV satisfies the conditions of a covering. To prove equality, we choose 9 E p-1(L hU) and then 3g E V : p(g) = h . p(g), which is equivalentto 9;1 9g-1 = d E D. Hence 9 = 90dg, i.e., 9 E LgodV. Since obviously UdEDLgodV C p-1(L hU), we thus have equality. To show that they are mutually disjoint, suppose 9 E Lgod 1 V n Lgod 2 V {:=} 391,92 E V : 90d191 = 9 = 90d292, hence

§S. Invariant vector fields and forms

1 1 d2" d 1 = g2g1 . But f(V X V) c 1d must have d2" 1 = ec and thus d 1

315

V and thus = d2 .

1 d2" d 1 E

V.

Since p(d2"1dd = eH, we

IQEDI

7.9 Proposition. Let G and H be A-Lie groups with A-Lie algebras 9 and ~ respectively and G simply connected. Ifr: 9 ---4 ~ is an A-Lie algebra morphism, there exists a unique A-Lie group morphism p : G ---4 H such that TeP = r. Proof Uniqueness follows from [2.17], so we only have to show existence. Therefore we consider the A-Lie group G x H with its A-Lie algebra 9 x ~ and the canonical projections 71"1 : G x H ---4 G and 71"2 : G x H ---4 H. Inside 9 x ~ we have the sub algebra .5 = {(x,r(x)) I x E g}. According to [4.7] we thus have an associated connected A-Lie subgroup j : S ---4 G x H. Now Tej is an isomorphism from TeS to the subalgebra.5, hence Te(71"1 oj) : TeS ---4 TeG is a bijection. Since 71"1 oj is an A-Lie group morphism, it is a covering [7.8]. Since S is connected and G simply connected, 71"1 0 j must be a diffeomorphism, i.e., an isomorphism of A-Lie groups. We now define P = (71"20 j) 0 (71"10 j)-1 : G ---4 H, which obviously is a homomorphism satisfying TeP = r. IQEDI

8 . INVARIANT VECTOR FIELDS AND FORMS In this section we define the notions of invariant vector field and invariant differential form on an A-manifold on which an A-Lie group acts smoothly. This generalizes the notion of left/right-invariant vector field on an A-Lie group. The main results of this section are that on a connected A-manifold a vector field is invariant ifand only ifit commutes with the fundamental vector fields and that a differentialform is invariant ifand only if the Lie derivative in the direction ofthefundamental vector fields is zero. To prove these results we generalize [V.5.15] and [V.7.27], which are essentially the case ofthe action of a I-dimensional A-Lie group.

8.1 Definition. Let : G x M ---4 M be a smooth left action of an A-Lie group G on an A-manifold M and let Y be a (not necessarily smooth) vector field on M. Extending the notion of a (left/right) invariant vector field on an A-Lie group [1.8], we will say that Y is invariant under the G-action if it satisfies the condition T 0 (Q x Y) = Yo . Using the generalized tangent map T g' g E G [Y.3.l9], we can reformulate this definition as Vg E G : Tg 0 Y = Yo g, which means that for g E G and m E M we have Tg(Ym) = Yg(m)' If: M x G ---4 M is a smooth right action, then we will say that Y is invariant under the G-action if it satisfies the condition T 0 (Y x Q) = Yo . In terms of the generalized tangent map Tg this also reads as Tg(Ym) = Yg(m), but here the map 9 is different from the one in the case of a left action. And of course the

316

Chapter VI. A-Lie groups

left/right invariant vector fields on an A-Lie group G are special cases of these definitions when viewing the multiplication m : G x G ----> G as left/right action of G on itself.

8.2 DiscussionINotation. We intend to show that for a connected A-Lie group G with its A-Lie algebra g, the vector field Y is invariant under the G-action if and only if Y commutes with the fundamental vector fields. The actual proof is a bit long, but the idea behind it can easily be explained. For x E 9 we have the fundamental vector field xM, whose flow is given by exp(tx). Using [Y.S.lS] we deduce that Y is invariant under the action of the subgroup formed by the exp( tx) if and only if Y and x M commute. Since the elements of the form exp( x) generate G the result follows. A first problem with this idea is that vector fields must be smooth to be integrable, which restricts attention to x E Bg. However, the main problem is that for odd elements x there is no guarantee that the odd vector field x M satisfies [xM, xM] = 0, a condition necessary for x M to be integrable. Since the even elements in Bg do not generate g, we can not reach the whole group G. To overcome this problem, we note that [V.S.lS] concerns the flow of a vector field. And the flowcPx ofa vector field X can be seen as the action of the I-dimensional A-Lie group A.s(x) on M (apart from the fact that the domain W x need not be the whole of A.s(x) x M). We thus generalize this result to the setting of general group actions: the flow cP x will be replaced by the group action , the time parameter t will be replaced by a group element g E G and the vector field X will be replaced by a fundamental vector fieldx M . In order to prepare the actual statement, we use the generalized tangent map to form the G-dependent vector field 1/Jy on M defined by

Ifwe introduce the function (1) : G x M ----> G x M, (g, m) t--> (g-1, g(m)) (note the analogy with the flow of a vector field), then we can write the definition of 1/Jy as

1/Jy = T (Q x Y) 0

0

(1) :

G

xM

---->

TM .

It is immediate that Y is invariant under the G-action if and only if 1/Jy is independent of g, i.e., 1/Jy(g, m) = Ym . If Z is a smooth vector field on G, we can form the vector field Z x Q on G x M. By abuse of notation we will denote this vector field also as Z. It is immediate from the definition that 1/Jy (g, m) E T mM. And thus the map 1/Jy and the vector field Z satisfy the requirements of [V.3.1O]. Hence it makes sense to talk about the derivative of 1/Jy in the direction of Z, i.e., about the map Z1/Jy : G x M ----> T M. Now recall that for x E 9 we have defined a corresponding fundamental vector fieldx Mon M and that in the same vein we have the right-invariant vector fieldx c on G [S.l]. Moreover, the right-invariant vector field xC and the fundamental vector fieldx M are related by <1>: Tm xClg = xMIg(m) [S.2]. Since x M and xC are smooth if x E Bg, the following statement makes sense.

§S. Invariant vector fields and forms

317

8.3 Proposition. Let If> : G x M ---> M be a smooth left action of an A-Lie group G with associated A-Lie algebra 9 on an A-manifold M, let x E Bg and let Y be a smooth vector field on M. Then

Proof This proof is a close copy of the proof of [V.S.lS]. We first note that the result is additive in the vector field Y and the A-Lie algebra element x, so we may assume that Y and x are homogeneous. Copying the approach of [V.S.lO], we choose a point (g, h, m) E G x G x M and we imagine that 9 is close to the identity e E G. If gr are coordinates around e E G, there exist AT such that xCle = LT ATOgrie = -x. Since x E Bg, the A are real (and thus even); since x is supposed to be homogeneous we have c(Ogr) = c(x) for all Ogr contributing to the sum. Finally we choose local coordinates systems x~ around m and xl around (gh, m). With these coordinate systems we are going to establish a certain number of identities that will be needed later. Since the xl are coordinates around iii =
xt

(8.4)

Using the generalized tangent map for the equality

(g, iii) =
olf>~k (( h)-l -) g, m J'l

uX b

=

~ olf>lk(g-1, m- ) . olf>~i (h- 1 ,m. ~)

L

j

J'l

J'l ..

uX b

UX'b

If we assume for the moment that h is also close to e so that we can use the coordinate system xl also around m, and if we take 9 = h = e in (8.S) (implying m = iii = in), we obtain immediately (8.6)

o
--k oX b

·

Since this is valid for all iii in the coordinate neighborhood, we can apply [111.3.13] to obtain (8.7)

We now introduce thefunction X: G x G x M ---> TMby X(g, h, m) = 1jJy(gh, m). It follows easily that (xc x Q x Q)I(e,h,m)X = (xc X Q)I(h,m)1jJy == (x c 1jJy )(h, m), simply

Chapter VI. A-Lie groups

318

because ((xClgIITRd X is given by

= XCI9h

by right invariance. In the given trivializations the map

1/Jy(gh, m) == X(g, h, m) = (8.8)

ak~ ((gh)- 1 ,(gh, m)) . -a ai 1m = ""' L.., Ybk ((gh, m)) . k

(8.9)

.

aX b

Xa

"

""' y;k(( ~)) . al ( -1 ( g, m ~)) . a~ b g, m a kg, ax{ i,j,k Xb b

= L..,

(h-1~) a Im, ,m' axi a

where we used (8.5) to obtain the last equality. In the same trivialization we now compute (xc x Q x Q)I(e,h,m)X using (V.3.l2) and the fact that in does not depend upon g :

a~. (h- 1,m ~). ~I a· m·

aXb

x~

To obtain the last equality we applied (8.4), (8.6) and (8.7). The additional sign appears because we have to change the order of the partial derivatives agr and axk. Since we b

have c:(y;'k) + c:(x~) = c:(Y), we see that the expression in parentheses represents the coefficients of [X, YJ (in) with respect to the basis xi. Comparing this result with the b local expression (8.8) for X, which is essentially the local expression for 1/Jy, we see that IQEDI this is the local expression of 1/J[X,YJ (h, m).

a

8.10 Corollary. Let : M x G ---> M be a smooth right action of an A-Lie group G with associated A-Lie algebra 9 on an A-manifold M, let x E Bg and let Y be a smooth vector field on M. Then

x 1/Jy =

1/J[x M

,YJ .

§S. Invariant vector fields and forms

319

Proof For a right action 1/Jy is defined as for a left action: 1/Jy (g, m) = T g-1 Y 9 (m)' but now 9 : M ---> M is defined as 9 (m) = (m, g). If is a right action, the map 111 : G x M ---> M defined by 111 (g, m) = ( m, g-1) is a left action of G on M. Looking at the definition of a fundamental vector field, it is immediate that the fundamental vector fieldxM, associated to x E 9 forthe(right) -action is the same as the fundamental vector fieldxM,w for the (left) 1l1-action [5.4]. Defining ~ (g, m) = T1l1 g-1 YWg(m) we have by [8.3] xC;j;y = ;j;[xM, Y] but also ;j;y (g, m) = 1/Jy (g-1 , m), i.e., 1/Jy = ;j;y 0 (Inv x id). Leaving it to the reader to prove that 1/Jy makes sense, i.e., that the condition of [V.3.10] is satisfied, we compute it according to the official definition:

x

(x 1/Jy )(g, m)

= 7ra(T(g,m)1/Jy(xg)) = 7ra(T(g-1,m);j;y(T(g,m) (Inv x id)(xg, Qm))) ~

C

= 7ra(T(g-1,m)1/Jy(xg-1,Qm))

~

= 1/J[xM,y](g

-1

,m)

= 1/J[xM,y](g,m)

,

where for the third equality we used [1.10] and the fact that xC is the right-invariant vector IQEDI field on Gsatisfying x~ = -x [5.1].

8.11 Corollary. Let : G x M ---> M be a smooth left action of an A-Lie group G on an A-manifold M and let Y be a smooth vector field on M. (i) IfY is invariant under the G-action, then "Ix E Bg : [xM, Yj = O. (ii) If G is connected and if "Ix E Bg : [xM, Yj = 0, then Y is invariant under the

G-action. Proof It is obvious from the definition that 1/Jy is zero if and only if Y is zero. Now if Y is invariant, then 1/Jy is independent of the G-coordinates, and thus xC 1/Jy = 0 for all x E Bg. By [8.3] we conclude that [xM, Yj = 0, proving the first part. For the second part we first invoke [8.3] to conclude that x C1/Jy = 0 for all x E Bg. Since the vector fields xC with x E Bg span the tangent space TgG at each point g E G, we conclude by [Y.3.20] that there exists afunction f : M ---> T M such that 1/Jy = f 0 7r M. Since 1/Jy (e, m) = Ym , it follows that f equals Y. In other words, Y is invariant under IQEDI the G-action.

8.12 Definition. Let : G x M ---> M be a smooth left action of an A-Lie group G on an A-manifold M and let w be a (not necessarily smooth) k-form on M, i.e., a section of I\k *TM. Using the generalized tangent map Tg, g E G [V.3.19], we define the generalized pull-back ;w by the same formula (V.7.20) as for smooth k-forms and smooth maps. For Xi, ... ,Xk E TmM the k-form (;W)m is defined by

By definition of an action we have e(m) = m. It then follows directly from [111.3.13] that Tme = id. Hence we always have ;w = w. If : M x G ---> Mis a smooth right action, we define the generalized pull-back ;w by the same formula, but, as for invariant vector fields, it is the definition of the map 9 that changes.

Chapter VI. A-Lie groups

320

The k-forrn W is said to be invariant under the action ofG if for all g E G we have ;w = w. As a particular case we mention that a k-form W on G itself is said to be left/right-invariant if it is invariant under the naturalleft/right action m : G x G ....... G of G on itself. Recalling that (forleft actions) Tm g(X) = T(g,m) (Qg, X), it is elementary to see that w is invariant under the G-action if and only iffor all (not necessarily smooth) vector fields Xl, ... , X k on M we have

(8.13) where 7r M G x M ....... M denotes the canonical projection. In fact, in order to be invariant, (8.13) need only be verified for a set of vector fields that span the tangent space at each point m E M. Multilinearity then does the rest. For example it is sufficient to verify (8.13) for smooth vector fields (even if w is not smooth). Yet another way to say the same is to note that the above definition is a particular case of the generalized pull-back given in [V.7.23]. This means that we look at the map 1/Jw : G x M ....... N *T M by 1/Jw(g, m) = (~W)m. And then the definition of w being invariant becomes the condition 1/Jw = w 0 7rM. For right actions (8.13) changes in the obvious way; the defining formula of 1/Jw does not change, but as before it is 9 that changes.

8.14 Lemma. Let w be a smooth k-form. dw.

If w is invariant under the G-action, then so is

Proof Using [V.7.22], the equality (Q x X)(f07rM) = (Xf)o7rM and the equality [Q x X, Q x Y] = Q x [X, 11, it follows immediately from (8.13) and the definition of the IQEDI exterior derivative that dw is invariant ifw is.

8.15 Lemma. The map w t---> wefrom left-invariant k-forms on an A-Lie group G to /\k * 9 = /\k *Te G is a bijection. Moreover, w is smooth if and only if We has real · . We E B/\k * g. coordmates, I.e.,

Proof. Suppose w is left-invariant and We

= 0, then for Xl, ... , X k

E TgG we have by

left -invariance: ~(XI'

... ' Xk)wg = ~(XI' ... ' Xk)(L;-lW)g = ~(<<XdTLg-l

», ... , ((XdTLg-1 )))we = o.

Hence w = 0, proving injectivity . • To prove surjectivity, we construct the global section w from the given value of We by specifying all k-linear graded skew-symmetric maps Wg : (TgG)k ....... A. For Xl, ... ,Xk E TgG we write (8.16)

§S. Invariant vector fields and forms

321

The global k-form W defined by this formula is left-invariant because ~(XI""

,Xk)(L'hw)g

= ~(<<XdTLh»"'" =

((XdTLh)))Whg ~(((XIIIT Lh <> T L(hg)-1 », ... , ((Xk I T Lh <> T L(hg)-1 »)we by (8.16) and (1.2.14)

= ~(((XdTLg-1),

... ,((XdTLg-1)))We

• If X I, ... ,Xk are smooth vector fields, the function

g f--+

~(Tm(Qg-1,

= ~(XI,

~(X I, ... , X k)W

... ,Xk)Wg .

is given by

XIl g), ... ,Tm(Qg-1, Xklg ))we .

Since g f--+ g-l f--+ Qg-1 is smooth, the map g f--+ Tm(Qg-1, Xilg) is smooth. Since We is a smooth multi-linear map if and only if We has real coordinates [11.6.22], [III. 1.27], the conclusion follows from [IV.5.9]. IQEDI

8.17 Corollary. A k-form W on G is left-invariant if and only iffor all left-invariant vector fields Xl"'" X k on G the junction L(X I , ... , Xk)w is constant. Proof A direct consequence of (S.16) and the equality ((Xg I T L g-1)) invariant vector fields.

Xe for left-

IQEDI

8.18 DiscussionINotation. Let If> : G x M --; M be a smooth left action of an A-Lie group G with associated A-Lie algebra 9 on an A-manifold M and let W be a smooth k-form on M. We want to show a statement analogous to the statement for invariant vector fields, i.e., that W is invariant if and only if the Lie derivative of W in the direction of the fundamental vector fields is zero. Here the basic idea is to use [V.7.27] to prove this, but the objections are the same as in the vector field case. Luckily the solution to this problem is also the same: the flow rP x will be replaced by the group action , the time parameter t will be replaced by a group element g E G and the vector field X will be replaced by a fundamental vector field zM. The preparations for the actual statement are quite analogous to those for [V.7.27]. In [S.12] we have seen that the invariance of W can be stated in terms of the G-dependent k-form 1/Jw : G x M --; N*TM, 1/Jw(g,m) = (;a)m [V.7.23]. Since we have 1/Jw (g, m) E (N *T M)m, it follows that any vector field Z on G extended by Q to G x M and the map 1/Jw satisfy the conditions of [V.3.10]. We thus can speak about the derivative of 1/Jw in the direction of Z. If one now recalls that for z E g, zG is aright-invariant vector field on G and zM the associated fundamental vector field on M, then [S.19] makes sense; [S.20] and [S.21] are proved as [S.10] and [S.11].

8.19 Proposition. Let : G x M --; M be a smooth left action of an A-Lie group G with associated A-Lie algebra 9 on an A-manifold M, let z E Bg and let W be a smooth k-form on M. Then

Chapter VI. A-Lie groups

322

Proof This proof is a very close copy of the proof of [V.7.27], so we will skip some of the motivations for the computations. Since the result is clearly additive in both z and W we may assume that z and W are homogeneous. We choose a coordinate neighborhood V around (g, m) with coordinates yi, and let W c G be a neighborhood of 9 with coordinates land U a coordinate neighborhood around m with coordinates x] such that (W x U) c V. On the local chart V there exist smooth functions wit, ... ,ik : V --4 A such that the k-form W has the form W = L'tIl'" ,tk. Wil ... ik . dyil 1\ ... 1\ dyik. There also exist smooth functions Ze on Wand Xi on V such that the vector fields zG and zM have the local form z7} = Le Ze(g)ogllg and zt::) = Li Xi(Y)Oyi I(Y). The relation

Tmz7}

= Z~g,m) between zG and zM . X'((g,x))

[5.2] then translates as the equation

e oi = ""' L Z (g). a e(g,x).

e

9

Moreover, it follows directly from the definition of the generalized pull-back ; that we have ~(oxi);dyi = ~(((oxil(g,x)IITi)(g, x). Hence all terms of the form ; dyi are of the form iF.* '!! 9

d i _ " " ' d] oi ( ) Y - L X ' ax] g, x . ]

With these preparations we compute:

zG ( * dyi)

= ""'( _l)(€(z)I€(xi » dz] . Ze(g). 02i. (g, x) L

9

age ax)

],e

""' dxJ.

= L

a (""' Z e Oi) ,,",' O( Xi 0
0

)

=

)

~

2: dx]. ~ j,k ux

k J

i

(g,x).

i

~Xk ((g,x)) = 2: ;dyk . ;oXk uy

k

oy

= ;(2: dyk . ~x:) = ;(dXi) = ;(.C(zM)dyi) . k

Y

For the second equality we used that the functions Ze do not depend upon the coordinates x J and for the last equality we used that Xi = ~(zM)dyi. For the coefficient function Wil ... ik we note that ;wil ... ik(X) = Wil ... ik((g,X)). Hence ZG(;Wi1 ... ik) = (zMWil ... ik) 0 = ;(zMWil ... ik)' The final computation to knit these results together is an exact copy of the analogous computation in the proof of [V.7.27] IQEDI

8.20 Corollary. Let : M x G --4 M be a smooth right action of an A-Lie group G with associated A-Lie algebra 9 on an A-manifold M, let z E Bg and let W be a smooth k-form on M. Then

§9. Lie's third theorem

323

8.21 Corollary. Let : G x M ---> M be a smooth left action ofan A-Lie group G on an A-manifold M and let w be a smooth k-form on M. (i) Ifw is invariant under the G-action, then "Ix E Bg : .c(xM)w = o. (ii) If G is connected and if "Ix E Bg : .c(xM)w = 0, then w is invariant under the G-action.

9. LIE's

THIRD THEOREM

In this section we prove that for each finite dimensional A-Lie algebra 9 there exists a unique (up to isomorphism) simply connected A-Lie group G with 9 as its A-Lie algebra. For this proof we need to introduce the notion differential forms with values in an A-vector space, a notion that will be generalized and studied in more detail in chapter VII.

9.1 Definitions (forms with values in an A-vector space). Let M be an A-manifold and E an A-vector space. We have seen that a smooth k-form on M can be interpreted as a left k-linear (over C=(M)) skew-symmetric map from vector fields on M to (ordinary, smooth) functions on M, i.e., an element of HomLk(r(TM)k; C=(M)). In analogy, we define a smooth k-form with values in E or smooth E-valued k-form as being a left k-linear (over C=(M)) skew-symmetric map from vector fields on M to smooth functions on M with values in E, i.e., an element ofHomLk(r(T M)k; C=(M; E)). Playing around with the various identifications [1.5.5], [1.8.8], [III. 1.24], [V.7.1], this space is isomorphic to *(N r(T M» <59 C=(M; E) ~ n,k(M) <59 C=(M; E), where the tensor product is taken over the graded ring C=(M). In analogy with ordinary k-forrns, the set of all smooth .E-valued k-forms will be denoted as n,k(M; E), i.e., n,k(M; E)

== HomLk(r(TM)k; C=(M; E»)

If w is a k-form on M and if v : M (isomorphic to) the E-valued k-forrn

--->

~ n,k(M) <59 C=(M; E) .

E is an element ofC=(M; E), then w <59 v is

(9.2)

from k vector fields Xl,"" X k to a smooth E-valued function. Conversely, let w be a smooth E-valued k-form, let (ei) be a basis for F, and let (le) be the associated left dual basis of *E. We then can form the map

wj

:

(XI,,,,,Xk)

f-+

((t(XI, ... ,Xk)wpe)

from k vector fields on M to a smooth function on M. Since this map is obviously skew-symmetric and k-linear over C= (M), it represents an ordinary k-forrn on M. Since v = Lj «vpe)) ej for all vEE, we can write (9.3)

Chapter VI. A-Lie groups

324

where we see each ej as a constant (smooth because ej E BE) E-valued function on M. In the particular case k = 0 the w j are smooth functions on M. It then is customary to omit the tensor product symbol 0 and to write w = Lj w j ej. Formally, a O-form is an element ofHom£(Coo(M); Coo(M; E)) [1.5.3]. Omitting the tensor product symbol amounts to applying the identification Hom£(Coo(M); Coo(M; E)) ~ Coo(M; E) [1.2.19], [1.8.8], [111.4.21], where one should not forget that in this context the basic graded ring is Coo (M). In short, a O-form with values in E is just an E-valued function. We thus have shown that for a fixed basis of E, an E-valued k-form is uniquely represented by dim E ordinary k-forms. However, these k-forms depend upon the choice of the basis for E. More precisely, if (ei) is another basis (in its equivalence class) for E, j it is related to the old basis by (real) matrix elements Ai by ei = Lj Aijej. In the new j . Since je = Li ie Ai j , we also basis the E-valued k-form defines ordinary k-forms have

w

One usually fixes a basis for E and then a smooth E-valued k-form is just a useful abbreviation for a system of dim E ordinary k-forms. However, one should be careful with parities: w is even if c(w i ) = c( ei) for all i and w is odd if c(w i ) = 1 - c( ei). Once we know that a smooth k-form with values in E is completely determined by dim E ordinary smooth k-forms, we can define a not necessarily smooth k-form with values in E as being determined by dim E ordinary (not necessarily smooth) k-forms for which (9.3) is valid (by definition). Since the arguments of an ordinary k-form can be any vector field, smooth or not, we can use (9.3) and (9.2) to extend an E-valued k-form to act also on arbitrary vector fields, yielding an E-valued function (not necessarily smooth) on M. Seen this way, a smooth E-valued k-form is one that maps smooth vector fields to smooth i-valued functions (see also [IV.S.ll]). As for ordinary k- forms, we define the notions of contraction with a vector field, exterior derivative, and pull-back by the natural formula;. More precisely, ifw = Li wi 0 ei is an E-valued k-form and X a vector field on M, we define L(X)W by

Using the natural action of vector fields on smooth E-valued functions [V.1.24], the exterior derivative of an E-valued k-form is defined by (V.7.4) (and thus in particular by [Y.7.6]). Since the ei are seen as constant functions on M, it is immediate that we have the formula (9.4)

If g : N as

->

M is a smooth map, the pull-back g*w is the E-valued k-form on N defined

§9. Lie's third theorem

325

Finally, if an A-Lie group G acts on M, we say that w is invariant under the G-action if all wi are invariant under the G-action. We leave it to the reader to verify that these definitions are independent of the choice of the basis for E. It is a simple consequence of the fact that the Ai j are real constants.

9.5 Discussion. In [Lie, Ch 25] Sophus Lie stated three fundamental properties of Lie groups, of which the third (in § 115) says that to every finite dimensional Lie group is

associated a finite dimensional Lie algebra and to each finite dimensional Lie algebra a finite dimensional Lie group. For the second part of this statement, several essentially different proofs by various authors have been given. Here we will present, in the context of A-Lie groups, a geometrical proof based upon a proof of Elie Cartan [B.Ca] (see also [vE], [Gor], [Tu1]). The idea of this proof is as follows. Let 9 be a finite dimensional A-Lie algebra. If the adjoint representation ad R : 9 ---; EndR(g) is injective, we can interpret 9 as a subalgebra ofEndR(g). Since we know that Aut(g) is an A-Lie group, we can apply [4.7] to conclude that there exists an A-Lie group with associated algebra g. If the adjoint representation is not injective, it has a kernel c = ker(ad R), called the center of g. Since ad R is smooth, cis a graded subspace ofg in the sense of A-vector spaces. Moreover, for any two x, y E c we have 0 = adR(x)(y) = [x, y], i.e., c is a subalgebra of 9 with trivial bracket. Standard arguments then show that the quotient I) = gj c inherits the structure of an A-Lie algebra from 9 and that the image 6 = adR(g) C EndR(g) is a subalgebra of End R (g) isomorphic to I). It follows that 9 is a central extension of I) (extended by c), and that a simple algebraic ingredient, a Lie algebra 2-cocyc1e, allows us to reconstruct 9 from I) and c. Since I) is (isomorphic to) a subalgebra of EndR(g), we can apply as before [4.7] to obtain an A-Lie group H whose A-Lie algebra is I). Cartan's construction then proceeds to transform the Lie algebra 2-cocycle into a group 2-cocyc1e, which allows us to construct a new A-Lie group G out of Hand c, and whose A-Lie algebra is (isomorphic to) g. We will not develop the theory of Lie algebra and Lie group cohomology, nor will we go into the details of central extensions and their relation to cohomology in dimension 2. The interested reader is referred to [Br], [Sta], [TW] and references therein for more details. Of course these references do not treat A-Lie algebras and A-Lie groups, but the generalization is straightforward. In the actual proof we will introduce only the strict minimum needed to construct the A-Lie group G. In [9.18] we will give some comments on how the various parts fit into a larger context.

9.6 Theorem. Let 9 be a finite dimensional A-Lie algebra. Then there exists a simply connected A-Lie group G, unique up to isomorphisms, such that 9 is (isomorphic to) the A-Lie algebra associated to G.

Proof We start as indicated in [9.5] by defining c = ker(ad R : 9 ---; EndR(g)), I) = gjc, and 6 = adR(g) C EndR(g). Since EndR(g) is the A-Lie algebra of the A-Lie group Aut(g), there exists a connected A-Lie subgroup (i, if) of Aut(g) whose A-Lie algebra

Chapter VI. A-Lie groups

326

6 [4.7]. We denote by H the simply connected cover of if. By [7.6] and the definition of a covering, H is an A-Lie group with A-Lie algebra I). If c were {O}, the A-Lie algebra I) would be isomorphic to 9 and G = H would be a simply connected A-Lie group with A-Lie algebra g. If c is not {O}, we choose a smooth injective even linear map a : I) ----> 9 such that 7r 0 a = id(I)), where 7r : 9 ----> g/c = I) is the canonical projection. Such maps certainly exist as can be seen by using a basis (see [11.6.24] and [11.6.23]). Associated to this a is a left bilinear map : I) x I) ----> c defined by I) is isomorphic (by Ti) to

L(X, y) = a([x, y]~) - [a(x), a(Y)]g . indeed takes its values in c because 7r is an A-Lie algebra morphism. By definition this is smooth, bilinear, even, and graded skew-symmetric (because the brackets in I) and 9 are). Given this and the A-Lie algebra I), we can reconstruct the A-Lie algebra 9 in the following way. On the A-vector space I) ED c we define the bracket

[(x, a), (y, b)] = ([x, yh, -L(X, y)
(9.7)

and we define the map rf; : I) ED c ----> 9 by rf;(x, a) = a(x) + a. It is easy to show that this rf; is an isomorphism of A-vector spaces, but also that rf; preserves brackets:

rf;([(x, a), (y, b)]) = rf;([x, y]~, -L(X, y) = [a(x), a(y)] = [rf;(x, a), rf;(y, b)] because c is the center. It follows that 9 and I) ED c with the bracket (9.7) are isomorphic A-Lie algebras. We now define D as being the smooth left-invariant c-valued 2-form on H whose value at the identity is [8.15], [9.1]. This 2-form is explicitly given by the formula

L(Xg, Yg)D

=

L(TLg-IXg,TLg-IYg), and more in particular for left-invariant vector

fields by

L(X, g)D :

(9.8)

g

1-+

L(X, y) .

But there is more. The graded Jacobi identity for the brackets on I) and 9 imply that for homogeneous x, y, z E I) we have (9.9)

L(X, [y, z])

=

L([X, y], z) + (_l)(€(x)I€(y» L(y, [x, z]) .

We will show that this implies (actually it is if and only if, but that is of no importance here) that D is closed. Using the definition of the exterior derivative [Y.7.3] or [V.7.6], and using that a vector field applied on a constant function yields zero, we find for three left-invariant homogeneous vector fields X, g, and ion H :

L(X, g, i)dD = L(X, [g, i])D - L([X, g], i)D -

(_l)(€(x)I€(y»

L(g, [x, i])D

=

L(X, [y, z]) - L([X, y], z]) -

(_l)(€(x)I€(y»

L(Y, [x, z])

=

0

by (9.9).

by [9.8]

§9. Lie's third theorem

327

Since homogeneous left-invariant vector fields generate r(T H) as a graded Coo (H)module [1.19], we conclude do' = 0 as claimed. Let (Vi) be a basis of I) and (iV) the associated left-dual basis of*l). As in [VII. 1.5] we define Wi to be the left-invariant I-form on H satisfying wile = iV. We then have in particular for two left-invariant vector fields x and iTthe identity ~(x, iT)dw i = (( [x, ylll iV). Now recall that for x E I) we have introduced the fundamental vector field x H on H associated to the natural left action of H on itself. This x H is the right-invariant vector field on H such that xHle = -x = -xe [5.1]. In orderto avoid this minus sign, we define x R as x R = _xH, i.e., x R is the right-invariant vector field on H such that xRle = x. Said differently, XRlg = TRg(x). We now invoke a theorem about ordinary Lie groups that is far from trivial: if Gis an ordinary simply connected Lie group, then the de Rham cohomology groups HI (G) and H2(G) are zero: Hl(G) = H2(G) = {a}. A proof can be found in [God, Ch xm, 6.3]; it relies on the Ktinneth formula and a theorem ofIwasawa [Iw] that any Lie group is topologically equivalent to the direct product of a maximal compact subgroup and a vector space (over R). Since H is (by definition) simply connected, the same is true for BH. Using [V.S.9] we conclude that HI(H) = H2(H) = {a}. This has several immediate consequences. We will prove in [9.13] that the I-forms ~(vf)o' are closed. Our first consequence is that these I-forms must be exact, i.e., there exist smooth functions Ii on H with values in c such that ~(vf)o' = dJ;. In the second place, the 2-form 0, is closed, and thus must be exact, i.e., there exists a I-form a on H, not necessarily left-invariant, with values in c such that da = o,. Now suppose that a e = Li ai dxi with respect to some coordinate system (xl, ... ,xn) in a neighborhood U of e. Since a is smooth, the coefficients ai are real. We thus can define the constant function A(x) = Li aixi on the neighborhood U. If we multiply this function A on U by a plateau function p around e in U, we get a global smooth function pA on H. Since p is constant 1 on a (smaller) neighborhood of e, (dP)e = 0 and hence (d(pA»e = Li ai dXi = a e. If we define a = a - d(pA), it satisfies da = 0" but it also satisfies ae = O. We conclude that we may assume without loss of generality that a e = O. We now reach the final stage of our preparations. Denoting by 7ri : H x H ---; H, i = 1, 2 the canonical projections on the first and second factor, we define the I-form W on H x H with values in c by the formula (9.10)

In [9.14] we prove that W is closed. Since H and thus H x H is simply connected, we have in particular HI (H x H) = {o}. This implies that there exists a smooth function F : H x H ---; c such that W = dF. In [9.15] we prove that the function F satisfies the relation (9.11)

Vg, h, k E H: F(g, h)

+ F(gh, k)

=

F(g, hk)

+ F(h, k)

.

We finally define the simply connected A-Lie group G as G = H x Co [7.2-i] with multiplication me defined by

me((g,a), (h,b» = (gh,a+b+F(g,h».

Chapter VI. A-Lie groups

328

Since the multiplication in H is smooth, as is F, the map me is smooth. From (9.11) we deduce first (putting h = e) that F(g, e) = F(e, k) = F( e, e), and then that me is a group structure with (e, - F( e, e)) as the identity. We claim that the A-Lie algebra of this G is (isomorphic to) g. By construction, the tangent space TeG is naturally isomorphic (as A-vector space) with TeH x C = ~ x c. Using [1.17] we compute the brackets of the vectors aXi Ie, aa j Ie where (x!, ... , xn) are (local) coordinates on Hand (a j ) (global) coordinates on Co. Since me depends only linearly on the coordinates any structure constant j with either i or j referring to a coordinate a is zero. On the other hand, whenever k ~ n, the structure constant j involves only m H, and thus for 1 ~ i, j, k ~ we recover the structure constants of~. The only remaining unknown structure constants are thus those with i,j ~ nand k > n, i.e., those involving the term a + b + F(g, h). Since we only derive with respect to the coordinates of g and h, we can forget about the a + band concentrate on F(g, h). We thus have to compute aian+jF - an+iajF at (e,e). We therefore compute

a,

c7

c7

(an+iF)(g, e)

n

= ~(an+il(g,e»)dF = ~((Qg, =

aile))w ~((Qg, aile))m*a - ~((Qg, aile)) 7r 2a

=

~(TLgaile)a - ~(aile)a = ~(§ilg)a

(because a e = 0),

where (as usual) ~ denotes the left-invariant vector field whose value atthe identity is ai. It follows that (ajan+iF)(e, e) = ajle(~(~)a). We thus find

(aian+jF - an+iajF) (e, e) = (aian+jF - (-1) (€(xi)I€(xj» ajan+iF) (e, e) = aile(~(§j)a)

- (-1)(€(xi)I€(xj»ajle(~(§i)a)

= -~(~Ie, §jle)da + [([8;, ~lle)a = -~(Vi' Vj) . Comparing these results with the bracket on ~ ED c given in (9.7), we see that they are the same. The conclusion is that the A-Lie algebra associated to the A-Lie group G is (isomorphic to) g. To finish the proof, we note that uniqueness up to isomorphisms follows from [7.9]. IQEDI

9.12 Lemma. Let G be an A-Lie group and let q; : G --; Aut(g) c EndR(g)o be defined byq;(g) = Ad(g-l). Then for x E g: Xgq; = -adR(x) 0 Ad(g-l) E EndR(g). Proof Consider the map : G x G --; Aut(g) defined as = q; 0 m. It is immediate that (( Xg I T q;)) = «(Qg, x) I T )). But we can define in a different way, using that Ad is a homomorphism: = Inv 0 m 0 (Ad x Ad), where here the multiplication is in Aut(g). U sing this interpretation of we can compute (( (~, x) I T
xe)

f---' f---'

(Te Ad)(x)) = -TRAd(g-l) adR(x) ,

(QAd(g) ,

(QAd(g) ,

adR(x))

f---'

T LAd (g) adR(x)

§9. Lie's third theorem

329

where we used [2.15] and [1.6]. In the identification T Aut(g) = Aut(g) x EndR(g) [1.20] we have, according to [V.3.2], «Xg II Tr,b)) = (r,b(g),Xgr,b). Using [1.23] we find in the same identification (( (Qg, x) II T
9.13 Technical lemma. The 110rms ~(vf")D on H are closed. Proof Using the definition of D, we find for an arbitrary left-invariant vector field

z:

Using the definition [V.7.3] or [V.7.6], we compute for two smooth homogeneous leftinvariant vector fields and y:

x

- ~(Xg, Yg) d(L(Vf")D)

= Xg(~(Y, vf")D) - (-l)(€(x)I€(Y))fjg(~(x, vf")D) - ~([x, mg, vf"lg)Dg = Xg(~(Y, Ad(g-l)Vi)

=

(_l)(€(x)I€(Y)) ~(y,

xg(Ad(g-l)Vi)) - ~(x, Yg(Ad(g-l)Vi))

- ~([x,y],Ad(g-l)Vi) = -( _l)(€(x)I€(Y)) ~(y,

[x, Ad(g-l)Vi]) + ~(x, [y, Ad(g-l)Vi])

- ~([x,y],Ad(g-l)Vi) =0,

where we used [9.12] to obtain the fifth line and (9.9) for the final result. Since smooth homogeneous left-invariant vector fields generate reT H) [1.19], the result follows. A faster proof of this result, which is less direct because it invokes [8.21], goes as follows. Since D is left-invariant, £(xH)D = 0 for all x E BI), and thus in particular £(vfI)D = O. But £(X)D = d~(X)D + L(X)dD = d~(X)D because D is closed. Since vfI = -vf" we obtain d~( vf")D = 0 as wanted. IQEDI

9.14 Technical lemma. The I-form W (9.10) on H x H is closed. Proof Using [V.7.9-iii] and [Y.7.22] we find

The tangent space at each point in H x H is generated by vectors of the form (Xl, X2) where the Xi are (smooth) left-invariant vector fields on H. In order to prove dw = 0, it is thus sufficientto prove ~((Xl' X2), (Yl, Y2) )dW = 0 for all Xi, Yi E I). By graded skewsymmetry and bilinearity it is sufficient to verify the three cases ~((Q, x), (Q, y) )dw = 0, ~((x, Q), (Q, y))dw = 0, and L((X, Q), (y, Q) )dw = 0 for x, y E I).

Chapter VI. A-Lie groups

330

Before we start these three verifications, we recall that for x E I:J we have T Lgfih = Xgh )

andTRhxg = TLgTRhx = TLgh(Ad(h-l)x) = Ad(h-l)xlgh' In the same spirit we I find for x R the equality xRlh = TRhx = Ad(h-l)xlh' Using [V.7.20] we now turn our attention to the three equalities we have to prove. • The first equation yields

• The second one yields

L((X9,Qh),(Qg,Yh))dW = = =

i L(Ad(h- 1 )x, y) - 0 - 0 + Li 0 + Li (_1)(€(y)!€(w »(<<x I iV)) . L(Yh, vflh)D L(Ad(h-l)x,y) + Li (-1)(€(y)!€(v'»(((xll i v)L(y,Ad(h- I )Vi)

= L(Ad(h- 1 )x, y) + (-1) (€(x)!€(y» L(y, Ad(h- l ) (Li (((x I iV)) . Vi)) = L(Ad(h-l)x,y) + (-1)(€(x)!€(y»L(y,Ad(h- l )x) = O. • Finally for the third equality we find

L((X9,Qh)' (:Q'g,Qh))

dw

= L(Ad(h-l)x,Ad(h-l)y) - (x,y) + Li (L(xg,:Q'g)dwi). (fi(h) - fi(e)) = L(Ad(h-l)x,Ad(h-l)y) - (x,y) + Li (([x,y]lliv)). (fi(h) - fi(e)) . To prove that this is identically zero, we introduce the functions Xl, X2 : H --; c by

= (Ad(h-l)x, Ad(h-l)y) - (x, y) X2(h) = Li(([X, y]11 iv)) . (fi(h) - fi(e)) . XI(h)

We then compute for an arbitrary homogeneous left-invariant vector field Z on H the functions ZXi (using [9.12]):

(zXI)(h)

=

-L([Z, Ad(h-l)x], Ad(h-l)y) - (_1)(€(z)!€(x» L(Ad(h-l)x, [z, Ad(h-l)y])

(zX2)(h)

= Li (_1)(€(z)!€(x)+€(Y)+€(Vi»(([X,yWv)). zhfi = Li (_1)(€(z)!€(x)+€(Y)+€(Vi»(([x,y]ll i v)). L(Zh)L(vf)D = Li (_1)(€(z)!€(x)+€(Y)+€(Vi»(([x,y]ll i v). L(z,Ad(h-I)Vi) = L(Z, Ad(h-I)Li (( [x, yWv)) . Vi) = L(Z, Ad(h-l)([x, y])) .

It follows from [2.19] and (9.9) that Z(XI + X2) = 0, and then it follows from [V.3.2l] and [1.19] that Xl + X2 is a constant function. Since XI(e) + X2(e) = 0, we conclude IQEDI that Xl + X2 is identically zero.

§9. Lie's third theorem

331

9.15 Technical lemma. Thefunction F satisfies the relation (9.11).

Proof We introduce the function K : H x H x H ----; c by

K(g, h, k) = F(g, h)

+ F(gh, k)

- F(h, k) - F(g, hk) ,

which means that we have to prove that K is identically zero. We do this by noting that K (e, e, e) = 0 and proving that dK = 0 as a I-form. The result then follows from [V.7.11]. If we denote by 71"12 : H x H x H ----; H x H the map (g, h, k) f---+ (g, h), by m12 : H x H x H ----; H x H the map (g, h, k) f---+ (gh, k), and similarly for 71"23 and m23, the function K can be written as

K = F 0 71"12

+ F 0 m12 -

F 0 71"23

-

F 0 m23 .

We thus find for dK, using dF = wand [V.7.19],

Now Wcontains terms with a and terms with a sum over fiS. Using the easily proved relations for functions from H3 ----; H: m 0 m12 = m 0 m23 (associativity of the multiplication), m 0 71"12 = 71"1 0 m12, m 0 71"23 = 71"2 0 m23, 71"1 071"12 = 71"1 0 m23, 71"2 0 m12 = 71"2 0 71"23, and 71"1 0 71"23 = 71"2 0 71"12, the reader can easily show thatthe parts involving a in dK cancel. We thus have to prove that the remaining terms given by

(9.16)

L 71"i Wi . (71"2fi -

+ (71"10 m12)*w i . (71"3fi

fi(e))

- fi(e))

- 71"2Wi . (71"3fi - fi(e)) - 71"iw i . ((71"2 0 m23)* fi - fi(e)) add up to zero. Ifwe introduce the matrix elements Ad(k)/ by Ad(k)vj = Li Ad(k)jiVi (the Nh matrix elements [1104.1] of the even automorphism Ad(k), seen for the occasion as left linear), then we can compute for three left-invariant vector fields X, y, and on H

z

l (usingthatTRhxg = Ad(h- 1)xl gh)

~((Xg, Yh, Zk)) (71"1 om12)*w i =

= ~((Xg, Yh))m*w i

~((((Qg'Yh)llTm))

+ (((xg,Qh)IITm)))w i

= ~(ihh)Wi + ~(Ad(h-1)xlgh)Wi =

blliv))

+ Lj ((xlljv». Ad(h- 1)/

= ~(Xg, Yh,

ik)71"2W i + Lj L(Xg, Yh,zk)71"iwj . Ad(h- 1)/ .

We conclude that

(71"1 omd*wil(g,h,k) = 71"2Wil(g,h,k)

+ Lj 71"iwjl(g,h,k)

. Ad(h- 1)/ .

Substituting this in (9.16) and evaluating this I-form in the point (g, h, k) E H3 we find

Chapter VI. A-Lie groups

332

Since the Wi are independent, we concentrate on the functions Xi : H x H

---->

c defined as

(9.17) In order to prove that these are identically zero, we want to apply [V.3.20]. Therefore we compute (Qh, Xk)Xi = ~((~, Xk) )dXi for an arbitrary left-invariant vector field x.

(Qh,Xk)Xi = -xhdi

+ Lj

(_l)(€(x)I€(vi)+€(Vj» Ad(h-I)i j

= -~(Xhk)~(vf)n + Lj (_l)(€(x)I€(vi)+€(Vj»

.

xdj

Ad(h-I)i j

.

~(xk)~(vf)n

= -~(x, Ad((hk)-I )Vi) + ~(Xk)~( (Lj Ad(h- I )ijvf))n =

=

-L( (x, Ad( (hk) -1 )Vi)) + ~(x, Ad(k- I ) (Ad( h -1 )Vi))

°

because Ad is a homomorphism.

We conclude by [1.19] and [Y.3.20] that Xi(h, k) is independent of k. But if we take k = e, we find Xi(h, e) = 0, and thus Xi is identically zero. This proves that the terms in (9.16) add up to zero, which proves that dK = 0, and thus that K is a constant function. Since K (e, e, e) = 0, we have shown that K is identically zero, i.e., that F satisfies the IQEDI relation (9.11).

9.18 Comments on the proof of [9.6]. In order to put some of the items of the proof of [9.6] in a wider perspective, we give some remarks for the interested reader. • The graded skew-symmetric bilinear function is a 2-cocycle in A-Lie algebra cohomology, the cocycle condition being given by (9.9). Changing the section a changes this cocycle with the coboundary of a 1-cochain. The reconstruction of 9 as the A-Lie algebra I) EDc is part ofthe standard isomorphism between cohomology classes in dimension 2 of A-Lie algebra cohomology and equivalence classes of central extensions of I) by c. • The function F is a 2-cocycle in A-Lie group cohomology, the cocycle condition for this cohomology being given by (9.11). • The functions Ii can be seen as a generalization of a momentum map known from symplectic geometry; here n plays the role of the symplectic form and the vf' the role of the fundamental vector fields of the group action on the symplectic manifold. • The functions Ii - Ii (e) can be put together to form a function on H with values in *1) 0 c. The fact that the functions Xi in (9.17) are identically zero then says that this new function can be seen as a 1-cocycle on H with values in the H-module * I) 0 c. More details can be found in [So, Thm 11.17].

9.19 Examples. An A-Lie algebra of dimension 111 has a basis VO,VI in which V€ has parity c. Since the bracket is even, we have [vo, vol = 0, [vo, VI] = AVI, and [VI, VI] = f-lVo, where A, f-l are real numbers because the bracket is supposed to be smooth. The graded Jacobi identity applied to Vo, VI, VI tells us that Af-l = 0. We conclude that, up to rescaling, there exists three A-Lie algebras of dimension 111 : an abelian one (A = f-l = 0),

333

§9. Lie's third theorem

°

one with [vo, vol = [VI, VI] = and [vo, vd = VI (A = 1, f..L = 0), and a third with [vo, vol = [vo, VI] = and [VI, VI] = Vo (A = 0, f..L = 1). We intend to apply the construction of the proof of [9.6] to find the corresponding A-Lie groups. • In the abelian case we find C = g, and thus ~ = {O}. The corresponding simply connected A-Lie group H is obviously {e} (of dimension 0). Since on an A-manifold of dimension 0 there are no non-zero k-forms with k > 0, we have a = 0, Ii = 0, W = 0, and hence F = O. We conclude that G = Co with group law

°

me (a, b) = a + b . In other words, G is the additive abelian group Co ~ A. • For the second case with [vo, vol = [VI, VI] = and [vo, VI] = VI we find adR(avo + bvd : Vo f-+ -bVb VI f-+ aVl' It follows that c = {O}, and that the image adR(g) c EndR(g) is given by

°

adR(g)

= { ( ~b ~) I a, bE A} .

Using [3.6] and the proof of [4.7], the corresponding A-Lie subgroup G of Aut(g) can be found to be G={ ~) I a E A o, a E A o, Ba > O} .

(!

As an A-manifold this is an open subset of go ~ A, but whose group law is given by = (ab, a + a(3). This can be interpreted as the a~ + a group of affine transformations of the odd affine line AI. • The third case with [vo, vol = [VO, VI] = and [VI, VI] = Vo presents the most interesting application of [9.6]. It is easily seen that adR(avo + bvd : Vo f-+ 0, VI f-+ bvo. It follows that c is the graded subspace generated by Vo of dimension 110, that ~ = gj c is the abelian A-Lie algebra of dimension Oil with single basis vector VI, and that the image adR(g) c EndR(g) is given by

me((a, a), (b, (3))

°

adR(g) = {

(~ ~) I b E A} .

Again using (the proof of) [4.7] and [3.6], the corresponding A-Lie subgroup H of Aut(g) can be found to be

H

= {( ~

nI

a E

Ad .

In other words, H = Al with the usual addition as group operation. Using the section a : ~ ---; 9 defined by a(vl) = VI, we find for the map :

Denoting by by

~

the odd coordinate on H, we find that the left-invariant 2-form

n = -~d~ 1\ d~ = d(-~~d~)

.

°

n is given

°

Hence a = - ~~ d~, which satisfies indeed the condition a e = (because e = in this group). The right-invariant vector field associated to VI = ae Ie is the vector field ae, hence

334

Chapter VI. A-Lie groups

= -d~, and thus f(~) = -~ is a solution. Together with the left-invariant l-form on H we find, using coordinates (~, ry) on H x H, for Wthe l-form

~(8e)n d~

For F we thus find the function F(~, ry) = -~~ry; for the group G = Al gives us the multiplication mc((a, a), ((3, b))

= (a + (3, a + b -

X

Ao ~ A this

~a(3) .

This group is the simply connected covering of the A-Lie subgroup H = GS I X Al discussed in [4.14]. The difference in constants is explained by the fact that there the scaling is such that [WI, WI] = -4wo. In fact, the covering map is given by the morphism p: G --; GS I X AI, (a,a) f-+ (e- 4ia , a). We have also encountered this A~Lie algebra as the A-Lie algebra of the multiplicative group A* = {a E A I Ba =I- O} discussed in [2.8]. This is a non-connected A-Lie group, whose connected components are simply connected. The covering map from G to the connected component containing the identity of A * is given by p : G --; A *, (a, a) f-+ e- 2a +o: = e- 2a + e- 2a a, which is actually an isomorphism because both are simply connected.

9.20 Remark. The three groups of dimensions 111 are exactly the three special cases considered in [MS-V] in a more general approach to integration of (non-homogeneous) vector fields.

Chapter VII

Connections

In a direct product with the two projections on the separate factors, we know what horizontal and vertical directions are: those that project to zero under the tangent map of one ofthese two projections. Afiber bundle 7r : B ----+ M with typical fiber F and structure group G is locally a direct product, but only one of the two projections is independent of such a local trivialization: the one corresponding to the bundle projection 7r. By convention the directions in B that project to zero under the tangent map T7r are called vertical. It follows that on a fiber bundle we do not have a natural definition of what horizontal directions are; the local idea of horizontal directions is not independent of the choice of the local trivialization. A connection on a fiber bundle is an additional structure which provides the notion of horizontal directions. This additional structure can take various forms. The most natural one is to define exactly the horizontal directions, i.e., a subbundle 11. C T B which is a supplement to the subbundle of vertical directions V = ker T7r C T B. In this form it is called an Ehresmann connection. But otherforms for the additional structure are sometimes useful: a connection I-form on a principal fiber bundle, a covariant derivative on a vector bundle, orparallel transport. The notion of an Ehresmann connection is too general for most purposes. A much more interesting subclass of connections is formed by FVF connections, whose form is determined, in a sense to be made precise, by the fundamental vector fields of the structure group on the typical fiber. The connection Ilorm, the covariant derivative and linear connections allfall in this subclass. Moreover, for the subclass offiber bundles concerned (principal/vector), they are equivalent to FVF connections. In this chapter we define the above mentioned notions of a connection and we show how they are related. On principal fiber bundles the FVF connection is also described as the kernel ofthe connection Ilorm, whereas on vector bundles the covariant derivative ofa section describes how far the section is from being horizontal. Moreover, a (vector) bundle B can be seen as associated to a principal fiber bundle P: the structure bundle. Sections of B then can be seen as a special kind of functions on P and the covariant derivative gets 335

336

Chapter VII. Connections

transformed into the exterior covariant derivative on P associated to its FVF connection. This correspondence can be generalized to differentialforms with values in an A-vector space or in a vector bundle. And then a covariant derivative and the exterior covariant derivative can be seen as generalizations of the usual exterior derivative of (ordinary) differential forms. The last aspect of connections that is treated here is the notion of curvature. An Ehresmann connection 11. on a bundle B is in particular a subbundle ofTB. As such one can ask whether 11. is afoliation, i.e., is involutive. In general the answer will be negative, but there are several cases in which one can measure to what extent it is not involutive. For principal fiber bundles with a connection I -form w this is done by the curvature 2-form n = Dw, the exterior covariant derivative of the connection I form. For vector bundles with a covariant derivative V' this is done by the curvature tensor R. In these cases the statement is that the FVF connection is involutive if and only if the curvature is zero. Moreover, we show that nand R correspond under the identifications which link connections on principal fiber bundles with those on associated vector bundles.

1.

MORE ABOUT VECTOR VALUED FORMS

In this technical section we generalize operations concerning A-vector spaces (composition, evaluation, bracket, etcetera) to vector valued differentialforms. We prove some elementary but useful formula? and we introduce the all important Maurer-Cartan I form e Me on an A-Lie group.

1.1 Definition. Let E, F, and G be three A-vector spaces with homogeneous bases (ei), (Ij), and (gk) respectively, and let : Ex F --; G be an even smooth bilinear map. With these ingredients we define the q,-wedge product 1\<1>, which associates to an E-valued p-form a and an F-valued q-form (3, a G-valued (p + q)-form a 1\<1> (3, all on an Amanifold M. The construction is as follows. The forms a and (3 are uniquely determined by ordinary differential forms a i and (3j according to a = Li a i 0 ei, (3 = Lj (3j 0Ij [VI.9.l]. And then a 1\<1> (3 is defined by (1.2)

(2: a i

i

0 ei) 1\<1>

(2: (3j 0 Ij) = 2: a

i

1\ I[,,(e i ) ((3j)

0 ( ei,fj) .

i,j

j

Lk

Introducing matrix elements for by q,( ei, Ij) = 7j gk. the G-valued (p + q)-form 1\<1> (3 is defined by the ordinary (p + q)-forms (a 1\<1> (3)k given by

a

(a 1\<1> (3)k =

2: a

i

1\ I[e(e i ) ((3j)

. 7j .

i,j

It is elementary to check that the definition of a 1\<1> (3 is independent of the choice of the bases for E and F, thus guaranteeing a correct definition of the -wedge product.

§ 1. More about vector valued forms

337

1.3 Notation. Each map If> has its associated If>-wedge product which we denoted as 1\<1>. However, specific maps If> have their own notation for the associated wedge product. We will need the following four specific maps with the associated notation. • Multiplication by scalars: in this application the A-vector space E is A, F = G, and If> is (left) multiplication: If>(a, v) = a· v In this case the If>-wedge product 1\<1> is simply denoted as 1\ • • Applying a linear map to a vector: here F and G are arbitrary A-vector spaces, E = HomR(F; G), and If> is the evaluation map: If>(A, v) = A(v). In this case the If>-wedge product 1\<1> is denoted as ~. • Composition of linear maps: here E, F and G are all equal to End R ( C), the set of (right linear) endomorphisms of an A-vector space C, and If> is composition: If>( A, B) = A 0 B. In this case the If>-wedge product 1\<1> is denoted as {} .

• The bracket in A-Lie algebras: here E = F = G = 9 are all equal to an A-Lie algebra 9 and If> is the bracket: If>(x, y) = [x, y]. In this case the If>-wedge product of a and (3 is denoted by [ a Ii- (3]. As is usual with notation, there is sometimes more than one way to write things. Here the exceptions all occur when either a or (3 is a O-form. The most obvious case is in the first case when g is a O-form on M, i.e., an ordinary function, and (3 an F-valued k-form. In that case the F-valued k-form g 1\ (3 is the same as g . (3. This is a direct consequence of the similar fact for ordinary k-forms [V.7.l]. Less obvious is the similar situation in the second case when A is a HomR(F; G)-valued O-form, i.e., a smooth function A : M ----; HomR(F; G), and (3 an F-valued k-form. Then it is customary to write A 0 (3 instead of A ~ (3. The idea is that at each point m EMit is the composition of the map (3lm from (TmM)k to F with the map Am from F to G. Similarly in the third case: if A is a HomR(F; G)-valued O-form and (3 a HOffiR(F; G)-valued k-form, then it is customary to write A 0 (3 instead of A {} (3. Coming back to the second case, if a is a HomR(F; G)-valued k-form and g an F-valued O-form, i.e., a smooth function s: M ----; F, then it is customary to write a(g) or a· g instead of a ~ g, the idea being that for fixed mE M and Xi E TmM it is the action of the homomorphism~(Xl"'" Xk)a m on the vector g( m).

1.4 Lemma. Let 9 be an A-Lie algebra, let a be an even g-valued Ilorm, (3 an even g-valued 210rm, and let X, Y, and Z be homogeneous vector fields on M. Then: ~(X, Y)[

a Ii- a]

=

-2[ ~(X)a, ~(Y)a]

and

~(X,Y,Z)[(3li-a] =

[~(X,

In case 9

Y)(3, ~(Z)a]

+ (_I)(€(Xll€(y)+€(Zll [~(Y, Z)(3, ~(X)a] + (-1) (€(Zll€(Xl+€(Y» [~(Z, X)(3, ~(Y)a]

= EndR(E) we also have the equality [ a Ii- a] =

2a {} a.

.

338

Chapter VII. Connections

Proof For the first equality we compute for homogeneous I-forms a and 'Y

ij

L (( _l)(€(X)I€(Y)+€(a)+€(vi» ~(Y)ai ~(Xhj

=

ij

- (_l)(€(Y)I€(a)+€(vi» ~(X)ai ~(yhj)

o (-1) (€(vi)I€(Vj)+€(-y»

[Vi, Vj]

= (-l)(€(X)I€(Y)+€(a»[~(Y)a,~(Xh]-

(-l)(E(y)I€(a»[~(X)a,~(Yh].

The special case follows immediately from this result because for even a we have the equality [~(Y)a, L(X)a] = -( _l)(E(X)IE(Y» [~(X)a, ~(Y)a]. For the second equality we compute for homogeneous a and (3: ~(X,Y,Z)[(3f,\a] =

=L

~(X,

Y, Z)((3i

1\

a j ) 0 (_l)(€(vi)I€(vj)+€(a» [Vi, Vj]

ij

=

L( (-1) (€(X)I€(y)+€(Z)+€(,6)+€(Vi» ~(Y, Z)(3i ~(X)aj ij

=

+ (_l)(€(Z)I€(X)+€(Y»

(_l)(€(Y)I€(vi)+€(,6»

+ (-1) (€(Z)I€(Vi)+€(,6»

~(X, Y)(3i ~(Z)aj) 0 (-1) (€(vi)I€(Vj )+€(a» [ Vi, Vj ]

(-1)(€(Z)I€(,6» [L(X, Y)(3, ~(Z)a]

+ (_l)(E(Y)I€(,6» For 9

= EndR(E)

~(Z,

X)(3i

~(Y)aj

+ (_l)(€(X)I€(Y)+€(Z)+€(,6» [~(Y, Z)(3, L(X)a]

(-1) (€(Z)I€(X)+€(Y» [~(Z, X)(3,

~(Y)a]

.

we have the canonical basis ei 0 ej and we compute:

ijpq

o (-1) (€(ei)+€(ej)I€(ep)+€(e q») ei 0

ej

0

- (-1) (€(ei)+€(ej )1€(Y)+€(ep)+€(e q))) ~(X)ai j ~(Y)aP q) ei

0

=L

ep 0 eq

(( -1) (€(Y)I€(ep)+E(eq») ~(X)aP q ~(Y)aij

ijpq

= (_l)(€(X)I€(Y» ~(Y)a 0 ~(X)a

ej

0

ep

0

eq

- ~(X)a 0 ~(Y)a = -[ ~(X)a, ~(Y)a] .

We thus have shown that ~(X, Y)(a {} a) = -[ ~(X)a, ~(Y)a] for all homogeneous X, Y. Combining this with the first result finishes the proof. IQEDI

§ 1. More about vector valued forms

339

1.5 Example (the Maurer-Cartan I-form). Let G be an A-Lie group, 9 its A-Lie algebra,

(Vi) a basis ofg, and (iV) the associated left-dual basis of*g = *TeG. Using [VI.S.I5] we define Wi to be the left-invariant I-form on G satisfying wile = iV. We then define the g-valued I-form 8 MC on G as

which is called the Maurer-Cartan I-form ofG. Since all wi are left-invariant, 8 MC is a left-invariant g-valued even I-form on G (that 8 MC is even follows from the fact that the parity of Wi is the same as that of Vi). Moreover, if x = Lxi. Vi (Xi E A) is a left-invariant vector field on G, the contraction L(x)8 MC yields

L(x)8 MC = LxjL(Vj)w i 0 Vi = Lxi. Vi = x, i,j

i.e., 8 MC is the tautological g-valued I-form on G. Another way to state the tautological nature of 8 MC is the following. Let Xg E TgG be an arbitrary tangent vector, then x = T L g-1 Xg E TeG == 9 satisfies by definition Xg = X g. It follows immediately that L(Xg)8 MC = x. Identifying 9 with the set of left-invariant vector fields on G, we conclude that L(Xg )8 MC is the left-invariant vector field on G whose value at g is the given tangent vector Xg E TgG. We also deduce that 8 MC can be defined by

L(Xg)8 MC = TL g-1Xg. We know that 8 MC is left-invariant, i.e., using the generalized pull-back we have MC for all g E G. To see its behavior under right translations, we first note that by definition we have R;8 MC = Li R;w i 0 Vi. And then we compute:

L;8 MC = 8

L(Xh)R;Wi = L(TRg TLhx)wilhg = L(TLhg TL g-1 TRgx)wilhg = L(Ad(g-l)x)w i . It follows that L(X)(R;8 MC ) = Ad(g-1)(L(X)8 MC ). In other words, using the notation of [1.3], we can write R;8 MC = Ad(g-l) o8 MC . Let us now consider the special case G = Aut(E) with 9 = EndR(E) for some A-vector space E. As explained in [VI. 1.20], we use the basis ei 0 e j for EndR(E). Using the left coordinates gij = eMR(g)i j , i.e., g = Li,j gij ei 0 e j , the Maurer-Cartan form can be written as

8

MC

=

L

dg P q >..~~ 0 er 0 e

S

p,q,r,s

with coefficients >.. that have to be determined. For X E 9 the corresponding leftinvariant vector field X is given by Xg = Li,j,k Xij gk i 8gk j [VI. 1.20]. The condition

(( X I 8 MC))

=

X thus leads to the equations "" Xi j gk i Aks \ir er L i,j,k,r,s

. '\" \ir 1.e., .LJi ,J. ,k Xi j gk i /lks

=

Xr" s lor a11'), r.

,0,

'C/

es = " L " xr s er

,0,

'C/

eS ,

r,s

Since this must be true for all X, we deduce

340

Chapter VII. Connections

that Lk gk i At = 5[ 5~, i.e., At = (g-l)r k 5~ (use (VI. 1.22)). We find for the MaurerCartan form 8 MC : 8 MC = L dg P q (g-lr P 0 er 0 eq = L(g-l 0 dgr q 0 er 0 e q = g-l 0 dg , q,r p,q,r where for the second equality we used (VI. 1.22) and the fact that gi j (and thus dg i j) has parity C(ei) + c(ej). To interpret the last equality, we note that the canonical inclusion Aut(E) ---> EndR(E), g r-+ g can be seen as an EndR(E)-valued O-form (function) on Aut(E), as can be the map g r-+ g-l, Aut(E) ---> EndR(E). The l-form dg thus is an EndR(E)-valued l-form, exterior derivative of the O-form g. The composition g-l 0 dg is the wedge composition of the O-form g-l with the l-form dg, where as usual we have omitted the wedge symbol because the first factor is a O-form [V.7.l], [1.3]. We now go back to the general case and we look at the exterior derivative of the Maurer-Cartan form d8 MC = Li dw i 0 Vi, which is aleft-invariant g-valued 2-form on G. Since the contraction of a left-invariant l-form with a left-invariant vector field is a constant [VI.S.17], the formula for the exterior derivative [Y.7.6] gives us -~(iJi,0)dwk

=

-~([iJi,iJj])wk

=

-C~j ,

where we have used the (real) structure constants C~j of 9 [VI. 1.16]. If we now consider the left-invariant 2-form ()k = ~ Lpq C;qwq A wP (beware of the order of the indices), we can compute q q ~(iJ· iJ)()k = ~(iJ·)(~(iJ)()k) = 1. ' " ck (5 5P _ (_1)(E( vi)I€(vj»5 5P ) = ck. " J 'J 2 L pq J , 'J 'J ' pq where we used that CJi = -( _l)(€(vi)I€(vj»c~j due to graded skew-symmetry of the bracket on g. Since the values of a basis ofleft-invariant vector fields generate the tangent space at each point [VI.1.1S], we deduce that dw k = ~ Lij c']iwi Aw j . In terms of d8 MC this gives us the formula

d8 MC

= ~

Lw i Aw j 0CJiVk ijk

= -~

Lw i Awj 0 (-l)(E(vi)I€(vj»[vi,Vj]. ij

Comparing this expression with the definition of the wedge Lie bracket shows that we can write this equality as d8 MC = -~ [8 MC Ii- 8 MC ] (remember that c( Vj) = c(w j )); it is called the structure equations of G.

1.6 Lemma. The Maurer-Cartan I-form 8 MC on an A-Lie group G satisfies the equation d8 MC = -~[ 8 MC Ii- 8 MC ].

Proof The proof of this result has already been given in [1.5]. Here we give another proof using [104] and (V.7.6). For homogeneous x, y E 9 we have the equalities

-*f,y)d8MC

= X(~(YJ8MC)

- (-1)(€(x)I€(Y»Y(~(x)8Mc)

- L([x,y])8 MC =

-~(

-------;

[x,y] )8 Mc = -[x,y]

and L(X, Y)[ 8 MC Ii- 8 MC ] = -2[ ~(X)8MC, ~(Y)8MC] = -2[ x, y].

§2. Ehresmann connections and FVF connections

2.

EHRESMANN CONNECTIONS AND

341

FVF

CONNECTIONS

In this section we introduce the notion ofan Ehresmann connection on an arbitrary fiber bundle and we show that there is a natural way to transport an Ehresmann connection to a pull-back bundle. We then introduce the more restrictive notion ofan FVF connection, which can be described by local I jorms r a with values in the A-Lie algebra of the structure group of the fiber bundle. We show that transporting an FVF connection to a pull-back bundle still gives an FVF connection and that it is described by the pull-backs ofthe local I-forms ra. Wefinish by showing that an FVF connection is integrable ifand only ifthe local 2jorms d r a + ~ [ra Ii- r a1 are all identically zero.

2.1 Discussion. If A and B are two sets, a function f : A ---> B is constant if and only if the image f (A) consists of a single point: f (A) = {b}. If we have a differentiable structure, f is locally constant if and only if its tangent map T f is zero [y'3.2l]. Thinking in terms of bundles, these elementary facts obtain a new formulation. If 7r : A x B ---> A denotes the projection on the first factor, there is a bijection between functions f : A ---> B and sections s : A ---> A x B of the (trivial) bundle 7r : A x B ---> A, the identification given by s( a) = (a, f (a)). In the direct product A x B it is customary to call the subsets {a} x B vertical and the subsets A x {b} horizontal. The reason for this choice is that it is customary to draw the target space A as a horizontal line and the source space A x B as a rectangle above it. Given the map 7r, the vertical subspaces {a} x B can also be described as 7r- 1 (a). If A and B are A-manifolds, we can also talk about vertical and horizontal directions: a tangent vector (X, Y) E TaA x nB ~ T(a,b)(A x B) [Y.2.2l] is vertical if X is zero, it is horizontal if Y is zero. Again using the map 7r, the vertical directions can be described as those tangent vectors that map to zero under T7r. The set of all horizontalJvertical directions forms a foliation whose leaves are the horizontal! vertical subsets (as long as they are submanifolds). In terms of these definitions, a section s : A ---> A x B is constant if and only if its image is a horizontal subset; it is locally constant if and only if its tangent map T s maps vectors X E TaA to horizontal vectors. Under the identification s(a) = (a, f (a)) this corresponds exactly to constant and locally constant functions f : A ---> B. We now generalize the above picture to a fiber bundle 7r : B ---> M with typical fiber F. Above a local trivializing chart U c M the bundle is isomorphic to the direct product U x F with projection on the first factor. As such we can speak about horizontal and vertical subsets and about horizontal and vertical directions. And as before, the vertical subsets can be described as 7r- 1 (m) and the vertical directions as those tangent vectors that map under T7r to zero. Obviously the notion of a vertical direction does not change when we change the local trivializing chart; it can be described intrinsically by the projection map 7r. On the other hand, there is no reason to think that what is horizontal in terms of one trivialization remains horizontal in another trivialization. Said differently, a local section s E ru(B) can be constant in one trivialization and non-constant in another, i.e., the notion of a (locally) constant section is not well defined. The purpose of a connection is to give a definition of what directions will be called horizontal, and thus what sections

342

Chapter VII. Connections

will be called (locally) constant. The fact that one concentrates on horizontal directions instead of horizontal subsets (submanifolds) is because the latter is too restrictive a notion. If the set of horizontal directions is involutive, we can recover the horizontal submanifolds by means of Frobenius' theorem. However, the set of horizontal directions need not be involutive at all. And indeed, the concept of curvature, which measures more or less the lack of involutivity of the horizontal directions, plays an important role in differential geometry and physics.

2.2 Definitions. On any fiber bundle 7r : B ---; M (locally trivial, with typical fiber F) we have the vertical sub bundle VeT B, which is defined as the kernel of T7r: V = ker(T7r). Its elements are called vertical (tangent) vectors. An Ehresmann connection 1-l on B is a subbundle of 11. C T B which is a supplement to V, i.e., 1-l ED V ~ T B [IVA.6]. A different way to characterize an Ehresmann connection is to require that for all b E B the map T7r : Hb ---; Trr(b)M is a bijection. Since T7r is an even linear map, this implies that Hb is isomorphic to Trr(b)M. Elements of 11. are usually called horizontal (tangent) vectors. An Ehresmann connection 11. on a bundle B automatically defines a projection h : TbB ---; Hb as the H-part in the direct sum TbB = Vb ED H b ; h(Y) is called the horizontal part of the tangent vector Y E TbB. Since the map n7r : Hb ---; TmM with m = 7r(b) is an isomorphism, the inverse map (Tb7r)-l : TmM ---; Hb exists. For any X E TmM the image X h = (Tb7r)-l(X) E Hb is called the horizontal lift of X at b. Similarly, if X is a vector field on M, its horizontal lift X h is the (unique) vector field on B such thatXr is the horizontal lift ofXrr(b) atb. In the context of connections some terminology changes: a (smooth) map f : N ---; B is said to be horizontal if it is tangent to 11. [V.6A]. In particular a (local) section s of the bundle B is horizontal ifTs(TmM) = Hs(m) for all m in the domain of definition of s, and a submanifold C C B (with its canonical injection) is called horizontal if TC c H. The connection 11. is said to be integrable or flat if the subbundle 11. C T B is integrable [Y.6.2].

2.3 Proposition. Let (U, 1jJ) be a local trivializing coordinate chartfor the fiber bundle

7r : B ---; M with coordinates Xi, i.e., in particular, 1jJ : 7r-l(U) ---; U x F is a diffeomorphism. If 11. is an Ehresmann connection, then there exist unique smooth functions 'Yi : U x F ---; T F, 'Yi (m,f) E Tf F with Chi) = c( xi) such that the restriction 11. Irr- 1 (U) in terms ofthe trivialization 1jJ is spanned by the dim M tangent vectors (2.4)

Conversely, ifa subbundle 11. C T B is spanned on local trivializing charts by dim M vectors oftheform (2.4), then 11. is an Ehresmann connection on B. Proof In terms of the trivialization, the projection map 7r : B ---; M is given as projection on the first factor 7rl : U x F ---; U. Since T7rl : H(m,f) ---; T mM is a bijection by definition of an Ehresmann connection, it follows that H(m,f) is spanned by vectors

§2. Ehresmann connections and FVF connections

343

ax'

of the form 1m - 'Yi(m,1) for uniquely determined functions "Ii U X F ---> TF, 'Yi(m, I) E TfF. To prove that these "I are smooth, we argue as follows. By definition of a subbundle, 11. is locally spanned by smooth vector fields Xi in a neighborhood of (m, I). Using the decomposition T(U x F) = TU x TF, each Xi can be written as Xi = Lj Xi for smooth functions and smooth maps Xi : U x F ---> T F

xi axj xi with Xi = Lj xi 'Yj. By definition of an Ehresmann connection the matrix xi must

be invertible. Hence, at least in a (small) neighborhood of (m, I), there exists a smooth inverse to it. This shows that "Ii is smooth in such a (small) neighborhood of (m, f). But smoothness is a local property and thus the "Ii are smooth on the whole of U. To prove the converse, it suffices to note that the conditions guarantee that it is locally generated by smooth independent vector fields, showing that 11. is a well defined subbundle, and that T7rl is a bijection between HI (m,f) and TmM, proving that it satisfies the condition for an Ehresmann connection. IQEDI

2.5 Proposition. Let 7r : B ---> M be a fiber bundle, let 11. be an Ehresmann connection on B, and let g : N ---> M be a smooth map. Using notation as in [lV2.2], there exists a unique Ehresmann connection g*H on the pull-back bundle g* B such that for all C E g* B we have Tg (g*1-l)c C Hg(c).

Proof In [IV.2.2] we have seen that for any trivializing atlas U for B, there exists a trivializing atlas V for g* B and an induced map (also denoted by g) from V to U such that the transition functions of g* B are given by 1/J g (a)g(b) 0 g when the 1/Ja(3 are the transition functions of B associated to the atlas U. Moreover, in these trivializations, the induced fiber bundle map 9 : g* B ---> B takes the form (n, I) r--+ (g( n), I). Now let yj be local coordinates on Va E V and let xi be local coordinates on Ug(a) E U. For any connection it. on g* B there exist local functions 'Yj : Va X F ---> T F such that it. is generated in the trivialization of g* B determined by Va by the vectors

The map T 9 maps these vectors to the vectors

in the trivialization of B determined by Ug(a). If we require that this image lies in which is generated by the vectors ax; Ig(n) + 'Yi(g(n), I), then we must have

H(g(n),j),

_ 'Yj(n, I)

=

L,

ag i ayj (n) . 'Yi(g(n), I) .

We conclude thatthe 'Yj are uniquely determined by the condition Tg (g*H)c C Hg(c) and that they are indeed smooth. Since the condition is independent of the local trivialization, we conclude that this condition determines a unique Ehresmann connection g*H on g* B.

IQEDI

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Chapter VII. Connections

2.6 Discussion. A special case of a pull-back bundle is the restriction of B to a submanifold N of M [IV.2.3]. We thus see that [2.5] tells us in particular that an Ehresmann connection 11. on B induces a unique Ehresmann connection on the restriction BIN = 7r- 1 (N) of B to the submanifold N. It is not hard to see that this induced connection is just the restriction 11.lrr-1(N) of11. toBIN. Slightly more general is the case of an immersion 9 : N ---> M, in which case again we have an induced Ehresmann connection on g* B. This application of [2.5] plays a fundamental role in the concept of parallel transport along a curve in M. Without going into details because that is outside the scope of this book, we briefly sketch the idea of parallel transport. We first note that a curve in M is an immersion 9 : N ---> M of a I-dimensional connected A-manifold N in M. Given a fiber bundle 7r : B ---> M we have a pull-back fiber bundle g*7r : g* B ---> N, and if'H is an Ehresmann connection on B, we have an induced Ehresmann connection g*11. on g* B. We now fix no E BN and bo E BBg(no). Parallel transport of bo along the curve is a smooth horizontal map pt : N ---> B with pte n) E Bg(n) and pte no) = bo, and thus in particular Tpt maps TnN into 11.pt(n), i.e., the vectors tangent to pt(N) are horizontal. A sufficient condition for such a map to exist (and then it is unique) is that g*11. is integrable and that its leaves are diffeomorphic to N via the projection map g*7r : g* B ---> N. Since N is I-dimensional, the induced Ehresmann connection g*11. on g* B is a I-dimensional subbundle ofT(g* B), and thus the integrability condition is automatically satisfied when N is even, i.e., of dimension 110. Given these conditions, the map pt is constructed as follows: since g*11. is integrable, there exists a leaf L passing through g-l(b) E (g* B)n (the map 9 is a diffeomorphism between fibers). Since this leafis diffeomorphic with N via g*7r, we can define pt = go (g*7rIL)-l. This map satisfies the given requirements. That the integrability condition alone is not sufficient is shown in the following elementary example. We take M = Ao with the global even coordinate x and the trivial bundle B = M x Ao == (Ao)2 with the global even coordinates (x, y) and the obvious projection 7r : B ---> M. On B we define the Ehresmann connection 11. by

As curve we choose the canonical embedding M ---> M, i.e., we see M itself as a curve in M. Now suppose that pt : M ---> B is a parallel transport map. Then it must be of the form pt(x) = (x, f(x)) for some smooth function f. And then the condition that Tpt maps TxM into 11.pt(x) implies that f must satisfy the condition (axf)(x) = - f(x)2 because Tpt maps ax to ax + (axf) (x) ay. This implies that f is of the form f(x) = (x - C)-l for some c E R. But such a map is not defined on the whole of M, and thus parallel transport along the whole of this curve does not exist (see also [5.10]).

2.7 Remark. In [Eh] C. Ehresmann introduced his general notion of a connection on an arbitrary fiber bundle. The definition he gave is slightly stronger than that of what here is called an Ehresmann connection. He added the requirement that parallel transport should always be defined. The underlying idea is that parallel transport provides an alternative

§2. Ehresmann connections and FVF connections

345

way to define a connection. Since this approach becomes highly unwieldy in the case of A-manifolds, we here only sketch the procedure in the case of R-manifolds, neglecting all questions about smoothness. Let g : [x, y] ---; M be a curve, let b E Jr-l(g(x)) be arbitrary and let 9 : [x, y] ---; M be a horizontal map satisfying Jr 0 9 = g and g(x) = b. Since 9 is uniquely determined by b, we obtain, by varying b, a well defined map Fg : Jr-l(g(X)) ---; Jr-l(g(y)), b = g(x) f--+ g(y). Running through the curve gin the opposite direction shows that Fg is bijective. Taking the derivative with respect to y at y = x gives us back the tangent vector of 9 at b. If parallel transport over all curves exists, we thus can recover the set of horizontal directions at b, i.e., we can recover the (Ehresmann) connection. This analysis also shows that to any curve g we have associated a diffeomorphism Fg between the fibers over the endpoints. Since in general there is no canonical way to compare, in a fiber bundle, fibers over different points, these diffeomorphisms are a useful tool when one wants to do so. The idea of comparing different fibers in this way is one of the main motivations for the introduction of a connection in the form of parallel transport. In the context of A-manifolds we have ignored this approach to a connection because not all points in a connected A-manifold can be connected by a smooth curve.

2.8 DiscussionlDefinition. The maps "Ii in (2.4) depend, obviously, upon the trivializing set U. Most, if not all, types of connections are special cases of an Ehresmann connection, eventually in disguise. They use special features of the bundle to impose restrictions on the form of the maps "Ii. We will restrict our attention to one special form of these maps, a form that will be sufficiently universal to cover all our examples. The idea is quite simple. We have a typical fiber F with a (pseudo effective) left action of the structure group G. We thus can require that all "Ii are fundamental vector fields associated to this action. More precisely, we define an FVF connection (for Fundamental Vector Field) to be an Ehresmann connection such that for each ma E M there exists a trivializing coordinate chart (U, 1/J) containing ma such that the map f f--+ "Ii (m, f) E Tf F is a fundamental vector field for all m E U. The next results show that the notion of being a fundamental vector field has nice smoothness properties and is independent of the choices that can be made.

2.9 Lemma. Let (U, 1/J) be a trivializing coordinate chart with coordinates (xi) such that the map f f--+ "Ii(m, f) E TfF is afundamental vectorfieldfor all m E U. Then there exists a unique map Ai : U ---; 9 such that "Ii(m, f) = Ai (m)f. Moreover, the map Ai is smooth. Proof By definition of a fundamental vector field, there exists an Ai (m) E 9 for each m E U such that "Ii (m, f) = Ai (m) It thus remains to prove that the map Ai is unique and smooth, which will be a consequence of the fact that the structure group acts pseudo effectively. We start with smoothness. In [2.3] we have seen that the maps (m, f) f--+ "Ii(m, f) == Ai(m)f are smooth. With respect to a basis (Vj) for 9 the map Ai takes the form Ai (m) = L j d (m) Vj for functions

f.

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Chapter VII. Connections

cj : U ----; A. Since the map TiJJ f is left linear, we have Ai(m).f = Lj c1(m) (Vj).f. With respect to a local coordinate system on F with coordinates (y, ry), each fundamental vector field (Vj)F has the form (Vj).f = L~~ F Ej (I) fA. Smoothness of Ai being equivalent to smoothness of the coefficients, we thus know that Lj c1 (m )Ej (I) is smooth for all k, and we want to show that this implies that all c1 are smooth. Since the Ej are smooth, we have Ej(y, ry) = LJ EJ,J(y)ryJ [111.3.17]. Taking derivatives with respect to the ry's of the smooth functions LJ(Lj c1 (m)Ej,J(Y) )ryJ, we obtain in particular that Lj c1 (m)EJ,J(Y) is smooth for all k and 1. Following the proof of [VI.6.6] we introduce the functions rP~(y) = LjjV' EJ,J(Y) E *g for real values of the (even) coordinates y. Varying also the local coordinate charts, we know from the proof of [VI.6.6] that there are £ :::; dim 9 independent elements rP~~ (yd, ... , rP~! (YR.), such that all other elements rP~ (y) are linear combinations with real coefficients of these £ elements. Since these independent elements define g,pO, and since g,pO = {O} [VI.6.l0], we conclude that £ = dim g. Changing the basis of 9 if necessary we may assume that the rP~: (Yr ) form the left dual basis, i.e., rP~: (Yr) = rv. It follows that E;''Jr (Yr) = 5j, and thus .

k

cr(m) = Lj c1(m)Ej,'JJYr) is smooth. To prove uniqueness ofthe Ai, suppose that A~ is another solution. Then Ai - A~ is a smooth family for which the associated fundamental vector field is identically zero. Since g,pO = {O}, this implies that Ai = A~. IQEDI

2.10 Lemma. Let (Ua , 1/Ja) be a trivializing coordinate chart with coordinates xi and let A~'x : U ----; 9 be smooth maps such that the Ehresmann connection 1-l is spanned in the trivialization 1/Ja by Oxilm +A~,X(m).f E TmM x TfP. j Ify is another system of coordinates on Ua, then 11. is also spanned in the trivialization 1/Ja by Oyj 1m + A~'Y(m).f E TmM X TfP ,

with Aj'y : U ----; ggiven by Aj'Y(m) = Li(Oyjxi)(m) . A~,X(m). If(Ub,1/Jb) is another trivialization with Ua = Ub, then 11. is spanned in the trivialization 1/Jb by 0Xi 1m + A~'x (m).f E TmM X TfP , with A~'x : U ----; 9 given by

A~,X(m) = Ad(1/Jba(m)) (A~,X(m) - TL,pba(m)-' T1/Jba ax; 1m) (2.11 )

= Ad(1/Jab(m)-l) A~,X(m) + TL,pab(m)-' T1/Jab 0Xi 1m ,

where 1/Jba : Ua = Ub ----; G is the transition function related to the change of trivialization from 1/Ja to 1/Jb. Proof The first part is a direct consequence of the fact that the tangent map is left linear: .

k

O~i 1m + TiJJf A~,X(m) = ~ ~~~ (m) (o~j 1m + TiJJ f (~ ~~j (m) A~'X(m)))

,

§2. Ehresmann connections and FVF connections

347

87.

because Lj axi yj . ayj xk = Since the matrix aXi yj is invertible, the result follows. To prove (2.11), we first note that (1/Jb ° 1/J;; 1)(m, f) = (m, ( 1/Jba (m), f)) by definition of the transition function 1/Jba : U ----; G. We then compute the image of the tangent vector ax, 1m + A~'x (m)r under the map 1/Jb ° 1/J;;1 :

T( 1/Jb ° 1/J;;1) (axi 1m + A~'x (m)n

= aXi 1m + T f T1/Jba aXi 1m + T,pba(m) A~'x (m)r

+ T,pba(m) (A~'X (m)r + T f T L,pba(m)-l T1/Jba axi 1m) axi 1m + T,pba(m) (A~,X(m) - TL,pba(m)-l T1/Jba aXi Im)~

= axi 1m

=

= axi 1m + (Ad(1/Jba(m)) (A~'X(m) - TL,pba(m)-l T1/Jba axi Im)):(,pba(m),f) , where the second equality follows from the equality f 0 L,pba(m) = ,pba(m)of as maps from G to F. This proves the first equality of (2.11). The second equality is obtained by interchanging the roles of a and b and using that 1/Jba (m) ·1/Jab( m) = e for all m E Ua

IQEDI

= Ub •

2.12 Discussion. We learn from [2.9] that being an FVF connection can be expressed in terms of (local) smooth maps with values in g. And then [2.10] tells us that the notion of being a fundamental vector field on a trivialization is independent of the chosen trivialization as well as the chosen coordinate system. Moreover, the explicit dependence of A;,y(m) in terms ofA~,X(m) also shows that the I-formr a with values in 9 on Ua defined as ra(m) = dx i 0 A~,X(m) = dyj 0 A;,y(m)

L

L j

is independent of the chosen coordinate system. This implies that if (Ua, 1/Ja) is a local trivialization, there exists a g-valued I-form r a on Ua such that, if Xi are coordinates on (a part of) Ua, the local functions A~'x can be recovered from r a by

The existence of r a is independent of whether there exists a global coordinate system on Ua or not, and 11. is given in the trivialization 1/Ja by

2.13 Corollary. Let 7r : B ----; M be a fiber bundle with typical fiber F and structure group G, and let ={ (Ua, 1/Ja) I a E I} be a trivializing atlas for B. If11. is an FVF connection on B, there exist unique g-valuedl -forms r a on Ua such that 11. is given in the trivialization 1/Ja by

(2.14)

348

Chapter VII. Connections

Moreover, on overlaps Ua n Ub the 110rms r a and rb are related by (2.15)

where 8 MG is the Maurer-Cartan 110rm on G. Conversely, ifwe have g-valued 110rms r a on Ua that are related on overlaps by (2.15), then (2.14) defines an FVF connection 11.. Proof IfH is given, (2.14) is a direct consequence of [2.12]. To prove (2.15) we choose (local) coordinates (Xi) on Ua nUb' Using (2.11) and [1.3] we obtain

rb(m) =

L dx

i

0 A~(m)

Ldx 0 (Ad(1,bab(m)-l)A~(m) +TL,pab(m)-l T1,bab oxi lm) = Ad(1,bab(m)-l)ora(m) + Ldx 0TL,pab(m)-l T1,babOxilm. i

=

i

To prove thatthe second term equals And then we use [1.5] to compute

1,b~b8 MG,

o

we first note that T1,bab xi 1m E T,pab(m)G.

Im)8 MG = T L,pab(m)-l T1,bab Oxi 1m . we always have a = 2:i dx i 0 L( 0xi )a, (2.15) follows.

~(Oxi Im)1,b~b8 MG = ~(T1,babOxi

Since for any I-form a To prove the converse, we first note that (2.14) obviously is the local expression of an FVF connection. It only remains to be shown that these local expressions coincide on IQEDI overlaps Ua nUb. But this is an immediate consequence of [2.10], (2.11).

2.16 Corollary. Let 7r : B ---> M be a fiber bundle, let H be an FVF connection on B, and let g : N ---> M be a smooth map. Then the unique Ehresmann connection g*1-l on the pull-back bundle g* B such that for allc E g* B we haveTg (g*H)c C Hg(c) [2.5J is also an FVF connection. In particular, if U and V are (trivializing) atlases as in the proof of[2.5 J, then g*H is determined by the local g-valued 110rms fa = g*rg(a) [2.13 J.

Proof Since H is an FVF connection, the maps 'Yi are given as 'Yi(m, f) = Ai(m)f for g-valuedfunctions Ai. Using the arguments and notation as in the proof of [2.5], g*H is determined by the functions "fj given by

"fj(n, f) =

ogi (Ogi )F L, oyj (n) . Ai(g(n))f = L oyj (n) . Ai(g(n)) f ,

This proves that g*H is an FVF connection. In terms of the g-valued I-forms associated to the trivializing atlases, we find:

r_a

=

"'"' ogij (n) . Ai(g(n)) = " , " , ' 0 Ai(g(n)) = g*rg (a) , L dyJ. 0 "'"' L a Lg*dx' j

i

Y

i

where rg(a) = 2:i dXi 0 Ai(m) is the g-valued I-form on Ug(a) associated to the connection H on B. IQEDI

§2. Ehresmann connections and FVF connections

349

2.17 Remark. In the context of general Ehresmann connections one could wonder why it is so easy to define a pull-back connection, because an Ehresmann connection is an object living on the tangent bundle, and for tangent vectors the notion of pull-back is not (directly) defined. A possible explanation is that the combination Li dxi 0 "Ii is independent of the chosen coordinate system, and for i-forms we do have a natural notion of pull-back. But a precise definition of this object is not easy. However, in the context ofFVF connections, the notion of a pull-back connection becomes natural. Such a connection is defined by local g-valued i-forms. And, as we have seen, the pull-back of these i-forms defines the pull-back connection.

2.18 Proposition. Let 7r : B ---> M be a fiber bundle with typical fiber F and structure group G. Let U = {Ua I a E I} be a trivializing atlas for B and let r a be g-valued I-forms on Ua defining an FVF connection 11. according to [2.13]. Then 11. is integrable ifand only if(all) the local2-fonns

dr

a

+ ~ [r a Ii- r a 1

are identically zero. Proof We have to show that 11. is an involutive subbundle. Since the value ofacommutator

of two vector fields at a point depends only upon the behavior of the two vector fields in a neighborhood of the given point, it suffices to verify that 11.17r-1(ua) is involutive for all a E I. But on 7r- 1 (Ua ) the connection 11.(m,f) is spanned by the vector fields 0Xi 1m + A~'x (m).f [2.12]. Using [V.1.19] it follows that it suffices to show that the commutator of two of these generating vector fields belongs to 11.. To compute such a commutator, we first compute the commutator [oxi , A F 1 for a smooth map A : Ua ---> g. To do so, we choose a basis Vj for 9 and (local) coordinates yk on F. It follows that there exist smooth functions Aj : Ua ---> A such that A = Lj Ajvj. There also exist smooth functions ~k : F ---> A such that (Vj)F = Lk ~k Oyk. And then we compute

jk

jk

We thus find for the commutator of two generating vector fields: [oxi

+ (A~,X)F , ox j + (A~'X)F 1 = (oxiA~,X)F _ (_l)(€(xi)I€(xj))(oxjA~'X)F

+ [A~'X,

A~,xlF ,

where we used [VI.S.2-iii]. Since this commutator is tangent to F, it projects to zero on Ua under the projection T7r. Since T7r is a bijection from 11.(m,f) onto TmM, the condition that this commutator belongs to 11. becomes the condition that this commutator must be zero (for all i and j). Looking at the definition of g,pO [VI.6.2], it follows that the image of the smooth functions Fi~'x : Ua ---> 9 defined by Fi~,X(m)

= oxiA~'x

- (_l)(€(xi)I€(xj))oxjA~'X

+ [A~'X,

A~,xl

Chapter VII. Connections

350

belongs to g1l>o. Since the action of G on F is pseudo effective, it follows from [VI.6.1O] that 11. is integrable if and only if all functions Ft/ are identically zero. On the other hand, using [Y.7.6], [104] and ~(Oxi )fa = A~'x [2.12], we find -~(oxi,oxj)(dfa+~[faf,\fa]) =Fi~'x.

Since any 2-form (3 on Ua can be reconstructed from its contraction with the 0Xi by 1 . i (3 = 2" Lij dx J 1\ dx . ~(OX" Ox j )(3, the result follows. IQEDI

3.

CONNECTIONS ON PRINCIPAL FIBER BUNDLES

In this section we show that an FVF connection on aprincipal fiber bundle can be described either as the kernel ofa so called connection I -form w on the bundle or as an Ehresmann connection that is invariant under the right action of the structure group on the bundle. We also show how the connection I -form w can be reconstructed from the local I forms fa defining the FVF connection. The description in terms ofa connection I form allows us to prove quite easily that there always exist FVF connections on a principal fiber bundle.

3.1 Lemma. Let 7r : P ---; M be a principal fiber bundle with structure group G and let U c M be open. Let T denote the map which associates to each local trivialization 1jJ : 7r-l(U) ---; U x G of7r- 1 (U) the local sections E fu(P), s(m) = 1jJ-l(m, e) with e E G the identity element. Then T is a bijection between the set ofall local trivializations of 7r- 1 (U) andfu(P). The inverse of T is given by the formula1jJ-l( m, g) = s(m) . g.

Proof If s is given as s( m) = 1jJ-l (m, e), then by definition of the right action of G on P we have 1jJ-l(m,g) = s(m) . g. Hence the given formula is a left inverse for T. To show that it also is a right inverse and that it indeed defines a local trivialization, we suppose that s E fu(P) is a local smooth section. We then define the smooth map U x G ---; 7r-l(U) by w(m, g) = s(m)· g. If (V, X) is any local trivializing chart for P, we obtain a map Sx : Un V ---; G such that (X ° s)(m) = (m, sx(m)) [IV. 1.20]. It follows that X ow: U n V x G ---; U n V x G is given by

w:

(3.2)

(xow)(m,g) = (m,sx(m) .g).

From this one deduces that Wis bijective, a bundle morphism and a local diffeomorphism. Hence 1jJ = w- 1 is also smooth. And then (3.2) shows that 1jJ is compatible with the IQEDI structure of the principal fiber bundle, i.e., (U, 1jJ) is a local trivialization.

3.3 Corollary. A principal fiber bundle 7r ifand only iff(P) is not empty.

:

P ---; M is (isomorphic to) the trivial bundle

Proof This is the special case U = Min [3.1].

351

§3. Connections on principal fiber bundles

3.4 Discussion. In [3.1] we have established a bijection between local sections and local trivializations of a principalfiber bundle 7r : P ---> M. We are thus allowed to speak of the (local) trivialization 1/J determined by the (local) section s. Now suppose that Sa E rUa(P) and Sb E rUb (P) are two local sections determining two local trivializations 1/Ja and 1/Jb. These two trivializations determine a transition function 1/Jba : Ua nUb ---> G by the formula (1/Jb 0 1/J;;1) (m, g) = (m, 1/Jba (m)g). Applying 1/Jb"1 and substituting the definition of 1/Ja and 1/Jb in terms of Sa and Sb gives us (3.5)

This formula gives us a way to determine the transition functions directly from the defining local sections: given Sa and Sb there exists for each m E Ua n Ub a unique 1/Jba (m) E G such that Sa (m) = Sb( m) ·1/Jba (m ),just because the right action of G on the fibers of P is free and transitive. (3.5) then tell us that it must be the transition function determined by the two associated trivializations 1/Ja and 1/Jb.

3.6 Remark. In the physics literature a local section of a principal fiber bundle is often called a (local) gauge and changing a local section S to a local section by the formula s(m) = s(m) . cjJ(m) is called a (local) gauge transformation.

s

3.7 Definitions. Let 7r : P ---> M be a principal fiber bundle with structure group G whose A-Lie algebra is g. In a local trivialization 7r-l(U) S:! U x G the vertical subbundle V is just the tangent space to the second factor TG. It follows that V is spanned by the left-invariant vector fields on G. Since the right action of G on P corresponds to right multiplication on the second factor and since the fundamental vector fields of right multiplication are exactly the left-invariant vector fields [VI.SA], we conclude that V is spanned by the fundamental vector fields of the right action of G on P, independent of the choice of a local trivialization. More precisely, Vp = {x: I x E 9 }. • An FVF connection on a principal fiber bundle P is called a principal connection, or simply a connection. Contrary to the vertical directions, the fundamental vector fields used in the definition of a principal connection on P are right invariant because the action of the structure group G on the typical fiber G is left multiplication . • A connectionl-formon the principal fiber bundle P is an even g-valued I-form won P satisfying the following two conditions. (i) "Ix E 9 : L(XP)W = x. (ii) Vg E G: ;w = Ad(g-l) ow.

Since the meaning of (i) is rather obvious, we concentrate on the precise meaning of (ii). On the left hand side ;w indicates the generalized pull-back as defined in [V.7.23], [VI.S.12]. On the right hand side we have an example of the alternative notation [1.3] for the evaluation-wedge product of an EndR(g)-valued O-form with a g-valued I-form. Writing all definitions explicitly, condition (ii) says that for all (p, g) E P x G and all Xp E TpP we must have (ii)

L({(Xp,Qg) I T(p,g)
352

Chapter VII. Connections

3.8 Lemma. An Ehresmann connection 11. on a principal fiber bundle 7r : P ----; M with structure group G is a principal connection on P if and only if 11. is invariant under the right action ofGon P, i.e.,forall g E G: Tg11.p = 11.pg. Proof In a local tri vialization (U, 'IjJ) a principal connection on P is given in terms of the maps "Ii by [2.9] "Ii(m, g) = Ai(m)~ ,

for smooth maps Ai : U ----; g. Since Ai(m)~

=

-TRgAi(m) [VI.S.l], we conclude that

where now Rh denotes the natural right action of h E G on U x G [IY.2ol2]. Since this action corresponds to the natural right action of G on P, we conclude that Th 11.p = 11.ph. Conversely, if 11. is invariant under the G-action, it follows that

In particular we must have "Ii(m, g) = T Rg"li( m, e). Denoting Ai(m) = -"Ii(m, e) we conclude that 11. is an FVF connection, i.e., a principal connection on P. IQEDI

3.9 Lemma. Let7r: P ----; M be aprincipalfiber bundle with structure group G whose ALie algebra is g. If w is a connection Ilorm on P, then ker( w) is a principal connection on P. Conversely, if 11. is a principal connection on P, then there exists a unique connection I-form won P such that 11. = ker(w).

= {Y E TpP I ~(Y)wp = O}. Our first goal will be to prove that ker(w) is a (smooth) subbundle of TP and a supplement to V. We can interpret w as an even (because w is even) left linear map from the bundle T P ----; P to the trivial bundle P x 9 by Y E TpP r--+ ~(Y)wp. By condition (i) of a connection i-form this map is surjective. We thus satisfy the conditions of [IV.3olS], proving that ker(w) is a subbundle of TP. For any X E Vp there exists E 9 such that X = By condition (i) ofaconnection i-form we have ~(X)w = x, and thus Vnker(w) = {O}. On the other hand, for any Y E TpP we can define = ~(Y)w E g, and then Y E ker(w)p. This proves that V ED ker( w) = T P. Condition (ii) of a connection i-form implies that for a tangent vector Y E TpP we have

Proof Notethatker(w) isdefinedasker(w)p

x

x

x:. x:

It follows that Tg(ker(w)p) C ker(w) 9p. Since Tg is a bijection, we conclude that ker( w) is invariant under the G-action and thus is a principal connection on P [3.S]. For the converse, let 1-l be a principal connection on P. This implies that we have TpP = Vp ED 11.p. Since V is generated by the fundamental vector fields of the right action of G on P, condition (i) of a connection i-form together with 11. = ker(w) completely determine w, provided it exists. We thus define the i-form w by ~(1-lp)wp = 0

§3. Connections on principal fiber bundles

353

and for x E 9 : ~(x:)Wp = x. In this way w is a well defined g-valued I-form on P satisfying condition (i) of a connection I-form. To show that this w is smooth, we invoke [IV.S.9], [IY.S.1I] which says that w is smooth if it maps smooth sections to smooth sections. Now any smooth section of TP is the sum of a smooth section of

11. and a smooth section of V. Our w maps (smooth) sections of 11. to zero, which is smooth. If (Vi) is a basis for g, the corresponding v[ form a basis of the (globally) smooth sections of V. Since the Vi are real, their image under w is smooth (a real constant). And thus w is smooth. To prove condition (ii) of a connection I-form, we first suppose that X E 11. p , which implies (by invariance of 11. under the G-action) that ~(Tq,gX)wpg = 0 = Ad(g-l )~(X)wp. We then suppose that X E Vp and hence we have a unique x E 9 such that X = x:' Since TiPg(x P ) = (Ad(g-l )x)P [VI.S.4], we have ~(TiPgX)wpg =

Ad(g-l)x =

Ad(g-l)(~(X)wp).

IQEDI

3.10 Discussion. According to [3.8] we can define a principal connection either as being generated by the structure group or as being invariant under the (right) action of G on P. It is the latter definition which is usually given. However, the former is more intrinsic because it is a description which can be used for an arbitrary fiber bundle whereas the description in terms of the action of G on P is not easily generalized to arbitrary fiber bundles. According to [3.9] we can describe a principal connection also by a connection I-form. Since it depends upon the context which description is the most useful, it is customary to mix both approaches and to switch between the description in terms of a sub bundle of TP with its horizontal vectors and the description in terms of the I-form. And in fact, both description can be given in terms oflocal g-valued I-forms r a satisfying (2.IS). The subbundle 11. can be recovered via [2.12] and the I-form w via [3.11].

3.11 Proposition. Let 1r : P ---> M be a principal fiber bundle with structure group G. Let 11. be a principal connection on P and let w be the associated connection I-form. Let {(Ua ,1/Ja) I a E I} be a trivializing atlasfor B and let Sa : Ua ---> P be the local section defining 1/Ja [3.1]. Finally, let r a be the g-valued I-fonns on Ua defining the principal connection 11. [2.13]. Then s~w = r a and WI 7r -l(Ua ) can be reconstructed from r a in the trivialization determined by Sa by

(3.12) Proof Since w is completely determined by the conditions ker(w) = 11. and condition (i) of a connection I-form, it suffices to verify these condition in the trivialization determined by Sa. In this trivialization the fundamental vector field x P is given as x [3.7], and thus

by definition of the Maurer-Cartan I-form. Introducing local coordinates Xi on Ua , we know that the connection 11. is generated by the vector fields oxilm + A~,X(m); with

354

Chapter VII. Connections

= ~(oxilm)ra [2.12]. Using the properties of the Maurer-Cartan I-form [1.5] and the factthat A~,X(m)~ = -TRgA~,X(m) [VI.5.1], we then compute

A~,X(m)

~(oxilm

+ A~,X(m);)(eMClg + Ad(g-l) oralm)

=

= Ad(g-l)L(Oxilm)ralm - ~(TRgA~,X(m))eMClg = Ad(g-l)A~,X(m) - ~(A~,X(m))R;eMC = 0 . We conclude that the given expression for W in the local trivialization determined by Sa has the required properties and thus must coincide with w. From this local expression for wand the fact that in this trivialization the local section Sa is given as Sa (m) = (m, e), it IQEDI follows that s~w = ra.

3.13 Discussion. A natural question is whether there always exists a principal connection on a given principal fiber bundle 7r : P ---> M. The answer is affirmative and relies on a partition of unity argument. LetU = {(Ua,1,Ua) I a E I} be a trivializing atlas. For each a E I we choose the I-form r a == O. These choices do not (in general) satisfy (2.15), and thus do not define a global principal connection. But on the restriction Plua = 7r-l(Ua ) they do define a principal connection. According to [3.11] we thus have a connection I-form Wa = e MC on the local trivialization 7r- 1 (Ua ) ~ Ua x G. Let {Pa I a E I} be a partition of unity associated to the open cover U. For each a E I we then have the global I-form Pa Wa. This global I-form obviously satisfies condition (ii) of a connection I-form, but condition (i) is replaced by ~(XP)Pa Wa = pa . x. It follows that LaEI pa Wa is a well defined global connection I-form on the principal fiber bundle P.

4.

THE EXTERIOR COVARIANT DERIVATIVE AND CUR V A TURE

In this section we continue the study of FVF connections on principal fiber bundles. We define the exterior covariant derivative D and apply it to the connection I -form W to obtain the curvature 2-form n = Dw. We then prove the structure equations of Cartan n = dM.J + ~ [w Ii- w] and the Bianchi identities dn = [n Ii- w]. We also show that n is determined by the local 2-forms d r a+ ~ [ra Ii- r a] if the FVF connection is determined by the local I forms ra. Not surprisingly, we can prove that the FVF connection is integrable ifand only if the curvature n is zero.

4.1 Definition. Let 11. be a principal connection on a principal fiber bundle P ---> M and let W be the associated connection I-form. For any k-form a on P with values in an A-vector space E one defines the exterior covariant derivative Da (with respect to the connection I-form w) by the formula

355

§4. The exterior covariant derivative and curvatnre

where h denotes the projection on the horizontal part h : TpP ---> 11. p. In particular the curvature 2-form f2 is defined as the exterior covariant derivative of the connection I-form w: f2=Dw.

4.2 Lemma. For any two vector fields X and Y we have ~(X, Y)f2 = ~([ hX, hY])w. As a consequence, the principal connection 11. on the principal fiber bundle P ---> M is integrable if and only if its curvature 210rm f2 is identically zero. Proof Using 01.7.6) we have for homogeneous X and Y : -~(X, Y)f2 = -~(hX,

hY)dw

= (hX)(~(hY)w) - (-l)(€(X)IE(Y»(hY)(~(hX)w) - ~([ hX, hY])w

=

-~([hX,hY])w.

According to Frobenius, the connection 11. is integrable if and only iffor all X, Y E 11. we have [X, YJ E H. Since 11. = ker w, this means that 11. is integrable if and only if have the implication ~(X)w = ~(Y)w = 0 ~ ~([X, Y])w = O. Using the definition of the projection h : T P ---> 11., this means that 11. is integrable if and only if for all vector fields X and Y we have ~([hX, hY])w = O. According to our previous computation, this is the IQEDI case if and only if f2 = O.

4.3 Lemma. Let w be a connection I-form on a principal fiber bundle P f2 = Dw be its curvature 2-form. Then we have thefollowing identities:

Dw

== f2

=

dw

+ ~ [w Ii- w]

df2=[ f2 li- w ]

--->

M and let

(the structure equations ofCartan) (the Bianchi identities).

Proof Interpreting k-forms as skew-symmetric k-linear maps on smooth vector fields

(IV.5.l5), it suffices to evaluate these identities on smooth vector fields. Since they are 2- and 3-additive, it suffices to show that we have equality when evaluating on homogeneous vector fields. Moreover, since a vector field splits as a sum of a horizontal and a vertical vector field, we may restrict attention to smooth vector fields which are either horizontal or vertical. And finally, since k-forms are linear over C=(P) and since vector fields of the form x P with x E Bg generate the module of smooth vertical vector fields [VI. 1.19], [3.7], it suffices to use this kind of vertical vertical vector field. We start with the structure equations of Cartan, for which we evaluate both sides on two homogeneous vectors X, Y. We distinguish three cases: both vertical, both horizontal, and X horizontal and Y vertical. If both X and Y are horizontal, we have ~(X, Y)f2 = ~(X, Y)dw. Since by [1.4] ~(X, Y) [w Ii- w] = 0, we have equality for two horizontal vector fields.

Chapter VII. Connections

356

If both X and Y are vertical, we may assume, as argued above, that X = x P and Y = yP for some x, y E Bg. It follows that [x P, yP] = [x, y]P is vertical. By definition of 0, ~(X, Y)O = O. On the other hand we have

and ~(xP, yP)[ w f,\ w] = -2[ ~(xP)w, ~(yp)w] = -2[x, y] [104], which shows that for two vertical vector fields we also have equality. If X is horizontal and Y vertical, we may assume that Y = yP for ayE Bg. By definition of a connection l-form we have the equality ;w = Ad(g-l) ow. From [VI.8.20] we know that we can take the derivative of the left hand side in the direction of y and that at g = e we obtain £(yp)w (<1>: = id). To compute the derivative of the right hand side in the direction of y at g = e, we first note that it depends on g only via Ad(g-l). And then [VI.9.l2] gives us Ye(Ad(g-l) ow) = - adR(y) ow, and thus £(yp)w = - adR(y) ow. We then compute ~([X,yp])w

= ~(X)£(yp)w - (-l)(€(X)IE(y»£(yp)~(X)w = -~(X) adR(y) ow = -( _l)(€(X)I€(y» adR(Y)(~(X)w) = 0,

because ~(X)w = O. In other words, if X is horizontal, then [X, yP] is also horizontal. Since Y = yP is vertical, ~(X, Y)O = O. Moreover, again by [104], we have the equality ~(X, Y)[ w f,\ w] = -2[ ~(X)w, ~(Y)w] = O. Finally,

because ~(yP)w is constant and X and [X, yP] are horizontal. We conclude that also in the third case we have equality, i.e., we have proven the structure equations of Cartan. To prove the Bianchi identities, we first note that dO = ~d[ w f,\ w] according to the structure equations of Cartan. With respect to a basis (Vi) of 9 we have ordinary l-forms Wi defined by w = L i Wi 0 Vi. And then we have dw = L i dw i 0 Vi and [w f,\ w] = Lij Wi 1\ wj 0 (_l)(€(vi)I€(vj» [Vi, Vj], and thus:

ij

= [dw f,\ w]-

2:) _l)(€(vi)I€(vj»dw

j 1\ Wi

0 (-[ Vj, Vi])

= 2[ dw f,\ w] .

ij

Again using the structure equations of Cartan we thus have dO

= [dwf,\w] = [Of,\w]-

~[[wf,\w] f,\w]

The result now follows because of the Jacobi identity, [104] and the following computation

357

§4. The exterior covariant derivative and curvatnre

for three homogeneous vectors: ~(X,Y,Z)[[wf,\w]

f,\w]

= [~(X,Y)[wf,\w],~(Z)w]

+ (-1) (€(X)!€(y)+€(Z» [~(Y, Z)[ w f,\ w ], ~(X)w ] + (_l)(€(Z)!€(X)+€(Y» [~(Z, X)[ w f,\ w], ~(Y)w] = -2( -1) (€(X)!€(Z»

(( _l)(€(X)!€(Z» [[

l.(X)w, l.(Y)w], ~(Z)w]

+ (_l)(€(X)!€(y» [[ l.(Y)w, ~(Z)w], ~(X)w] + (-1) (€(Z)!€(y» [ [~( Z)w, ~(X)w ], ~(Y)w ])

=

0.

IQEDI

4.4 Corollary. On any principal fiber bundle with a connection I-form w we have the equality DD == D 2 w = o.

Proof Using [1.4] and the Bianchi identities [4.3] we have ~(X, Y,

= Since

Z)DD = l.(hX, hY, hZ)dD = [~(hX,

~(h W)w

hY)D, l.(hZ)w]

~(hX,

hY, hZ)[ D f,\ w]

+ (_l)(€(X)!€(Y)+€(Z» [~(hY, hZ)D, ~(hX)w] + (-1) (€(Z)!€(x)+€(Y» [~( hZ, hX)D, ~(hY)w]

.

is zero for all W, the result follows.

4.5 Remark. If G is an A-Lie group, we can see it as a principal fiber bundle over a point M = {mo} with (trivial) projection 7r : G ---> {mo}. Since there is no non-zero I-form on a zero dimensional A-manifold, it follows from [3.11] that any connection I-form w on this principal fiber bundle necessarily is the Maurer-Cartan I-form: w = e Me. And then the structure equations of Cartan [4.3] tell us that [1.6] can be interpreted as saying that the curvature of this connection is zero.

4.6 Discussion. Let w be a connection I-form on a principal fiber bundle 7r : P ---> M, let 11. be the associated principal connection, let {( Ua, 1/Ja) I a E I} be a trivializing atlas for P determined by local sections Sa' and let Xi be (local) coordinates on some Ua . In [3.11] we have seen that the local I-forms s~w are the local I-forms r a defining 11., which are related on overlaps Ua nUb by (2.15). Moreover, the local functions A~'x describing the fundamental vector fields 'Yi are given by A~'x = ~(OXi)S~W [2.12]. We intend to give a similar description of the curvature2-form D = Dw. We thus define the g-valued2-forms s~D on Ua and the homogeneous smooth functions Pt/ : Ua ---> 9 by Pi~'x (m) = -~( oxi, ox j )(s~D)m with parity C(Pi~'X) = c(Xi) + c(x j ). Since a 2-form is graded skew-symmetric in its entries, the functions Pi~'x are graded skew-symmetric in their indices: pa.'x = _(_l)(€(x i )!€(x j » pa.'x. With these functions the 2-form s*D can J' 'J a be written as (s~D)m = -~ dx j 1\ dx i 0 Pi~,X(m) .

L ij

358

Chapter VII. Connections

4.7 Remark. The minus sign in the definition of Ftt is conventional. One could say that it is a consequence of our way to identify the dual of an exterior power [1.7.22], [Y.7.14]. In the ungraded case it would allow us to write s~O = ~ Lij dXi 1\ dx j 0 Ftt with the indices in the same order.

4.8 Proposition. The 2-form s~O is detennined by the i-fonn

s~w

= ra

as

in terms of the functions A~'x and Ftt this equality is given as (4.9) 017l'-1(Ua )

can be reconstructedfrom s~O in the trivialization determined by Sa by

(4.10)

O(m,g)

=

Ad(g-l) 0 (S~O)m

= -~ Ldxj

1\

dx i 0 Ad(g-l)F;',t(m) .

ij

On the intersection Ua n Ub of two local trivializations we have (4.11)

Proof The structure equations of Cartan tell us immediately that we have the equality s~O = d(s~w) + ~[s~w Ii- s~w]. Substituting (3.12) in this equation gives us:

ij

j

ij

ij

= _.! " " dx j 2 ~

1\

dxi,o, I6f

{8 ·Aa,x xt

J

(_1)(€(x i )I€(x j »8Xl·Aa,x ~

+ [Aa,X

Aa,X]}

~'J

.

ij

From this (4.9) follows immediately. To prove the local form of 0, we first recall that the Adjoint representation is indeed a representation of g, i.e., [Ad(g)x, Ad(g)y] = Ad(g) [ x, y]. Using the local expression w(m,g) = Ad(g-l)(s~W)m + 8 MC Ig and [1.6] we compute:

O(m,g)

= dw + ~ [w Ii- w] = d(Ad(g-l)(S~W)m) + ~[Ad(g-l)(s~W)m Ii- Ad(g-l)(s~w)m]

+ ~[Ad(g-l)(S~W)m Ii- 8 MC lg] + ~[8MClg Ii- Ad(g-l)(S~W)m] = Ad(g-l)(S~O)m + (dAd(g-l))

1\ (s~W)m

+

[Ad(g-l)(s~W)m

Ii- 8 MC lg] .

359

§4. The exterior covariant derivative and curvature

The last line is a consequence of the proof of [1.4], from which one can deduce that for even g-valued I-forms a and 'Y we have [a Ii- 'Y] = ['Y Ii- a]. We thus have to show that the last two terms cancel. We will do this by evaluating on tangent vectors. Since each term is a product of a term which only acts on vectors in the M-direction and a term which only acts on vectors in the G-direction, we only have to show that for homogeneous X E TmM and Y E TgG we have

For any Y E TgG there exists y E 9 such that Y =

~.

We then compute:

where the minus sign comes from the fact that L(y) is a derivation and has to be commuted with the even I-form Ad(g-l )(s:w)m. Using [VI.9.12] we compute the second term: L(X,y)(dAd(g-l)) 1\ (S:W)m

= (_l)(€(Y)I€(X»(yg Ad(g-l)) . L(X)(S:W)m = -( _l)(€(Y)lo(X» adR(y) Ad(g-l)L(X)(S:W)m = -( _l)(€(Y)I€(X» [y, Ad(g-l)L(X)(S:W)m] = [Ad(g-l)L(X)(S:W)m, y]. 0

This proves that the two terms cancel and thus we have proven (4.10). To prove (4.11), we note that in the trivialization determined by Sa, the section Sb takes the form sb(m) = (m,1/Jab(m)) (a direct consequence of (3.5)). Using the local expression (4.10) we find immediately

4.12 Proposition. Let 7r : P --> M be a principal fiber bundle with structure group G, and let W be a connection I-form on P and D its curvature 21orm.

c M be open and S E ru(p) a local section. Then S is a horizontal section if and only if s*w = 0, and either of these two conditions implies that DI7r-1(u) = 0 and that the horizontal submanifolds in 7r-l(U) ~ U x G are exactly the horizontal sets U x {g}, g E BG (see [2.1]). (ii) D = Oon P if and only if there exists a trivializing atlas U = {(Ua,1,Ua) I a E I} determined by local sections Sa E rUa (P) such that for all a E I we have s:w = O. (i) Let U

Proof The local section s is horizontal if and only ifTs(Xm) E Hs(m) for all m E U and all Xm E TmM [2.2]. Since Hs(m) = ker ws(m)' this is the case if and only if o = L(((XmIITs)))ws(m) = L(Xm)(S*w)m,which is the case if and only if S*W = O. Using the structure equations of Cartan, this implies directly that s*D = o. And then

360

Chapter VII. Connections

(4.10) tells us that nl 7r -l(U) = O. (3.12) shows that WI 7r -l(U) is given in the trivialization determined by the local section s as 8 MC ' This implies that in this trivialization H(rn,g) is given as TrnM x {O}g C T(rn,g)(U x G). And the integral manifolds of this subbundle are obviously the subsets U x {g} as announced. For (ii), if U exists, then by (i) n = O. We thus assume that n = 0, which means that 11. is integrable [4.2]. Now let p E BP be arbitrary, then through p passes a (unique) leaf (i, L) of the involutive subbundle 11. C T P [V.6.9]. Let £ E L be the unique point such that i( £) = p. By definition of a leaf, Ti : TeL ---; Hp is an isomorphism, and by definition ofaconnection, Trr : Hp ---; T7r(p) is a bijection. Using [V.2.l4] we deduce that there exist neighborhoods V oU and Urn of m = 7r(p) such that 7r 0 i is a diffeomorphism from Vto Urn. We then define s: Urn ---; Pby s = i O ((7r o i)lv)-l. Composing on the left with 7r shows that 7r 0 s = id, and thus s is a local section. Since (i, L) is a leaf, s is horizontal, i.e., s*w = O. Since p E BP is arbitrary, we conclude that every m E BM admits a neighborhood Urn on which there exists a local horizontal section s. Since for all m E M we have m E UBrn by definition of the DeWitt topology, these neighborhoods cover M. The corresponding local sections thus define a trivializing atlas as desired.IOEDI

5. FVF

CONNECTIONS ON ASSOCIATED FIBER BUNDLES

In this section we show that there is a natural way to introduce an FVF connection on an associated fiber bundle starting with one on the original bundle. Defining the structure bundle as the principalfiber bundle with the same structure group and transition functions as the original bundle allows us to show a close relationship between FVF connections on general fiber bundles and those on principal fiber bundles. Using this relationship, we show that a leafofan integrable FVF connection is a covering space ofthe (connected) base space. In particular, a fiber bundle B over a simply connected base space admits an integrable FVF connection ifand only ifB is trivial. We end this section by giving an intrinsic description of an associated fiber bundle to a principal fiber bundle, which allows us to give alternative descriptions of some constructions concerning associated bundles.

5.1 Proposition. Let 7r : B ---; M be a fiber bundle with typical fiber F and structure group G. Let H be another A-Lie group with apseudo effective action on an A-manifold E. Let p : G ---; H be a morphism ofA-Lie groups. Finally, let 11. be an FVF connection on B defined by local g-valuedI -forms r a relative to a trivializing atlas U = {( Ua, 1/Ja) I a E I} for B. Then the local ~-valued I -forms ria = TeP 0 r a define an FVF connection HP on the associatedfiber bundle BP,E ---; M [lV.2.l].

Proof Let us denote by 1/Jba the transition functions associated to the trivializing atlas U. Then, according to its definition [IV.2.l], the transition functions of the associated bundle

§5. FVF connections on associated fiber bundles

are given by 1/J~a

= po 1/Jba' fib

361

According to [2.13] we only have to prove that

== Tepo (Ad(1/Jab(m)-l) ofa(m) + (1/J~be~dlm) = Ad(1/J~b(m)-l) ofla(m) + (1/J~beZ.c)lm .

To prove this, we start with the equality po Ig = Ip(g) 0 p as maps from G to H, where Ig denotes, as in [VI.2.12], the map x r--+ gxg- 1 . Taking the tangent map at the identity and using the definition of the Adjoint representation gives us TeP 0 Ad(g) = Ad(p(g)) 0 TeP, This proves that Tepo Ad(1/Jab(m)-l) ofa(m) = Ad(1/J~b(m)-l) ofla(m). Next we note that po Lg = Lp(g) 0 p as maps from G to H. Taking the tangent map of this identity gives us the equality

Comparing this with the equality 1/J~be~c = Li dXi 0 T L..pab(m)-l T1/Jab aXi 1m given IQEDI in the proof of [2.13], we can conclude thatTepo1/J~be~c = 1/J~beZ.C'

5.2 Definition. In [IV.2.14] we have seen that any fiber bundle can be seen as being an

associated fiber bundle to a principal fiber bundle. We formalize this by defining the structure bundle trs : SB ---> M as being this principal fiber bundle, i.e., SB = Bid,G is the (principal) fiber bundle with typical fiber G and structure group G associated to the fiber bundle 7r : B ---> M with typical fiber F and structure group G by the identity representation id : G ---> G. The underlying idea is that both Band SB are defined by the same transition functions 1/Jab associated to a trivializing atlas U.

5.3 Corollary. SBid,F = B and if H is another A-Lie group with a pseudo effective action on an A-manifold E and if p : G ---> H is a morphism of A-Lie groups, then BP,E = SBP,E, i.e., constructing an associated fiber bundle from the original fiber bundle B or from its structure bundle yields the same result.

5.4 Remark. The structure bundle is a generalization of the frame bundle for vector bundles. If 7r : B ---> M is a vector bundle with typical fiber the A-vector space E, we can define for each m E M the set F m of all bases, also called frames, of the 2t-graded A-module Bm. Since two bases are related by an element of Aut(Bm) [11.2.6], the set Fm is isomorphic to Aut(Bm ). In order to give the disjoint union FB = lImEMFm the structure of a (principal) fiber bundle over M, we proceed as follows. We fix a basis (ei) of E. In a local trivialization (Ua,1/Ja) we define a map {m} x Aut(E) ---> Fm by (m, V) r--+ (1/J~1(Vei))1~lE. This gives us an isomorphism X~l between Ua x Aut(E) and U mEUa F m. To see how this isomorphism depends upon the chosen local trivialization, we choose another one (Ub, 1/Jb), which gives us an isomorphismX;l between Ubx Aut(E) and lImEubFm by (m, W) r--+ (1/J;1(Wei))1~lE. For m E Ua n Ub we get the same

362

Chapter VII. Connections

basis ofB m iffor all i we have '!jI;l(Vei) = ~bl(Wei)' Applying ~b and the definition of the transition function, this happens if and only if for all i we have Wei = ~ba(m) Vei, i.e., if and only if W = ~ba(m) 0 V. This implies that Xb 0 X;l is given by the map (m, V) f---+ (m, ~ba (m) . V). We conclude that :FB is a principal fiber bundle with structure group Aut(E) and the same transition functions as B. In other words, the frame bundle of the given vector bundle B is exactly the structure bundle as defined in [5.2].

5.5 Corollary. Let B ---> M be any fiber bundle and let SB ---> M be the associated structure bundle. Then there exists a canonical bijection between the set of FVF connections on B and the set of FVF/principal connections on SB. In particular, any fiber bundle B ---> M admits an FVF connection.

Proof According to [IV.2.14] and [5.2] the fiber bundle B ---> M and the principal fiber bundle SB ---> M are associated to each other by the identity representation. According to [5.1] an FVF connection on one of these two bundles defined by local i-forms r a determines an FVF connection on the other one by the same set of i-forms. Finally, according to [3.13] any principal fiber bundle admits a principa1JFVF connection. IQEDI

5.6 Corollary. Let P ---> M be a principal fiber bundle with structure group G, let H be an A-Lie group with a pseudo effective action on F, let P : G ---> H be a morphism of A-Lie groups, and let B == pp,F ---> M be the associated fiber bundle (associated to P by the representation p). If H P is a principal/FVF connection on P with associated connection I-form w, and if HB is the associated FVF connection on B given in [5.1], then HB is integrable ifand only ifTeP 0 n is zero, where n is the curvature 210rm on P.

Proof According to [2.18] HB is integrable if and only if the local 2-forms TeP 0

(d r a + ~ [ra f,\ r a])

are zero (use that [TeP(x), TeP(Y) 1 = TeP([ x, y]), i.e., TeP is a morphism of A-Lie algebras). On the other hand, using (4.10), we find

Since s~ n = d r a + ~ [r a f,\

r a1, the result follows.

5.7 Corollary. Let HB be an FVF connection on afiber bundle B ---> M, let H S be the associated FVF/principal connection on the structure bundle SB ---> M [5.5J, and let w be its associated connection I-form. Then the following four statements are equivalent: (i) HB is integrable; (ii) H S is integrable; (ii) the curvature 2-form non SB is zero; (iii) there exists a trivializing atlas U = {(Ua, '!jIa) I-forms r a determining H B are zero.

I a E I} for B such that all local

§5. FVF connections on associated fiber bundles

363

Proof The equivalence between (i) and (ii) has been shown in [4.2], and the equivalence between (i) and (iii) is an immediate consequence of [5.6] because SB is related to B by the identity representation. Since Band S B are determined by the same local I-forms IQEDI r a , the equivalence between (iii) and (iv) is a direct consequence of [4.12].

5.8 Proposition. Let 7r : B ---; M be a fiber bundle with typical fiber F, let 11. be an integrable FVF connection on B and let (i, L) be a leaf of 11.. Then 7r( i( L)) is open and closed in M and 7r 0 i : L ---; 7r( i(L)) is a covering (§ VI. 7).

Proof Suppose that mo E M belongs to the closure of7r(i(L)). According to [5.7], there exists a local trivialization (U, 1jJ) containing mo (part of a trivializing atlas) such that the local I-form ron U is zero. Taking a smaller U if necessary, we may assume that U is connected. By definition of closure, there exists a point m E Un 7r( i(L)). Let f!. E L be such that 7r(i(f!.)) = m, then 7r(i(Bf!.)) = Bm E U because 7r 0 i is smooth and U open. Let f E F be such that 1jJ(i(f!.)) = (m, f). Since the local I-form r on U is zero, the local section S : U ---; B, s(m') = 1jJ-1(m',Bf) is a smooth horizontal section. But then U is connected, sis tangentto 11. and i(Bf!.) = Bi(f!.) E s(U) n i(L) and thus by [V.6.9] s(U) C i(L). Since s is a section, this implies that U C 7r(i(L)). We thus have shown that an arbitrary point mo in the closure of7r(i(L)) admits an open neighborhood U contained in 7r( i( L)), and thus 7r( i( L)) is open and closed. To show that 7r 0 i is a covering, we have to find for all m E 7r( i( L)) an open neighborhood with certain properties [VI.7.1], [VI.7.3]. Since 11. is integrable, there exists (as above) a local trivialization (U,1jJ) of B on which r is zero with U connected, contained in 7r( i(L)) and containing m. Let (7r 0 i)-1 (U) = UaEJ Va be the decomposition of (7r 0 i) -1 (U) in connected components. Then by definition the Va are pairwise disjoint. Since (7roi)-1(U) is open in L, it follows from [111.1.3] that each Va is open in L. Moreover, 7r 0 i is smooth and Te( 7r 0 i) is a bijection for each f!. E L because Ti : TeL ---; 11.i(e) is a bijection by definition ofaleaf, and becauseT7r: l-l i(e) ---; T 7r (i(e» is a bijection by definition of a connection. If we can show that 7r 0 i is a diffeomorphism from each Va to U, we will have shown that 7r 0 i is a covering map. Using the projection 7rF : U x F ---; F, we define the map g = 7rF 0 1jJ 0 i : 7r-1(U) ---; F. Since i is tangent to 11. and since r = 0 on U, i.e., T1jJ(11.) = TU x {o}, it follows that Tg = O. By [V.3.21] g is constant on each Va. Since g is smooth, there thus exist fa E BF such that i(Va) C 1jJ-1(U X {fa}). We now define the local smooth sections Sa(m') = 1jJ-1(m',Ja). Since r = 0 on U, this means (as before) that Sa is tangentto 11.. Since i( f!.) E Sa (U) n i( L) for any f!. E L, it follows from [V.6.9] that there exist smooth maps ga: U ---; Lsuchthat Sa = iog a . Since 7r O Sa = id(U), ga(U) C (7roi)-l(U), and since U is connected, ga (U) is contained in one of its connected components V,a. Moreover, again because 7r 0 Sa = id(U), i(V,a) = 1jJ-1(U x {fa}). Since i is an injective immersion, it follows that i : V,a ---; 'ljJ-1 (U x {fa}) is a diffeomorphism (because Land U have the same dimension). But i(Va) C 'ljJ-l(U X {fa}), and thus by injectivity of i we deduce that a = (3 and that i : Va ---; 'ljJ-l (U x {fa}) is a diffeomorphism. Since the same is true for 7r : 1jJ-1 (U x {fa}) ---; U, we have shown that 7r 0 i : Va ---; U is a IQEDI diffeomorphism.

364

Chapter VII. Connections

5.9 Corollary. Let M be a simply connected A-manifold and let 7r B : B ---> M be a fiber bundle. Then B admits an integrable FVF connection if and only ifit is trivial. Moreover, if B is a vector bundle, then triviality is as a vector bundle. Proof If B admits an integrable FVF connection, then by [5.7] the corresponding FVF connection 11. s on 7r : SB ---> M is integrable. By [5.8] any leaf (i, L) of 11. s is a covering of M. Since M is simply connected, this implies that 7r 0 i : L ---> M is a diffeomorphism. But then i 0 (7r 0 i)-l is a global section of SB and thus SB is trivial. Adding this global trivialization ofSB to a trivializing atlas, the corresponding trivializing atlas of the associated bundle B also contains a global trivialization, and thus B is trivial. In particular, if B is a vector bundle, this global trivialization is compatible with the vector bundle structure and thus B is trivial as a vector bundle. Conversely, if'lb: B ---> M x F is a global trivialization, then (T'lb)-l(TM x {O}) is an integrable FVF connection, determined by the (global) i-form r == O. IQEDI

5.10 Remark. Since any i-dimensional connected A-manifold N is simply connected (it is an interval [VA.l]), it follows from [5.8] that any leaf of an integrable FVF connection on N is diffeomorphic to N via the projection map. A particular consequence is that parallel transport along an even curve always exists for FVF connections (see [2.6]).

5.11 RemarkIDiscussion. The definition of an associated fiber bundle and the construction of the FVF connection on an associated fiber bundle both use local trivializations. Even though this works quite well, one would like to have a more intrinsic/global description. Such a more global description can be given for arbitrary fiber bundles associated to a principal fiber bundle. The idea is as follows. Let trp : P ---> M be a principal fiber bundle with structure group G and let P denote the right action of G on P. Let H be an A-Lie group with a pseudo effective action F on an A-manifold F, and let p : G ---> H be an A-Lie group morphism. These data allow us to define an associated fiber bundle 7r B : B ---> M with structure group Hand typical fiber F. The more intrinsic construction of B starts with the observation that the map W : G x F ---> F given by W(g, f) = F (p(g), f) is a left action of G on F (not necessarily pseudo effective). With this action we define an effective right action x of G on P x Fby

The projection 7r pO 7rl : P x F ---> M (with 7rl : P x F ---> P the projection on the first factor) is constant on G-orbits because trp is constant on G-orbits in P. We thus have an induced map 7ro : (P x F)jG ---> M from the orbit space (P x F)/G to M. now define a map 7rx : P x F ---> B in terms of the localtrivializations 'lba for P and 1/Ja for B relative to the same trivializing atlas by

'!Ie

§5. FVF connections on associated fiber bundles

365

or equivalently, still in local trivializations, by

7rx (m,g,f) = (m, iI!(g,f)) . We claim that this is a well defined map, independent of the chosen local trivialization, and that it is constant on G-orbits. Moreover, the induced map 7rx : (P X F)IG ---; B is a bijection verifying 7r B 0 7rx = 7ro, i.e., we have the following commutative diagram

PxF ,/ 7r x

P x FIG (-(- - - - 4

B.

M The verification of these claims is straightforward and is left to the interested reader. We do not say that 7rx is a diffeomorphism between the orbit space (P x F) /G and B because we have not defined how to induce the structure of an A-manifold on an orbit space (if possible at all; the only instance where we have defined the structure of an A-manifold on an orbit space is for homogeneous A-manifolds [VI.S.9]). Either by using the bijection 7rx or by more direct means, one can give the orbit space (P x F)/G the structure of an A-manifold and then the structure of a fiber bundle over M with structure group G and typical fiber F. Once we have this structure, the bijection 7rx becomes an isomorphism of fiber bundles. It follows that we can take the orbit space (P x F) /G, for which one also finds the notation P x G F, as the definition of the associated fiber bundle B. Once we have the description of the associated bundle B as the orbit space (P x F) /G, we can give a global description of the induced FVF connection. To that end, let H P be an FVF connection on the principal fiber bundle P. Then HB == T7r x (H P x {O}) is the FVF connection on the associated fiber bundle B == (P x F)IG defined in [5.1]. Note that the principal fiber bundle P here plays the role of the bundle B of [5.1] and that the bundle B here is the associated fiber bundle, associated to P by the representation p of [5.1]. Another property of associated bundles that now can be given a more intrinsic description is the following. The construction of an associated bundle starts with a trivializing atlas for the initial bundle, and then the associated bundle has the same trivializing sets. Adding more elements to the original trivializing atlas, it follows that for each local trivialization (U, 1/J) of the initial bundle there exists a corresponding local trivialization (U,1/J') for the associated bundle. If the initial bundle is a principal fiber bundle 7r : P ---; M, we also know that a local trivialization of P is completely determined by a local section s. It follows that for each local section s : U ---; P there is a corresponding local trivialization (U, 1/Js) of B. In terms of the intrinsic description ofB given above, this local trivialization IS gIven as (m, f)

r--+ 7rx

(s(m),f) .

366

6.

Chapter VII. Connections THE COVARIANT DERIVATIVE

In this section we introduce the notion of a covariant derivative on a vector bundle, which is a generalization of the derivative of a vector valued functions to sections of the vector bundle. We show that a covariant derivative is determined by local I -forms r a with values in EndR(E), the space of endomorphisms of the typical fiber E. We prove that these local I -forms behave exactly as the local I -forms defining an FVF connection, thus showing that a covariant derivative on a vector bundle is equivalent to an FVF connection on it.

6.1 Discussion. In [2.1] we discussed the idea of (locally) constant (local) sections of a fiber bundle in terms of Ehresmann connections. For vector bundles there is another approach in terms of a covariant derivative which is based more on the derivative of a function being zero than on the section being horizontal. Let M be an A-manifold, E an A-vector space and I : M -4 E a (smooth) function, corresponding to the section s of the (trivial)bundle 7r : M x E -4 M given by s(m) = (m,I(m)). If X is a vector field on M, we have an action of X on f giving a new function X I : M -4 E [V.1.24]. We can transform this action on functions into an action on sections by defining the section X s as being given by (Xs)(m) = (m, (Xf)(m)). This is a simple transcription of the action of vector fields on E-valued functions to sections. We then can say that s is a constant section if and only if X s is the zero section for all vector fields X on M. If we try to generalize this to arbitrary vector bundles, we encounter a problem: a (global) section s is represented by local functions s,p, but there is no guarantee that the new local functions X s,p glue together to form a new global section X s. This corresponds of course to the fact that being horizontal in one trivialization does not necessarily correspond to being horizontal in another trivialization. The idea of a covariant derivative is to extract the essential features of the above definition of the action of vector fields on sections of a trivial bundle, and to use these to define something meaningful on an arbitrary vector bundle. In [6.18] and [7.2] we will see that the approach to define constant sections via a covariant derivative is equivalent to doing it via an FVF connection.

6.2 Definition. Let 7r : B -4 M be a vector bundle over M with typical fiber the A-vector space E. Recall that r(B) denotes the graded COO (M)-module of smooth sections of the bundle B and that r(T M) denotes the graded COO (M)-module of all vector fields on M. A covariant derivative \7 on the bundle B is a map \7 : r(TM) x r(B) -4 r(B) satisfying the following conditions.

(i) \7 is bi-additive and even; (ii) for all I E COO(M), X E r(TM), s E r(B) we have \7(1 X, s) (iii) for homogeneous I, X, s we have

\7(X, Is)

=

= I\7(X, s);

(Xf)s + (_I)(E(X)IO'(l» I\7(X, s) ,

which can also be written as \7(X, sf) = \7(X, s)I + (_1)(0'(8)10'(1» (X f)s.

§6. The covariant derivative

367

Following custom, we will denote \7 (X, s) also as \7x s ; it is called the covariant derivative

of s in the direction of X.

6.3 Example. Let M be an A-manifold and E an A-vector space. In the identification between E-valued functions on M and sections of the trivial bundle M x E (with its canonical trivialization), the action of vector fields on sections (X, s) f-+ X s as in [6.1] is an example of a covariant derivative on this trivial bundle.

: B ---> M be a vector bundle, \7 a covariant derivative on B, and let V cUe M be two open subsets.

6.4 Proposition. Let 7r

= t\u, then\7(X,s)\u = (\7(X,t))\ufor any X E f(TM). (ii) If X, y E r(TM) are such that Xm = Ym for some m E M, thenfor any s E f(B) we have \7(X, s)(m) = \7(Y, s)(m). (iii) There exists a unique covariant derivative \7u on B \u such that for all s E f( B) we have \7(X, s)lu = \7U(Xlu, slu). (iv) (\7 U) V = \7 v . (i) /fs,t E f(B) aresuchthats\u

7r : Blu == 7r-l(U) ---> U [IV.1.13]. Apart from the difference coming from the presence of a bundle, the proof is a close copy of the proofs of [V.lo4], [V. loS], [IV.S.S]. • (i) Without loss of generality we may assume that sand t are homogeneous of the same parity. For any m E U, let p be a plateau function around m in U. It follows that p(s - t) = O. Using the properties of a covariant derivative, we obtain

Proof Recall first that Blu is the subbundle

0= \7(X, p(s - t)) = (Xp)(s - t)

+ p\7(X, (s -

t))

(p is even).

Since p(m) = 1 this gives us (\7x s)(m) = (\7x t)(m) . • (ii) Let (Xi) be local coordinates on a neighborhood W of m, then there exist functions Xi and yi on W such that Xlw = Li Xiaxi and Ylw = Li yiaxi. If p is a plateau function around m in W, then paxi is a global smooth vector field on M and pX i and pyi are global smoothfunctions. Since p is zero outside W, we have the global equalities p2 X = Li(PXi) . (paxi) and p2 = Li(Pyi) . (paxi). Using the properties of a covariant deri vati ve we find

and a similar equation with X replaced by Y. Evaluating these sections at m, and using = 1 and Xi(m) = yi(m), we find

p(m)

\7(X,s)(m) = L:Xi(m). \7(pax i,s)(m) = L:yi(m). \7(paxi,s)(m) i

= \7(Y, s)(m)

.

368

Chapter VII. Connections

• (iii) As for derivations, the main problem is that not every smooth section above U need be the restriction of a global smooth section. So let t E ru(B), Y E ru(TM) and m E U be arbitrary and let p be a plateau function around m in U. It follows that pt is a well defined global smooth section of B and that pY is a well defined global smooth vector field on M. Moreover, t and (pt) Iu are two local sections above U that coincide in a neighborhood of m and Y and (p Y) Iu are two vector fields on U such that Ym = ((PY)lu)m. Now, if\1u exists, we can combine (i) and (ii) with the defining property of \1u to obtain (6.5)

\1u (Y, t)(m) = \1u (pY)lu(pt)lu(m) = \1 (pY, pt)(m) .

This proves uniqueness of \1u, but we can also use (6.5) to define it. To see that (6.5) indeed produces a well defined \1u, independent of the choice for p, suppose p has the same properties as p. It follows that pt and pt coincide in a neighborhood of m, and (pY)m = (PY)m. And thus by the preceding result \1(pY, pt)(m) = \1(pY, pt)(m), i.e., (6.5) is independent of the choice for p. Since sand p( s Iu) coincide in a neighborhood of m, and Xm = (p(Xlu) )m, it follows that

\1(X,s)(m) = \1(p(Xlu),p(slu))(m) = \1 u (Xlu,slu)(m) , i.e., the covariant derivative defined by (6.5) has the desired property. To prove that \1u is a covariant derivative, we first note that it is obviously bi-additive and even, since the same holds for \1. Property (ii) of a covariant derivative is also a direct consequence of the corresponding property for \1, simply because p is even and thus commutes with any function. To prove property (iii), we only need to add the argument that p(m) = l. • (iv) This is a direct corollary of the uniqueness in (ii). IQEDI

6.6 Definition. The covariant derivative \1u on the subbundle Blu == 7r-l(U) is called the induced covariant derivative. As is customary, we will usually omit the superscript u and use the same symbol \1 to denote the induced covariant derivative on the restriction to an open subset U c M. Worse, in most cases we will not even mention that we use the induced covariant derivative.

6.7 Discussion. If (U, 1jJ) is a local trivialization of B, then the structure of a free graded A-module on each fiber is defined by declaring that the map 1jJ : 7r-l(U) --; U x E is even and linear on each fiber. In other words, for a, b E 7r- 1 (m) and A, J-t E A, if 1jJ(a) = (m, e) and 1jJ(b) = (m, f) then 1jJ(aA + bJ-t) = (m, eA + fJ-t). As in §IV.3 and §IY.S we introduce the local sections fi E ru (B) associated to a basis (ei) of E by the formula

In [IV.lo20] we have shown that there is a bijection between local sections s E ru(B) and functions s1/1 : U --; E. Using (left) coordinates with respect to the basis (ei), each

§6. The covariant derivative

369

(smooth) function 8,p defines ordinary (smooth) functions 8 i : U ---> A by the equality 8,p( m) = Li 8i (m) . ei. Using the free graded A-module structure on each fiber, the local sections fi' and the definition of the function 8.p we thus have the equalities

valid for any local section similar looking formula;:

8 :

U

--->

B. The (local) functions 8 i thus define

8

and 8,p by

(6.8)

An explicit example, though slightly hidden, of the use of the local sections fi is given by the local vector fields ai == aXi associated to local coordinates Xi on an A-manifold [V.1.16]. As a consequence, even though it is not said explicitly, the local I-forms dXi, as well as the local k-forms dxil /\ ... /\ dXik are examples of the use of the local sections fi.

6.9 Discussion. Let (U, 1jJ) be a local trivializing coordinate chart for B with coordinates Xi. Using the local sections fi : U ---> 7r- 1 (U) introduced in [6.7] and the covariant derivative \7 (officially we should say the induced covariant derivative \7 u ), we define homogeneous smooth functions f i j k : U ---> A of parity c(fij k) = c(Xi) + c(ej) + c(ek) by \7(aXi, fk)(m) = f i j k(m) . fj(m) ,

L j

where we used that the local sections fi form a basis of the fiber at each point. We can put these functions together in homogeneous maps fi : U ---> EndR(E) of parity c(f i ) = c(Xi) by fi(m)

= ,L , "f,i'J k

.

k

ej 0 e ,

jk where as usual the e k denote the right dual basis. In terms of the matrix representations given in [11.4.1] this means that we use left coordinates eMR: f i j k = eMR(fi)j k. To show that f i is independent of the choice of the basis (ei) of E, let (ej) be another basis with the associated local sections fj and functions i q p defined by the equality

r

q \7 (axi, fp) (m) = Lq ri p( m) . §.q (m).

By definition the basis (ej) is related to the basis (ei) byej = Li eiaij for some real valued matrix (aij) (real valued because we remain in the equivalence class). It follows that the right dual bases are related by e R = Lk aRk? From the definition offj we deduce the relation fj = Li fi aij and thus, using [6.2-ii] and the fact that the a i j are real constant, we obtain

q

jq

Chapter VII. Connections

370

Comparing coefficients of fj gives us the equality 2:k f i j k . a k p which we compute

= 2: q

j

i\qp . a q

with

and thus f i = f i is independent of the choice of a basis. If we change the coordinates xi to ye, we get new functions I'e j k defined by the j equality 'V(fJyi, fk)(m) = 2: j I'e k(m) . fj(m). Since fJyi = 2:i(fJyi xi)fJxi,it follows from [6.2-iiJ that these functions are related to the functions f i j k (m) by (6.10)

fi by I'e = 2:i( fJyi

Xi)

==

I'e j k

. ej ® ek :

EndR(E) are related to the . fi. We conclude that the even EndR(E)-valued I-form f on U

As a consequence, the maps I'e

2: j k

U

---t

defined by

f

=L

j dxi ®fijk' e ® ek

=L

i dx ®

fi

ijk

=

L

dye ®

I'e

e

is well defined. It follows that this I-form exists on U, even if there does not exist a global coordinate system on U. With respect to the basis (ej ® e k ) of EndR(E), the EndR(E)-valued I-form f can be written as (see (VI.9.3)) (6.11)

f

=

L

fj k ®

k (ej ® e )

jk

with ordinary I-forms fj k 2:i dxi . f i j k. As for the EndR(E)-valued I-form f, it follows from (6.10) that these ordinary I-forms are well-defined, independent of the chosen local coordinates.

6.12 Remark. In the particular case that the vector bundle 7r : B ---t M is the tangent bundle B = TM and that the covariant derivative is derived from a metric, in that case the functions f i j k are called the Christoffel symbols associated to the metric.

6.13 Lemma. Let (U, 'Ij;) be a local trivialization, let S E fu(B) be a local section with its associated local function s..p and let X = 2:i Xi fJxi be a vector field on U. Then the localfunction ('Vxs)..p associated to the local section 'Vxs is given by (6.14)

§6. The covariant derivative

371

Proof Since (6.14) is additive in s, we may assume that s is homogeneous. The local section \7X s can then be written as

\7x s

= L(Xsk)~k + LXi(_I)(€(skl!€(xi))sk\7(Oxi, ~k) k

ik

= (L(Xsj) + LXi(_l)(€(Sk)!€(Xil)skrijk) ~j' j

ijk

Using that ek(s,p) = ek (2: p sPe p ) = (_l)(€(sk)!€(e k)) sk, the function (\7x s),p is given by

(\7x s),p = (L(Xsj)

+ LXi(_l)(€(Sk)!€(Xi))skrijk)

=

X (L sj ej)

+ L Xi r ij k ej (_l)(€(skl!€(e k)) sk ijk

j

= XS,p

ej

ijk

j

+ L Xi rijk ej' ek(s,p) = XS,p + L Xi r i · s,p ijk

= X s,p + t(X)r . s,p

.

6.15 Lemma. Let (Ua , 1/;a) and (Ub, 1/;b) be two local trivializations of B with transition function 1/;ab : Ua n Ub ----t Aut(E) and let r a andr b be the EndR(E)-valued I-forms on Ua and Ub associated to the covariant derivative \7. Then r a and rb are related on Ua n Ub by (6.16)

Proof Throughout this proof we will use left coordinates eMR [11.4.1] for endomorphisms A E EndR(E). The left coordinates Aij == eMR(A)i j can be defined by the formula Aej = 2:i Aij ei. This is compatible with the definition of the endomorphisms r i in terms of the functions r i j k [6.9]. The transition function 1/;ab is defined by the equality (1/;a o 1/;b 1 )(m,J) = (m,1/;ab(m)J). Associated to each trivialization we have local sections f:.i and f:.~. On Ua n Ub they are related by:

f:.~(m) = 1/;b 1 (m, ej) = (1/;;;10 1/;a 0 1/;b 1 )(m, ej) = 1/;;;l(m, 1/;ab(m) ej)

= L 1/;ab(m)ij . 1/;;;l(m, ei) = L 1/;ab(m)ij . ~nm) , i

w here the first equality of the second line follows from the fact that the1/; a is compatible with the A-module structure (actually, the A-module structure on each fiber is defined in this way [IV.3.2]). Using coordinates Xi on Ua n Ub we compute:

L ~

r~jk' 1/;ab P j' f:.~

= L j

r~jk' f:.~

= \7(Oxi,

f:.~)

= L i

i

\7(Ox i ,1/;ab k' f'l)

Chapter VII. Connections

372

Comparing coefficients of f~( m) and using that 1/Jba (m) is the inverse of 1/Jab( m) [IY.1.S] as well as (VI. 1.22) for matrix multiplication in terms of left coordinates, we obtain the relation

r ,bj k (m)

P f, j p + "'(_l)(E(xi)IE(et)+E(ek».f, . •f, j L..- o1/Jab ox i k . •'f/ba L..'f/ab i k . rap , £ 'f/ba p . p pi

= '"

Substituting this in the definition of r~ and using (VI.1.21) several times for matrix multiplication in terms of left coordinates for homogeneous but not necessarily even endomorphisms, we compute:

L dx i ® r~j k . ej ® ek ijk i = L dx i ®

rb=

(_l)(E(x )IE(e t )+E(e k »1/Jab £k

. r~p i . 1/Jba j p . ej ® ek

ijkpi P

" . o1/Jab k . k + 'L..-dx' ® ~. 1/Jba Jp' ej ®e ijkp

=L

dx i ®

i

(_l)(E(x )IE(e t )+E(e k »1/Jab i k

. (1/Jba °

rn i . ej ® ek j

ijki

'"

.

.

= L..-dX'®(1/Jbaor~o1/Jab)Jk·ej®e

k + 'L..-1/Jba " . o1/Jab o (dx'®( ox i ))

0k

i

= 1/Jba ora ° 1/Jab + 1/Jba ° d1/Jab

,

where the last equality follows (among others) from the identity df = 2:i dXi . oXi f, valid for any smooth function. Note that the composition symbol ° in the last line (and some in the line before that) is the alternative notation of the ,{} symbol. A faster proof, though less direct and demanding more explanation to justify all steps, is the following computation. Let S be a section, with local representative functions Sa == s"Ij;a and Sb == s"Ij;b' and let X be a vector field. Then Sa and Sb are related by Sa = 1/JabSb. Since a similar relation holds for the local representatives for \7x s, we have, using (6.14),

1/Jab(\7x S)b

=

(\7x s)a

= X Sa + t(X)ras a =

X(1/JabSb)

+ t(X)ra1/JabSb

+ t(X)ra1/JabSb = 1/Jab(XSb + t(X)(1/Jbaora o1/Jab + 1/JbaOd1/Jab)Sb)

=

(X 1/Jab)Sb + 1/JabX Sb

Comparing this with the expression (\7x sh result.

=

XSb

+ t(X)rb Sb

.

also gives the desired

IQEDI

6.17 Proposition. LetU = {(Ua, 1/Ja)} be a trivializing atlasfor the vector bundle Band suppose that on each Ua we have an even EndR(E)-valued I-form ra. If these r a are

§6. The covariant derivative

373

related to each other by (6.16), then there exists a unique covariant derivative \7 on B such that the fa are determined by \7 as in [6.9 J. Proof To prove existence, we construct \7 as follows. For any s E f(B) and any vector field X E f(T M) we define the section t = \7x s piecewise on each trivializing chart Ua by (cf. (6.14))

ta = XSa

+ t(X)fa sa . = 1/Jba (m )ta (m)

These local functions glue together if and only if they satisfy tb( m) (IV.1.21). We thus compute:

tb

=

X tb + t(X)fbtb

=

X( 1/JbaSa)

+ t(X) (1/Jba

0

fa o1/J;;;}

+ 1/J;;b1 d1/Jab) 1/Jbasa 0

1

=

(X1/Jba)Sa +1/JbaXsa +1/Jba t (X)fa sa +1/J;;b (X1/Jab)1/Jba Sa

=

((X 1/Jba)1/Jab + 1/JbaX 1/Jab) 1/Jba sa + 1/Jba (X Sa

+ t(X)fa Sa)

=

1/Jbata ,

where the last line follows because 1/Jba o1/Jab = id, implying that X (1/Jba 0 1/Jab) = O. This proves that t is indeed a well-defined section and thus that we have defined a map \7 : f(T M) x r(B) ---. f(B). That this map has the properties of a covariant derivative follows from the fact that the local defining formula has these properties. Let us show this for the third condition, leaving the others to the reader. For X, f, and S as required we define sections t = \7x sand U = \7x f s. In order to show that we have U = (X j)s + (-1) (E(X)IE(f)) ft, we show this for all local representative functions, using that the local representative functions respect the A-module structure:

U..p

= =

+ t(X)f fs..p = (Xj)s..p + (_l)(E(X)IE(f)) f(Xs..p + t(X)fs..p) (Xj)s..p + (_l)(E(X)IE(f))ft..p .

X(Js..p)

To prove uniqueness, it suffices to note that the action of \7 is completely determined by IQEDI the local f via (\7x s)..p = Xs..p + t(X)fs (6.14).

6.18 Corollary. Let 1[" B : B ---. M be a vector bundle with typical fiber E and let 1[" s : S B ---. M be the associated structure bundle. Then the following four objects (i) (ii) (iii) (iv)

an FVF connection on B, a covariant derivative on B, a (principal) connection on SB, a connection I-form on SB

are four incarnations of a same concept: all four objects are determined by local even EndR(E)-valued I-forms fa associated to a trivializing atlas and satisfying (2.15)/(6.16). Proof The only thing that has to be proven is that (2.15) and (6.16) are the same. But that is a direct consequence of [VI.2.13] and that 8 MC = g-l odg on Aut( E) [1.5]. IQEDI

374

Chapter VII. Connections

6.19 Definition. Just as FVF connections on a principal fiber bundle have the special name principal connection, so are FVF connections on a vector bundle usually called linear connections. For a linear connection one can also find the name affine connection in the literature. However, since there is nothing really affine in such a connection (the I-forms r a take their values in the linear group EndR(E), not in the affine group of the typical fiber E), the name linear connection should be preferred.

7.

MORE ON COVARIANT DERIVATIVES

In this section we study covariant derivatives on vector bundles in more detail. We start by showing that the link between a covariant derivative and an FVF connection is given by the fact that the covariant derivative measures how far a (local) section is away from being horizontal with respect to the FVF connection. Knowing that an FVF connection defines an FVF connection on a pull-back bundle, we show that the associated covariant derivative on a pull-back bundle again measures, in a sense to be made precise, how far away a lift is from being horizontal. In a similar way we can construct a covariant derivative on an associated vector bundle from one on the original bundle. Of this phenomenon we investigate several examples: the dual bundle, the bundle of homomorphisms and the second tensor power.

7.1 Discussion. In order to get a better understanding of the link between a covariant derivative and a linear connection, we have to delve deeper in the (coordinate) structure of a vector bundle. So let E be an A-vector space, e1, ... ,en a basis, and let (ie) and (e i ) be the associated left and right dual bases. As an A-manifold, E is modeled on E~ and it follows easily from [III. 1.26] (see also [Y.3A]) that for any vector vEE its (left) coordinates are given by the values yi(V) = ((vallie)) and yi(V) = (vdie>, where v = Va + V1 denotes the splitting into even and odd parts. And indeed we have

v=L

(vii ie))

. ei = L (va

+ vd ie)) . ei =

L(yi(V)

+ yi(V)) ei

.

To compute the fundamental vector field AE on E associated to A E EndR(E) (EndR(E) is the A-Lie algebra of Aut(E)), we have to be very careful in the use of coordinates. According to the definition ofa fundamental vector field, we have A{f = -(AIITv>, where denotes the left action of Aut(E) on E, and thus v(g) = g . v. According to [VI. 1.20], on Au t( E) we have to use left coordinates gij = eMR(g)i j [I1A.l] defined by g = I : i j gij ei ® ej . And then the tangent vector A E EndR(E) = Te Aut(E) is given j as A ~ I : i j Ai j 09i j ' when A itself is given as A = I : i j Ai j ei ® e . In terms of the coordinates gij on Aut(E) and yi, yi on E ~ Eg, the coordinates of gv are given by (g is even) j

pq

§7. More on covariant derivatives

375

and similarly j

pq

This gives us for the fundamental vector field the result

Now let 7r : B ----t Mbe a vector bundle with typical fiber E, let Hbe a linear connection on B and let \7 be a covariant derivative on B. Suppose furthermore that H and \7 are determined by the same local EndR(E)-valued I-forms a associated to a trivializing atlas U = {(Ua, 'l/Ja) I a E I}. Then H is given in the trivialization 'l/Ja by (see [2.12])

r

With these preparations we can state a more direct relationship between the linear connection H and the covariant derivative \7.

7.2 Proposition. Let 7r : B ----t M be a vector bundle with typical fiber E, let H be a linear connection on B and let \7 be a covariant derivative on B. Suppose furthermore that Hand \7 are determined by the same local EndR(E)-valued I-forms a associated to a trivializing atlas U = {(Ua, 'l/Ja) I a E I}. Thenfor any section S E r(B) and any vector field X on M we have

r

x

(\7 s)(m)

(7.3)

= 7ra( ((Xm I Ts)) - X~(m) ) .

Roughly in words: \7x s measures how far away ((XIITs)) is from being horizontal. Conversely, if Xm or s is homogeneous, then X~(m) is determined by (\7x s) (m) via (7.3). Proof Let Ua be the trivializing set containing m and let Sa : Ua ----t E be the function determining the section s in the trivialization 'l/Ja, i.e., 'l/Ja(s(m)) = (m, Sa(m)). Let furthermore (yi) be the (global) coordinates on Eo with their associated coordinates (It) on El (see [Y.3.4]). Then X~(m) is given in the trivialization 'l/Ja by -h X(m,Sa(m))

=

'( . 8 Xm - " ~ ((t(Xm)ra((sa)o)ll'e)) 8yi ISa(m)

,

+ {t(Xm)ra((sah)llie> ~Isa(m))

,

376

Chapter VII. Connections

and (( Xm I Ts)) is given in that same trivialization by (see [Y.3.4])

((XmII Ts )) = Xm

. 8iISa(m) 8 . +" L-( ((Xm(sa)oll'e> + ((Xm(sahll'e> Y

i

8 ) . ~Isa(m) uy

It follows that (( Xm I Ts > - X~(m) projects to zero under T7r and thus we can apply [Y.3.7]. This gives us in the trivialization determined by 'lj;a

7r{)

7r{)({XmIITs)) -X~(m)) = 2:({Xm(sa)olli e )) + ((Xm(sahll ie >+ i

((t(Xm)ra((Sa)O) Ilie))

+ ((t(Xm)ra((sah) Ilie> )ei

= Xmsa + t(Xm)ras a = (\7x S)a(m) , where the last equality follows from [6.14]. Since this is the local expression of (\7x s)(m) in the trivialization determined by 'lj;a, we have proven (7.3). If we know (\7x s)(m), we know the coefficients ((Xmsa + t(Xm)ras a Ilie>, and thus we know the coefficients (( t(Xm)ras a I ie whereas for X~(m) we need to know

>,

the values of {t(Xm)ra((Sa)a) Ilie)), for a = 0,1. If S is homogeneous, it is obvious that the former determines the latter. When it is Xm that is homogeneous, we can decompose (( t(Xm)ra Sa I ie)) in its homogeneous components which are given by the formula ((( t(Xm)r aSa I ie)) )E(ei)+E(X",)+a = (( t(Xm)r a( (Sa)a) I ie )), and thus again we obtain the desired result. IQEDI

7.4 Discussion. Let 7r : B ---+ M be a vector bundle with typical fiber E and let \7 be a covariant derivative on B. From [6.18] we know that \7 is equivalent to a linear connection H on B. But for FVF connections we have two ways to construct new ones: on pull-back bundles [2.16] and on associated bundles [5.1]. It follows that we have two ways to create new covariant derivatives out of the given one. We first concentrate on pull-back bundles. We thus consider a smooth map g : N ---+ M and the pull-back bundle g*7r : g* B ---+ N and the associated vector bundle map 9 : g* B ---+ B [IV.2.2]. Given the covariant derivative \7 on B and the associated linear connection H [6.18], we apply [2.16] to obtain an induced linear connection g*H on g* B. And thus by [6.18] we have an induced covariant derivative g*\7 on g* B associated to this linear connection. If U and V are (tri vializing) atlases as in the proof of [2.5] and if \7 is determined by the local EndR(E)-valued I-forms r a, then the induced covariant derivative g*\7 is determined by the local EndR(E)-valued I-forms a = g*rg(a). The natural question that arises is whether there is another way to define g* \7 which does not involve the local I-forms ra. According to the definition of a covariant derivative, if S is a section of g* B and X a vector field on N, then we have to define (g* \7) X S as a section of g* B. The natural idea that comes to mind is the following. Find a vector field Y on M related to X by g, i.e., \:In EN: Yg(n) = {Xn I Tg and find a section t of B such that tog = 9 0 s. And then define (g* \7) X S as

t

>,

g((g*\7)XS)

=

(\7y s)(g(n)) ,

§7. More on covariant derivatives

377

which is well defined because 9 is a bijection between (g*7r) -1 and 7r- 1 (g( n)). For a generic smooth map g there is no hope that such Y and t exist, but perhaps we might circumvent this problem. For Y we indeed can: in view of [6.4-ii] we could use any Y such that, at a fixed point n EN, we have Yg( n) = (( X n II T g)) in order to define (g* \7) X 8 at n EN. How such a Ybehaves elsewhere is of no importance for (\7y 8)(g(n)). However, no such trick is available for the section t. If g is injective, t must be given on g(N) by t(g(n)) = g(8(n)), and thus we have to extend this t outside the image g(N). This might not be easy depending on g(N). But when g is not injective, there is no hope to find a section t for a generic section 8 ofg* B. We thus have to be more subtle in our tentative to give a definition of g*\7 without using the local I-forms

ra.

7.5 Remark. In [6.4-iii] we have shown that there exists an induced covariant derivative on the restriction of the bundle to an open subset. But such a restriction is a particular case of a pull-back bundle [IV.2.3]. The fact that the proof of the existence of this induced covariant derivative is not immediate confirms that extending t outside of g(N) (as described in [7.4]) will not be an easy task, iffeasible at all.

7.6 Definition. Let 7r : B ----+ M be a fiber bundle with typical fiber E and let g : N ----+ M be a smooth map. A map (7 : N ----+ B will be called a lift of g if 7r 0 (7 = g, i.e., if we have a commutative diagram

B

(J"/

17r

---+ 9

M.

N

The set of all smooth lifts (7 : N ----+ B will be denoted as Lift g (B). A particular case of a lift is a section: a section 8 : M ----+ B is a lift of th identity map id : M --+ M, and thus r(B) = Liftid(B). If B is a vector bundle, the set Liftg(B) is in a natural way a graded COO(N)-module: for f E COO(N) and(7,T E Liftg(B) we define f '(7+T E Liftg(B) as

(J. (7 + T)(n)

=

f(n) . (7(n)

+ T(n)

.

7r : B ----+ M be a vector bundle with typical fiber E, let g : N ----+ M be a smooth map, and let g*7r : g* B ----+ N be the pull-back bundle with associated vector bundle map 9 : g* B ----+ B. Then the map J : r(g* B) ----+ Liftg(B) defined by J(8) = go 8 is an isomorphism of graded COO (N)-modules.

7.7 Lemma. Let

Proof Let us first show that J is a morphism of graded COO (N)-modules. For that choose f E COO(N) and 8, t E r(g* B). Then:

J(J.

8

+ t)(n) =

(go (J.

8

+ t))(n)

=

g(J(n)· 8(n)

+ t(n))

= f(n) . g(8(n)) + g(t(n)) = (J. J(8) + J(t))(n) ,

378

Chapter VII. Connections

where the second equality follows from the fact that 9 is a vector bundle morphism [IVA.I9]. To prove that 'J is injective, suppose s, t E f(g* B) are different, i.e., 3n EN: s(n) =1= t(n). Since both belong to thefiber (g*7r)-l(n) and since 9 is an isomorphism when restricted to a fiber, g(s(n)) =1= g(t(n)), i.e., 'J(s) =1= 'J(t). Finally to prove surjectivity, let lJ E Liftg(B) be arbitrary. Using the set theoretic definition of g* B, we define s : N ----+ g* B by s( n) = (n, lJ( n)), which indeed belongs to g* B because 7r(lJ(n)) = g(n) by definition ofa lift. Obviously g(s(n)) = lJ(n) by definition of 9 in terms of the set theoretic definition of g* B. To show that this s is smooth, we look at its representative in a local trivialization. Let V be a trivializing atlas for B and let U be an atlas for N such that for each Ua E U there exists Vg(a) E V such that g(Ua) C Vg(a). According to [IV.2.2] we have a local trivialization of g* B given by 'lj;a : (g*7r)-l(Ua) ----+ Ua x E of g* Band Xg(a) : 7r- 1 (Vg(a) ----+ Vg(a) x E of B. In terms of this trivialization the lift lJlua is given as lJlua(n) = (g(n), S(n)) for some smooth functionS: Ua ----+ E. And then slua is given as slua(n) = (n, S(n)), which is obviously smooth. IQEDI

7.8 Proposition. Let 7r : B ----+ M be a vector bundle with typical fiber E, let g : N ----+ M be a smooth map, and let g*7r : g* B ----+ N be the pull-back bundle. Let furthermore \7 be a covariant derivative on B and let H be the associated linear connection defined by the same local EndR(E)-valued I-forms. then the induced covariant derivative g*\7 on g* B is given by thefollowing procedure. For X a (smooth) vector field on Nand s a smooth section of g* B, denote lJ = 'J(s) = go s E Liftg(B) andT = 'J( (g*\7)(X, s)). Thenfor any n E N we have (7.9)

Roughly in words: T( n) measures how far (( Xn I TlJ)) is from being horizontal. Proof We use notation as in the proof of [7.7]. According to that proof, we have (locally) 'lj;a(s(n)) = (n, sa(n)) for some smooth function Sa : Ua ----+ E and simultaneously Xg(a)(lJ(n)) = (g(n), sa(n)). Iffg(a) is the EndR(E)-valued I-form on Vg(a) defining \7 and H, then g*\7 is determined by g*fg(a) :=:: fa [2.16]. Thus in the trivialization (Vg(a), Xg(a») the local expression for (g*\7)(X, s) is given as

((g*\7)(X, s)t(n) = Xnsa

+ t(Xn)fa . Sa(n) = XSa + t(((Xn IITg> )fg(a) . Sa(n)

.

On the other hand, with notation as in the proof of [7.2], we have

((XnIITlJ))

. 87lsa(n) 8 . ifijiISa(n) 8 ) = ((XnIITg)) + " L-( ((Xn(sa)oll'e> + ((Xn(sahll'e)) ;

y

y

and -------

h

((XnIITg))(g(n),Sa(n))

= ((XnIITg)) -

L,

(

. 8 ~t(((XnIITg)))fa((sa)o)II'e)) 8yilsa(m)

+ ((t«XnIITg»fa((sah)lli e> ~Isa(m») and then the result follows as in the proof of [7.2].

,

§7. More on covariant derivatives

379

7.10 Discussion. [7.7] tells us that the sections of the pull-back bundle are a natural generalization of sections of the original bundle. And then comparing (7.9) with (7.3) shows that the induced covariant derivative g*\1 is the natural generalization of \1 to these generalized sections. Having treated induced covariant derivatives on pull-back bundles, we now turn our attention to covariant derivatives on associated bundles. As before we let 1[" : B ----+ M be a vector bundle with typical fiber E and \1 a covariant derivative on B. But now we consider a representation, i.e., a homomorphism of A-Lie groups, p : Aut(E) ----+ Aut(F) for some A-vector space F. If H is the linear connection on B having the same local EndR(E)valued I-forms a as \1, then we know from [5.1] that there exists a linear connection HP on the associated vector bundle 1["P : BP,F ----+ M, associated to B by the representation p. And thus we have a corresponding covariant derivative \1P on BP,F. Moreover, HP and thus \1P is determined by the local EndR(F)-valued I-forms a = Tep(ra) [5.1]. As for the situation with pull-back bundles, the natural question is whether there exists a way to define \1 P which does not involve the local I-forms ra. The answer is positive for A-vector spaces F that are constructed out of E by natural operations such as F = E* or F = EndR(E). However, the answer depends upon the representation p, so we will give several examples to show the idea.

r

t

7.11 Proposition. IfF = A and p the trivial representation, i.e., Vg E Aut(E) : p(g) = 1, then BP,A is the trivial bundle BP,A = M x A and in the identification r(BP,A) ~ Coo (M), the induced covariant derivative is given by \1I s = X s (see [6.3 J).

If F = E* (the right dual of E) and p the natural representation, i.e., Vg E Aut(E) : p(g) = (g-l )*, then BP,E' = B* is the right-dual bundle and, using the identification r(B*) ~ r(B)* [IV.5.14], the induced covariant derivative \1P is the unique covariant derivative on B* such that for s E r(B) and for homogeneous a E r(B*), X E r(TM) we have

7.12 Proposition.

(7.13)

7.14 Proposition. If F = EndR(E) and p = Ad the adjoint representation, i.e., Vg E Aut(E), VA E EndR(E) : p(g)(A) = Ad(g)(A) = gAg- 1 [VI.2.13], then BAd,EndR(E) = EndR(B) is the bundle of right-linear endomorphisms ofB and, using the identification r(EndR(B)) ~ EndR(r(B)) [IV. 5. 9], the induced covariant derivative \1P is the unique covariant derivative on EndR(B) such that for s E r(B) and for homogeneous cP E r(EndR(B)), X E r(TM) we have

(7.15)

Chapter VII. Connections

380

7.16 Proposition. IfF = E !& E and p the natural representation, i.e., Vg E Aut(E) : p(g) = g ® g, then BP,E0E = B ® B is the tensor product of B with itself and, using the identification f(B ® B) ~ r(B) ® f(B) [IV.5.l7], the induced covariant derivative \1P is the unique covariant derivative on B ® B such that for t E r(B) and for homogeneous

X

E

f(TM), s E f(B) we have

(7.17)

Proofs . • [7.11] If U is any trivializing atlas for B (e.g., the full vector bundle structure), it also is a trivializing atlas for the associated bundle. But for the associated bundle all transition functions are constant 1, and thus we have a natural global trivialization of BP,A as BP,A ~ M x A. Since TeP = 0, it follows that all induced local I-fonns r'a = TeP 0 fa are identically zero. This shows that on any local trivializing chart (U, 'Ij;) we have (\1{ s),p = X s,p, and thus the result holds globally . • [7.12] Ifthe transition functions for B are given by 'lj;ab with respect to some trivializing atlas, then the transition functions of BP,E* are given by p( 'lj;ab) = ('Ij;;;b1 ) *, which are exactly the transition functions of the right-dual bundle B* [IVA.16]. To compute TeP we first note that we have identified Te Aut(E) with EndR(E) and (thus) Te Aut(E*)with EndR(E*). The representation P is the composition of the map Inv : Aut(E) ----+ Aut(E) with taking the dual map: * : EndR(E) ----+ EndL(E*). We see that the natural target space of P is the space of left linear endomorphisms of E*, not the right linear ones. However, since g E Aut(E) is even, it is left and right linear and it is equal to its transpose (1.2.16), and thus we can as well define p as p(g) = 'I'(g-l)*. The map A ~ 'I'A* is an even isomorphism from EndR(E) to EndR(E*). Combining [VI.1.6] with [V.3.3] we find:

TeP: A

~

-'I'A* .

Using (6.14) we now compute the local expression of\1{a: in a trivialization (U, 'Ij;): (\1{a:),p

= Xa:,p + t(X)(Tepo r) a:,p = Xa:,p + (TeP(t(X)r)) (a:,p) = Xa:,p - ('I'([t(X)f]*)) (a:,p) = Xa:,p - (_l)(E(X)!E(a»((a:,pII [t(X)f]*))

,

where the last equality follows from (1.2.16) and the fact that f is even and thus that we have c(t(X)r) = c(X). Since coo(Ui E*) is naturally isomorphic to COO(U; E)*, we can evaluate (\1{a:),p E COO(U; E*) on s,p E coo(Ui E), yielding

(\1{a:),p(s,p) = (Xa:,p)(s,p) - (_l)~(X)!E(a» ((a:,pll [t(X)f]*))(s,p) =

(Xa:,p)(s,p) - (-l)(E(X)!E(a»a:,p«t(X)r)(s,p))

= (Xa:,p)(s,p) -

(-l)(E(X)!E(a»a:,p«\1xs),p - XS,p)

= (Xa:,p)(s,p) + (_l)(E(X)!E(a»a:,p(Xs,p) =

- (-l)(E(X)!E(a»a:,p«\1x s ),p)

X(a:,p(s,p)) - (-l)(E(X)!E(a»a:,p«\1x s),p),

381

§7. More on covariant derivatives

where for the second equality we used [1.2.20]. Looking carefully at the identification f(B)* with f(B*) (see also [IVA.5]) shows that we have a(s)lu = a1f;(s1f;) E COO(U). The above computation thus gives us the equality

Since U is an arbitrary trivializing chart, we have shown (7.13). Uniqueness of \!P follows immediately from the fact that via (7.13) we know the action of \!{ a on any S E f( B) and thus we know \!{a itself. • [7.14] The transition functions for B being 1/Jba, those for BAd,EndR(E) are given by A ~ 1/Jba . A· 1/J"ba1. Comparing this with (IVA.17) shows that these are exactly the transition functions of the bundle EndR(B) of right-linear endomorphisms of B. Using (6.14) and [VI.2.15] we now compute the local expression of \!{¢ in a trivialization (U, 1/J): (\!{¢)1f;

= X¢1f; + t(X)(Te Ad =

X ¢1f;

+ t(X)f

0

0

r) ¢1f;

= X¢1f; + ad R (t(X)f)(¢1f;)

¢1f; - (-1) (E(X)IE(¢)) ¢1f; 0 t(X)f .

Since coo(U; EndR(E)) is naturally isomorphic to EndR(Coo(U; E)), we can evaluate (\!{¢)1f; E coo(U; EndR(E)) on s1f; E COO(U; E), yielding (\!{¢)1f;(S1f;)

= (X¢1f;)(s1f;) + t(X)r(¢1f;(s1f;)) = (X¢1f;)(s1f;) + t(X)r(¢1f;(s1f;)) = X(¢1f;(s1f;)) + t(X)r(¢1f;(s1f;)) -

(-l)(E(X)IE(¢))¢1f;(t(X)f(s1f;)) (-l)(E(X)IE(¢))¢1f;((\!x s )1f; - XS1f;) (-l)(E(X)IE(¢))¢1f;((\!x s )1f;) .

Analyzing the identification f(EndR(B)) ~ EndR(f(B)) shows that we have the equality ¢1f;(s1f;) = (¢(s))1f; and thus the above computation gives us ((\!{¢)(S))1f;

= X(¢(s))1f; + t(X)r(¢(s))1f; - (-l)(E(X)IE(¢))(¢(\!x s ))1f; = (\!x(¢(s)))1f; - (_l)(E(X)IE(¢)) (¢(\!xs))1f; .

Since this is valid for an arbitrary trivializing chart (U, 1/J), we have shown (7.15). Uniqueness of \!P follows as for [7.12] . • [7.16] If the transition functions for B are given by 1/Jab, then the transition functions of BP,E0E are given by p( 1/Jab) = 1/Jba ® 1/Jba, which are exactly the transition functions of the tensor product bundle B ® B [IVA.9]. Since the tensor product is indeed a product (as can be seen by computing explicitly the matrix elements of p(g)), the tangent map is given by

TeP : A in a local trivializing chart (U, (\!{ (s ® t))1f;

1/J)

~

A ® id + id ® A .

we thus obtain

= X(s ® t)1/J + t(X)(Tepo r)(s ® t)1f; = X(s ® t)1f; + (t(X)f ® id + id ® t(X)r)(s ® t)1f; .

382

Chapter VII. Connections

The identification r(B ® B) ~ f(B) ® f(B) is such that we have (s ® t),p = s,p ® t,p, and thus we obtain:

+ (t(X)f ® id + id ® t(X)r)(s,p ® t,p) + (_l)(E(X)IE(s))s,p ®Xt,p + (t(X)fs,p) ® t,p + (_l)(E(X)IE(s)) S,p ® t(X)ft,p = ('Vx S),p ® t,p + (-1) (E(X)IE:(s)) s,p ® ('Vx t),p = (('Vxs) ® t),p + (_l)(E(X)IE:(s))(s ® 'Vxt),p .

('VI (s ® t)),p = X(S,p ® t,p) = (XS,p) ®t,p

Since this is true for an arbitrary trivializing chart, we have shown (7.17). Uniqueness of

'V P follows from the fact that the tensor products s ® t generate f( B ® B) (because of the IQEDI isomorphism between f(B ® B) and r(B) ® f(B)).

7.18 Remark. The representation Ad in [7.14] can also be written as Ad(g) = (g-l) * ® g (see [IVA.16]). Combining the ideas of the proofs of [7.12] and [7.16] yields for Te Ad the formula

Te Ad : A

1--+

id ® A - 'I'A * ® id .

A careful study of the identification given in [1.7.12] for the right linear case shows that for homogeneous elements the action of 'I'A * ® B on an endomorphism cP is given by

('I'A* ®B)(cP)

=

(_l)(E(A)IE(B)+E(¢))B°cP°A.

We thus obtain

+ t( X) (id ® f - 'I'f* ® id) cP,p = X cP,p + t( X) (f ° cP,p - cP,p = X cP,p + t(X)f ° cP,p - (-1) (E(X)IE(¢)) cP,p t( X)f .

('VI cP),p = X cP,p

0

r)

0

This provides an alternative approach to the proof of [7.14].

8.

FORMS WITH VALUES IN A VECTOR BUNDLE

In previous sections we have introduced the exterior derivative of differential forms, the exterior covariant derivative on a principal fiber bundle and the covariant derivative on a vector bundle. In order to show how these objects are related, we have to generalize vector valued differential forms to differential forms with values in a vector bundle. This generalization is presented in this section.

383

§8. Forms with values in a vector bundle

8.1 Definition. Let p : B ----+ M be a vector bundle with typical fiber E. A k-form on M with values in B is a section of the bundle (I\k *T M) 129 B, just as an ordinary k-form is a section of the bundle (I\k *T M). In analogy with ordinary k-forms and vector valued k-forms, the set of all smooth B-valued k-forms will be denoted as nk (M; B), i.e.,

As for ordinary differential forms, we define n(M; B) as the direct sum over all k : co

= E9nk(M;B).

n(M;B)

k=O

Let (ei) be a basis for E and let (U, 'Ij;) be a local trivializing chart for B (i.e., a chart for M with coordinates Xi and trivializing for B). According to [6.7] dXil 1\ ... 1\ dxik 129 fj

forms a basis for a local section of the vector bundle (I\k *T M) 129 B, i.e., a (smooth) k-form w with values in B has the local expression

wlu =

(8.2)

j

w

wt, . .

with j = 2:i 1 , ... ,ik Wt, ... ,i k dX i1 1\ ... 1\ dXik and where the ,i k are (smooth) functions on U. The parity of the B-valued k-form w is determined in the usual way: c( w) = c(w j )+c( ej), where c( wj ) is the standard parity of the local k-form wj . Obviously the local k- forms w j on U are defined independently of the local coordinates xi. However, they need not glue together to form global k-forms on M. More precisely, let (Ua , 'lj;a) and (Ub, 'lj;b) be two local trivializations with associated local sections f't and f~ and associated local k-forms w~ and w{ The transition function 'lj;ba : Ua nUb ----+ Aut(E) for the bundle B defines (left) matrix elements 'lj;ba (m) i j == •M R ( 'lj;ba (m)) i j E A by the formula 'lj;ba (m )ei = 2: j 'lj;ba (m )i j ej [11.4.1], [VI. 1.20]. Combining the definition of the transition function with that of the local sections f't and f; gives us the equality

f't(m) =

L 'lj;ba(m)i j f~(m) . j

Since 2:i w~ 129 f't and 2: j also have the equality

wt 129 f; both represent the same section on Ua nUb, we must

(8.3)

This implies that if we want the local k-forms w~ to glue together to form a global k-form j w , then the transition functions 'lj;ba must be the identity. Said differently, they glue together to form a global k-form if the bundle B is trivial. In the general case the (local) k-forms wi depend upon the local trivialization (U, 'Ij;), but they also depend upon the choice for the basis (ei) for E. If e1, ... , en is another

384

Chapter VII. Connections

basis for E, it is related to the former basis by immediately follows that fi

= 2: j

j

Ai fj

ei =

and thus

2:j

wlu

j Ai ej

for (real) constants

= 2:i wi ® fi = 2: j

w

j

Aij.1t

® fj with

(8.4)

This formula looks exactly as (S.3) but its interpretation is completely different: (S.4) concerns a single trivialization (U, 'Ij;) and two bases, whereas (S.3) concerns two trivializations and a single basis. Moreover, the matrix elements 'lj;ba (m )i j in (S.3) depend upon the choice of the basis for E, whereas the matrix elements Ai j in (S.4) do not depend upon the trivialization. A way to hide the dependence of the local k-forms w j on the choice of a basis is to form the local E-valued k-form w'Ij; = 2: j w j ® ej, which is independent of such a choice. We thus have the similar looking formulae (8.5)

wlu

=L

wj

®fj

j

where the first formula is a description of the B-valued k-form restricted to a local trivializing chart (U, 'Ij;) with the trivialization hidden in the local sections fj' and where the second is a local E-valued k-form which depends upon the trivialization 7/J, as indicated by the subscript 'Ij;. This should be compared with (6.S), which can be seen as the special case of a O-form. Using the local E-valued k-forms w'lj; we can give another interpretation of (S.3). If (Ua , 'lj;a) and (Ub , 'lj;b) are two local trivializations, we can form the local E- valued k-forms Wa == w'Ij;a = 2:i w~ ® ei and Wb == W'Ij;b = 2: j wt ® ej. We now interpret 'lj;ba as an even EndR(E)-valued O-form on the intersection Ua n Ub and we want to compute the wedge-dot product 'lj;ba t.-. Wa [1.3-ii]. Since the matrix elements 'lj;ba (m )i j are defined in such a way that we have 'lj;ba = 2:ij 'lj;bai j ej ® e i , we compute according to (1.2): 'lj;ba

t.-. Wa =

L

7/Jbai

j

1\ ([E(ej )+E(ei) (w~) ® (ej ® e i )( ek)

ijk

=

L

'lj;bai

j

.

([E(ej)+E(ei)(w~) ® ej

L w~ ®

W~

'lj;bai

j

r??)ej

0

0k =

oi = L

7/Jba ei .

To obtain the second equality we used that 'lj;bai j is a function (O-form) and thus the wedge product is just the ordinary product; to obtain the third equality we used that 'lj;ba is even, and thus that c('Ij;bai j ) = c(ej) + c(ei). Using the alternative notation for 7/Jba t.-. Wa [1.3] we thus can rewrite (S.3) as (8.6) We conclude that, given a trivializing atlas U = {(Ua , 'lj;a)}, the B-valued k-form W is represented by a system of local E-valued k-forms Wa on Ua satisfying the compatibility

385

§8. Forms with values in a vector bundle

condition (8.6). It is not hard to show that conversely a system oflocal E-valued k-forms Wa on Ua satisfying the compatibility condition (8.6) defines a (global) B-valued k-form w. This description of B-valued k-forms should be compared with the description of sections of (vector) bundles given in [IV .1.20].

{(Ua,1/;a) I a E I} for the bundle Band a basis e1, ... , enfor the typical fiber E, any B-valued k-form W is represented by local k-forms w~, 1 :::; i :::; non Ua which are related on overlaps by (8.3) and which change all j at the same time by (8.4) in case we change the basis (ei) to eej) given byei = j Ai ej.

8.7 Corollary. Given a trivializing atlas U

=

2:

8.8 Discussion. In [VI.9.l] we have defined k-forms with values in an A-vector space. This can be seen as a particular particular case of k-forms with values in a vector bundle as follows. We first construct the trivial bundle B = M x E with its canonical global trivialization 1/; : B ----t M x E. In this (global) trivialization a B-valued k-form is given by n global k-forms Wi as w = 2:i wi~ §.i' From this we can obtain the global E-valued k-form 2:i wi ® ei. Since they depend in the same way on the chosen basis for E, we get a bijection between B-valued k-forms and E-valued k-forms. In this sense an E-valued k-form is a particular case of a vector bundle valued k-form. However, whereas we are allowed to change the (global) trivialization when we consider B-valued k-forms, we are not allowed to change the trivialization for E-valued k-forms. E-valued k-forms correspond to B-valued k-forms in a given fixed global trivialization. Another way to interpret this particularity is to play with the various identifications as in [V.7.l]. This gives us thatr((N *TM)®(MxE)) is isomorphic to Homtk(*f(TM)k; f((MxE))). Using a (global, fixed) trivialization 1/; [IV. 1.20] we obtain an isomorphism between f(M x E) and COO(M; E) given by S ~ s'Ij;. Since the latter space represents the space of E-valued k-forms, we obtain an identification between k-forms with values in the trivial bundle M x E and E-valued k-forms. However, this identification obviously depends upon the chosen global trivialization 1/;. Hence the statement that E-valued k-forms are a particular particular case of k-forms with values in a vector bundle.

8.9 Discussion. By definition, a B-valued O-form is a section of the (vector) bundle (N *TM) ® B = (M x A) ® B, where N *TM ~ M x A comes with a canonically defined trivialization. Using the natural isomorphism A ® E ~ E [1.4.10] we obtain a natural isomorphism (M x A) ® B ~ B [IV.4.23]. It follows that a B-valued O-form can be identified with a section of B. In terms of local representations this amounts to omitting the tensor product symbol: if the B-valued O-formw is locally given by wlu = 2: j w j ®§'j, then the wj are O-forms, i.e., ordinary functions. It is identified with the (global) section of B whose local expression is given by 2: j w j . §'j' Now suppose that B is the trivial bundle B = M x A with its canonical trivialization. Using the same techniques as above, we obtain an identification between I\k *T M and (N *TM) ® (M x A). It follows that k-forms with values in the trivial bundle M x A are

386

Chapter VII. Connections

just ordinary k-forms, i.e., vector bundle valued k-forms are a generalization of ordinary k-forms.

8.10 More definitions. Playing around with the various identifications as in [Y.7.1], we *T M) ® B) is isomorphic to Hornik (*r(T M)k; f(B)) [8.8]. already argued that f( In the same way we can show that it is isomorphic to (N *r(TM)) ® r(B). The first isomorphism allows us to identify B-valued k-forms as skew-symmetric k-linear maps (over COO(M)) from vector fields on M (sections of TM) to sections of B. The second isomorphism allows us to define the contraction of a B-valued k-form w with a vector field X to yield a B-valued (k - I)-form t(X)w. There are at least two ways to define this contraction. The first is to apply [1.6.16], which does not yield a directly applicable formula. The second is to note that the contraction t( X)w is such that, as a (k - 1 )-linear skew-symmetric map, this B-valued (k - I)-form is given by

(N

However, the most useful formula to define this contraction is by using the local expression (8.2) and to note that the contraction is given by the local formula

(t(X)w)lu

=

L t(X)w1 ® f.j . j

Using the compatibility condition (8.3), it is not hard to show that one can use the above local expression to define the global contraction t( X)w, without any reference to the various identifications described above. Moreover, using the local E-valued k-forms w..p (8.5), it is immediate that we have t(X)(w..p) = (t(X)w)..p.

8.11 Remark. In [VI.9.1] we have defined the pull-back of a k-form with values in an A-vector space. This can be seen as a special case of a more general notion of a pull-back of vector bundle valued k-forms involving the pull-back bundle. Since we will never need this more general notion, we do not go into the details of the more general notion.

8.12 Definition. In [1.1] we have defined the -wedge product of vector valued differential forms. We now want to extend this definition to the case of arbitrary vector bundles. So let E, F, and G be three A-vector spaces with homogeneous bases (ei), (Ij), and (gk) respectively, and let : Ex F ---+ G be an even smooth bilinear map. Suppose furthermore that B ---+ M, C ---+ M, and D ---+ M are vector bundles over M with typical fibers E, F, and G respectively. If a is a B-valued p-form and f3 a C-valued q-form, the -wedge product a 1\4> f3 should be a D-valuedp+ q-form, all over M. The idea of the construction is as follows. In any common trivializing chart U c M for the bundles B, C, and D, the forms a and f3 determine ordinary differential forms ai and f3j on U according to

f3lu = Lf3 j

j

® f.j

.

§8. Forms with values in a vector bundle

387

We now introduce matrix elements for by ( ei, ij) = 2:k 7j gk. With these matrix elements, the D-valued p + q-form a !\ (3 should be defined on the trivializing chart U by

(a!\ (3)lu =

(8.13)

La

i

!\ 7j flk .

i,j,k

The main problem is that these local expressions need not glue together to form a globally well defined D-valued p + q-form.

8.14 Lemma. Let U = {Ua I a E I} be ajoint trivializing atlasfor the three bundles B, C, and D, and let 'l/;ba be the transition functions for B, Xba those for C, and cPba those for D. If the map intertwines these transition functions in the sense that for all mE Ua n Ub and all e E E, i E F we have

then the -wedge product a !\ (3 is globally well defined by (8.13). Proof On two trivializing charts Ua and Ub we have, as in [S.l], local p-forms a~ and a~ and local q-forms (3~ and (3{ They satisfy the relations (S.3) a~lm = 2:£ a~lm 'l/;ba(m)£i and (3tlm = 2: s (3~lm Xba(m)J With these we compute:

L

ai,lm!\ 7j gk

i,j,k

=

L

a;lm 'l/;ba(m)£i !\ (ei, ij)

i,j,i,s

=

L

a;lm!\ ('l/;ba (m)£i ei, Xba(m)sj ij)

i,j,i,s

= L a;lm!\ ('l/;ba(m)e£, Xba(m)is) e,s =

L a;lm !\ ((3~lm) ® cPba(m)(e£, is) e,s

= L a;lm!\ is

cPba(m)gr

i,s,r

=

L

a;lm!\ is cPba(m)/ gk .

i,s,r,k

To obtain the second equality, we have used, among others, that 'l/;ba(m) is even and thus that the parity of 'l/;ba (m )e i is c( ei) + c( ee). Replacing gk by g% in the first and last line of this computation and noting that 2:k cPba (m)r k fl% = fl~' we -may conclude that the local expressions on Ua and Ub coincide. IQEDI

Chapter VII. Connections

388

8.15 DiscussionIDefinition. If a given joint trivialization of the bundles B, C, and D satisfies the condition of [8.14], it is by no means guaranteed that this condition is satisfied for another joint trivialization. However, in the applications there usually is a natural choice for the joint trivializations such that the condition is automatically satisfied. (This is analogous to the situation in [IVA.23].) The first application we have in mind is the case that all three bundles are trivial: B = M x E, C = M x F, and D = M x G, all three with a fixed trivialization. In other words, the case of k-forms with values in A-vector spaces. In that case the local formula (8.13) becomes the global definition:

C~= oJ ® ei) i

!\

(2:,8j ® fj) = j

2: a i !\ ([E(e

i

)

(,8j) ® iP( ei, fj) .

i,j

Since the trivialization is fixed from the beginning, no problems arise here, no matter what map iP we use. And we recover the previous definition for vector valued differential forms. The second application is the case in which we start with a bundle P -> M with typical fiber E, and then form the Hom-bundle B = C = D = HomR( P; P) with typical fiber EndR(E) [IVA.16]. In that case the application iP is just composition o : EndR(E) x EndR(E) -> EndR(E), (cP, X) ~ cP 0 X. Since all three bundles are formed from one and the same bundle P, any trivializing atlas for P will do. Since the transition functions of the Hom-bundle are given by the formula (IV.4.17)

it is immediate that composition intertwines these transition functions. And thus in this application the iP-wedge product is well defined. The third application is the case in which C is an arbitrary vector bundle with typical fiber E and D an arbitrary vector bundle with typical fiber F, but where B is the Hombundle B = HomR(C; D) with typical fiber HomR(E; F). In that case the map iP is evaluation HomR(E; F) x E -> F, (A, e) ~ Ae. And again the particular form of the transition functions of the Hom-bundle guarantee that they intertwine with this iP. And thus again the iP-wedge product is well defined. The fourth application is the case in which iP : A x E -> E is left multiplication by elements of A. Here the bundle B is the trivial bundle B = M x A and C = D is an arbitrary vector bundle. Using the canonical global trivialization for B and the same transition functions for C and D, it is immediate that left multiplication intertwines these transition functions, and thus the iP-wedge product is, again, well defined.

8.16 Discussion. In the fourth application, the k-forms with values in the trivial bundle = M x A are just ordinary k-forms. We thus have defined a map

B

Ok(M) x Oi(M; C)

->

Ok+i(M; C) .

We can extend this map by bilinearity to a map O(M) x O(M; C) -> O(M; C). Since O(M) is a Z x Z2-graded commutative ring, the reader can easily establish the fact that this map gives O(M; C) the structure of a Z x Z2-graded O(M)-module.

§9. The covariant derivative revisited

389

8.17 Notation. Each (admissible) map has its associated -wedge product which we denoted as !\. For the maps of the explicit applications we will use the same notation as for vector valued forms [1.3]: for the composition map in the second application we will denote the -wedge product of a and (3 by a {}. (3 ; for the evaluation map in the third application we will denote it by a t.-. (3 ; for left multiplication in the fourth application we will denote it simply by a !\ (3. And just as for vector valued forms, there are sometimes several ways to write the same object: if g is a O-form on M, i.e., an ordinary function, and (3 a B-valued k-form, then the B-valued k-form g !\ (3 is the same as g . (3; if A is a HomR(C; D)-valued O-form and (3 an C-valued k-form, then it is customary to write A 0 (3 instead of At.-. (3; and if a is a HomR( C; D)-valued k-form and sa C- valued O-form, i.e., a section of C, then it is customary to write a( s) or a . s for a t.-. s.

9.

THE COVARIANT DERIVATIVE REVISITED

In this section we show that a covariant derivative on a vector bundle can be extended to an operation on differential forms with values in the vector bundle, the original covariant derivative being the special case of O-forms. Seen this way, the covariant derivative is a straightforward generalization of the exterior derivative as a covariant derivative on the trivial bundle M x A. At the end of this section we define the curvature tensor of a covariant derivative as the square of the covariant derivative and we prove that the FVF connection on the vector bundle associated to the covariant derivative is integrable ifand only if this curvature tensor is zero.

9.1 Definition. In [VI.9.l] we defined the exterior derivative of a k-form w with values in an A-vector space E by the formula

It is tempting to think that one can generalize this to vector bundle valued k-forms simply by using the local representation: if wi u = I:j w j ® f j ' then (dw) Iu = I:j (dw j ) ® f j ' However, M-dependent transition functions prevent these local definitions to glue together to a well defined global definition. Another idea might be to generalize the original definition of the exterior derivative for ordinary k-forms [V.7.3] in terms of contraction with vector fields and the action of vector fields on functions. However, when one tries to define the action of a vector field on a section of an arbitrary vector bundle, one encounters the same problem: local representations do not glue together to a global object. A solution to this problem, already discussed in [6.1], is given by the covariant derivative. In order to prepare the actual generalization of the exterior derivative to bundle valued k-forms, we first note that the map (X,!) 1--4 XI, which associates a vector valued function X I to a vector field X and a vector valued function I, is an example of a

390

Chapter VII. Connections

covariant derivative [6.3]. We thus could write 'Vx j = X j. By definition of the exterior derivative, the action of X on j can also be written as X j = t( X)dj. We thus can write

'VX j = t(X)dj . This analogy between covariant derivative and exterior derivative can be extended to arbitrary vector bundles as follows. Let 1[" : B ----+ M be a vector bundle with typical fiber E and let 'V be a covariant derivative on B. For a fixed section s E f(B), the map 'Vs : X ~ 'V(X, s) from f(TM) to f(B) is linear over COO(M) [6.2]. According to [8.10] we thus can see it as a B-valued I-form. It follows that we can see the covariant derivative 'V as a map OO(M; B) ----+ Ol(M; B), s ~ 'Vs, just as we can see the exterior derivative as a map from functions to I-forms. With these analogies, we are ready to extend the notion of a covariant derivative to a map 'V : O(M; B) ----+ O(M; B) of parity (1,0) using a formula analogous to that for the exterior derivative (V.7.4) (recall that O(M; B) is a Z x Z2-graded O(M)-module [8.16]). To that end we start with a B-valued k-form a on M, which we see as a k-linear graded skew-symmetric map overCOO(M) from vector fields to sections of B [8.10]. We then define 'Va as the B-valued (k + I)-form on M of the same parity as a, seen as a (k + I)-linear graded skew-symmetric map (over COO (M)) from vector fields on M to sections of B, by the formula (9.2)

(_I)k. t(Xo, ... , X k ) 'Va = i+ I:(E(Xp)IE(X i )) = (-1) p<' 'VXi (t(X o, ... , X i -

L

j+

+

(-1)

I: i
1,

Xi+l, ... , X k ) a)

1,

[Xi, Xj], X i+1, ...

(E(Xp)ldXj))

t(Xo, ... , X i -

... , Xj -

1,

Xj+l, ... , X k ) a .

And as for the exterior derivative, this formula needs some explanation. The vector fields Xi are supposed to be homogeneous; taking the sum over homogeneous parts we obtain a k + I-additive function of vector fields on M to sections of B.

9.3 Lemma. The map 'Va given by (9.2) is (k + I)-linear over COO(M) and graded skew-symmetric. It thus defines a B-valued (k + I)-form on M.

Proof The proof is a direct copy of the corresponding proof for the exterior derivative [V.7.S] and is left to the reader. IQEDI

9.4 Discussion. If U c M is open, it is itself an A-manifold. Restricting a and 'Va to U gives us a Blu-valued k-form alu and a Blu-valued (k + I)-form ('Va)lu on U. Denoting the induced connection on U by 'V u [6.6], we also have the associated map'V u acting on Blu-valued differential forms. As for the exterior derivative we are thus faced with the question whether ('Va)lu equals 'V u (alu).

391

§9. The covariant derivative revisited

9.5 Lemma. For any open U eM: (\7a)lu

= \7U(alu).

Proof The proof is left to the reader. It is a direct copy of the corresponding prooffor the IQEDI exterior derivative [V.7.7], now using [6.4] instead of [V.loS].

9.6 Discussion. Once we know that the induced map on B-valued differential forms commutes with restrictions, it is customary (again) not to attach the superscript U to the symbol \7 when restricting to an open subset U c M. As for the exterior derivative, (9.2) is not meant for easy calculation. Using local trivializations, [9.7] gives us formulae that do allow more explicit calculations.

9.7 Proposition. Let a be a B-valued k-form (3 an ordinary I!-form on M, and let (U, 'Ij;) be a local trivialization for the bundle B ----t M. Then:

(i) denoting by f the local EndR(E)-valued I-fonn on U defined by the covariant derivative \7 [6.9J, we have

(ii) \7((3/\ a)

= (d(3) /\ a + (-1)£(3/\ \7a.

Proof. Both sides of the equality (\7a),p = da,p + f ~ a,p are E-valued (k + I)-forms, which we interpret as (k + 1)-linear skew-symmetric maps from vector fields to functions. We thus show that both sides give the same result when evaluated on (k + 1) vector fields. We first concentrate on the term f ~ a,p :

I : (E(Xp)IE(Xk»

=

(-I)p
. t(Xk)f. t(Xo, ... , Xk-da,p

- t(Xo, ... , Xk-d(f ~ t(Xk)a,p) , where the first equality is a consequence of the fact that t(X k ) is a derivation of degree (-1, c(Xk)) [1.6.16] and that f is an even I-form; the second equality a consequence of the fact that t(Xk)f is a O-form of parity c(Xk). By induction this gives us

L O~i~k

i+ I:(E(Xp)IE(Xi»

(-1)

p
t(Xi)f· t(Xo, ... , Xi-I, Xi+I, ... , Xk)a,p .

392

Chapter VII. Connections

With this preparation, we compute:

(_l)k . t(Xo, ... , Xk)(\7a).;; =

H

(-1)

+

H

L

(-1)

2:

(E(Xp)IE(Xj»

t(XO, ... , Xi-I, [Xi, X j ], Xi +1 , ...

i
2: (E(Xp)IE(Xi»

(

Xi(t(XO, ... ,Xi-I,Xi+I, ... ,Xk)a.;;)

p
0:<:; i:<:; k

+ t(Xi)r . t(XO, ... , Xi-I, XHI , ... , X k ) a.;; ) H

+

=

(-1)

2:

(E(Xp)IE(Xj»

t(Xo, ... , Xi-I, [Xi, X j ], X HI , ...

i
(_l)k. t(Xo, ... , Xk)(da.;;)

+ (-l)k t(Xo, ... , Xk)(r t.-. a.;;) .

In this computation we used the general equality t(X)(w.;;) = (t(X)w).;; [8.10]. To obtain the second equality we used (6.14) and to obtain the last equality we used the definition of the exterior derivative (V.7.4) and (9.8). This proves (i) . • To prove (ii) we first note that (3 is a global I-form and thus for a local trivialization (U, 'Ij;) we have (3.;; = (3. As a I-form with values in the trivial bundle, we should officially write (3.;; = (3 ® 1 with 1 the canonical basis of A. Applying the definition of (3/\ a [1.3] this means that if a.;; = 2:i a i ® ei, then ((3/\ a).;; = 2:i((3/\ a i ) ® 1· ei. Using the basis (ej ® ek ) of EndR(E), the local even I-form r can be written as r = 2: j k r j k ej ® ek (6.11). Applying the definition of wedge-dot product we thus find:

r

t.-. ((3/\ a).;; =

Lr

j

k /\

r[E(ej0

ek

) ((3/\

a i ) ® (ej ® ek )(ei)

ijk ijk = (-1/(3/\ (r t.-. a.;;) ,

where we used that r k is a I-form of parity c( ej ® ek ) to obtain the second equality. We now use (i) to compute (\7((3/\ a)).;; in an arbitrary local trivialization (U, 'Ij;): j

+ r t.-. ((3/\ a).;; = (d(3) /\ a.;; + (_1)i! (3/\ da.;; + (_1)i! (3/\ r t.-. a.;; = (( d(3) /\ a + (-1) f (3 /\ \7 a).;; ,

(\7((3/\ a)).;; = d((3/\ a).;;

§9. The covariant derivative revisited

393

where we used [V.7.9-iiiJ and (VI.9.4) to obtain the second equality. Since the trivialization was arbitrary, we have proven (ii). IQEDI

9.9 Remark. Another approach to a covariant derivative consists in defining it as an Rlinear map of parity (1,0) on n(M; B) satisfying the Leibniz type rule [9.7-iiJ. Starting from this definition, one can prove the existence oflocall-forms r such that [9.7-iJ holds. In that way one recovers our original definition!construction of a covariant derivative. Apart from the minor difference that we use a bi-grading on n(M; B), this is the approach taken in [QuJ.

9.10 Theorem. Given a covariant derivative \7 on a vector bundle 1[" : B ----; M, there exists a (unique) even EndR(B)-valued 2-form R, such that for all E-valued k-forms we have

\7(\7a)=Rt.-.a. In a local trivialization (U,1j;)for E with r the local EndR(E)-valued I-form on U defined by the covariant derivative \7 [6.9], R is given by

Proof If R exists, then in particular for any s E f(B), any two vector fields X, Yon M and a point m E M we must have

(([(Xm, Ym)Rlls(m))) = ([(X, Y)\7(\7a))(m) . Since [(Xm, Ym)R E EndR(Bm) is uniquely determined by its action on elements in Bm, uniqueness of R follows. That it must be even follows from the fact that \7 does not change parities. To show existence, we proceed as follows. In any local trivialization (U, 1j;) we have

(\7(\7a)),p = d(\7a),p + r t.-. (\7a),p = d(da,p + r t.-. a,p) + r t.-. (da,p = d(r t.-. a,p) + r t.-. da,p + r t.-. (r t.-. a,p) = d r t.-. a,p - r t.-. da,p + r t.-. da,p + (r .(} r) t.-. a,p = (d r + r .(} r) t.-. a,p ,

+r

t.-. a,p)

where the minus sign after the fourth equality comes from the fact that r is a I-form. This computations shows that, if R exists, it is locally given by the announced formula

R,p = d r

+ r .(} r.

If we now take for a a .B-valued O-form s, i.e., a section of E, and if we fix two vector fields X, Yon M, we obtain a map R(X, Y) : f(B) ----; r(B), S

1---+

[(X, Y)\7(\7s) .

394

Chapter VII. Connections

The local expression for \7(\7s) shows that this map is right linear over COO(M), i.e., R(X, Y) is an element ofEndR(f(B)). Since this space is isomorphic to f(EndR(B)) [IV.S.9], [IV.S.13] we obtain a map R : r(TM)2 ---t f(EndR(B)), (X, Y) ~ R(X, Y). This map is obviously graded skew-symmetric and bilinear over COO(M). It thus defines

an EndR(B)-valued 2-form R as announced. Coming back to the local expression for \7 (\7 s) shows that the local expression of R is given by R..p = d f + f {} f. And then the local expression of\7(\7a) for an arbitrary B-valued k-form shows that we have the IQEDI general result \7(\7a) = R t.-. a.

9.11 DefinitionIDiscussion. The EndR(B)-valued 2-form R whose existence is guaranteed by [9.10] is called the curvature tensor of the covariant derivative \7. We have defined it in terms of the extension of \7 to an operator on n( M; B) as R = \7 2 . The usual approach to the curvature tensor, which does not make use ofthis extension, starts from the observation that \7x s is the generalization of the derivative of a function in the direction of the vector field X. Since the commutator of two vector fields is defined in terms of the action of vector fields on functions, it is natural to ask to what extent the covariant derivative in the direction of a commutator corresponds to the commutator of the covariant derivatives. More precisely, if X and Yare two vector fields on M and s a section of B, one wants to know whether [\7x ' \7y ]s == \7x (\7y s) - (-l)(o(X)lo(Y»\7y (\7x s) is the same as V[X,Yls, Looking atthe official definition of \7 applied to B-valued I-forms we find:

-[(X, Y)Rs = -[(X, Y)\7(\7s)

= \7x ([(Y)\7s) - (-l)(o(X)lo(Y»\7y ([(X)\7s) - [([X, Y])\7s = \7x (\7y s) - (-l)(o(X)lo(Y»\7y(\7x s) - V[X,Yl s = [\7x , \7y ]s - V[X,Yl s .

We conclude that the difference between [\7x ' \7y ]s and V[x,Yl s is linear over COO(M) in s and that it is measured by the curvature tensor. The usual definition of the curvature tensor takes the equation -[(X, Y)Rs = [\7x , \7y ]s - V[x,Yl s as the defining equation for R.

9.12 Proposition. Let \7 be a covariant derivative on the vector bundle 1[" : B ---t M and let H be the associated linear connection on B. The connection H is integrable if and only if the curvature tensor R of \7 is (identically) zero. Proof According to [9.10], R is (identically) zero if and only if the local expressions R..pa = df a + ~[fa f,\fa] are zero for a trivializing atlas U = {(Ua,'ljIa) I a E I}. According to [2.18] this is the case if and only if H is integrable. IQEDI

§ 10. Principal fiber bundles versus vector bundles

10.

395

PRINCIPAL FIBER BUNDLES VERSUS VECTOR BUNDLES

In this section we show that a special kind of differential forms on a principal fiber bundle are in 1-1 correspondence with differential forms with values in an associated vector bundle. Moreover, we prove that under this identification the exterior covariant derivative on the principal fiber bundle corresponds to the covariant derivative on the associated vector bundle.

10.1 Definition. Let 7r : B ----+ M be a fiber bundle and let E be an A-vector space. An E-valued k-form a on B is said to be horizontal if for allb E Band allX E Vb we have t(X)a = 0, where V == ker(T7r) C TB denotes the vertical subbundle [2.2]; in words: a is horizontal if the contraction with a vertical tangent vector yields zero. In particular, any E-valued function on B (a O-form) is horizontal.

10.2 Lemma. Let 7r : B ----+ M be a fiber bundle with typical fiber F, let (U, 1/;) be a trivializing coordinate chart with coordinates xi, let E be an A-vector space, and let a be an E-valued k-form on B. The restriction al 7r -l(U) is horizontal if and only if there exist E-valued functions fi, ... ik on U x F such that (10.3)

((1/;-l)*a)l(m,!)

=

L

dX i1 1\ ... 1\ dxik ® fi, ... ik (m, 1) .

il ... ik

Proof If yj are (local) coordinates on F, then a basis for the space of k- forms on U x F at (m,1) E U x F is given by the set of k-forms

with p, q E N, p+q = k and j1, ... , jp and i 1 , ... , iq increasing sequences as in [11.3.16]. It follows that ((1/;-1) *a) I(m,!) is a linear combination of these basis k-forms:

p=O j1 , ... ,jp il , ... ,iq

In the trivialization 1/;, the vertical space V is spanned by the tangent vectors Oyj. It follows immediately that if (10.3) is satisfied, then t( X)a = 0 for all X E Vb. We thus suppose that a is horizontal and we wantto prove that all coefficients ,jp;i , , ... , iq are zero for p > O. Since a is horizontal, it follows in particular that all contractions t( Oyj) (( 1/;-1 )*a) I(m,!) must be zero. For simplicity's sake we start considering contraction with Oyl:

f!t ,...

396

Chapter VII. Connections

where the integer n1 E N is such that 1 = j1 = ... = jnl < jnl +1. Since the sequences j2, ... ,jp are still of the form given in [11.3.16], the (k - 1)-forms dy12 1\ ... 1\ dyjp 1\

dX i1

1\ ... 1\

dx iq are independent. It follows that all coefficients

f!t, ... ,jp;il, ... ,iq with

j1 = 1 must be zero, i.e., we can restrict attention to increasing sequences j1, ... ,jp as in [11.3.16] with j1 ~ 2. But then we contract with Oy2 to conclude that we also can discard sequences j1, ... ,jp starting at j1 = 2. Continuing this way we conclude that horizontality of a implies that all coefficients JJl, ... ,jp;i1, ... ,i q with a sequence of j's must IQEDI be zero. In other words, we have proven (10.3).

10.4 Definition. Let 7r : P --+ M be a principal fiber bundle with structure group G, let E be an A-vector space, and let p : G --+ Aut(E) be a representation. An E-valued k-form a on P is said to be of type p if for all g E G we have ;a = p(g-l) 0 a, where g denotes the right action of g E G on P [IV.2.12]. The interpretation of this condition is analogous to that of condition (ii) of a connection I-form: ;a denotes the generalized pull-back as defined in [V.7.23], [VI.S.12]. p(g-l) 0 a is an example of the alternative notation [1.3] for the evaluation-wedge product of an EndR(E)-valued O-form with an E-valued k-form. More precisely, for Xl, ... , X k E TpP and g E G we must have

t( ((XdTg )), ... , ((XdTg)) )a p . g == t( (( (Xl, 0) I T
In particular afunction

f :P

--+

p(g-1)(t(X 1, ... , Xk)a p) .

E is of type p if f (pg)

= p(g-l) (J (p)).

10.5 Examples . • A connection I-form w on a principal fiber bundle is a g-valued I-form of type Ad according to condition (ii) [3.7]. According to condition (i), it can never be horizontal. However, if Wo is another connection I-form on the principal fiber bundle P, then the difference w - Wo is of type Ad and horizontal, because the fundamental vector fields x P generate the vertical subspace V. • The curvature 2-form n = Dw of a connection I-form w is by definition horizontal. It is also of type Ad as can be seen directly from (4.10).

10.6 Discussion/Construction. If 7r : B --+ M is a fiber bundle and E an A-vector space, then the set nk(B; E) of all (smooth) E-valued k-forms is a graded Coo(B)-module. If a E nk (B; E) is horizontal, its homogeneous components are also horizontal. It follows easily that the subset n~or (B; E) of all horizontal E-valued k-forms is a graded submodule. Now let 7r : P --+ M be a principal fiber bundle with structure group G. If an E-valued k-form a E nk(p; E) is of type p for some representation p : G --+ Aut(E), then the product fa of a with an arbitrary f E Coo (P) will not in general be again of type p. On the other hand, since 7r is surjective, the map Coo(M) --+ Coo(P), f ~ f 0 7r is an injective morphism of graded R-algebras. We thus can interpret Coo(M) as a subalgebra

§ 10. Principal fiber bundles versus vector bundles

397

of Coo (P) and we can see nk (P; E) as a graded Coo (M)-module. Since Jr 0 9 = Jr, it follows easily that if a is oftype p and f E Coo(M), then the product fa is again of type p. We conclude that the set n~(p; E) of all (smooth) E-valued k-forms of type p is a graded COO (M)-module. A particular consequence of the preceding discussion is that the intersection n~or,p(P; E) = n~(p; E) n n~or(P; E) of all smooth horizontal E-valued k-forms of type p is a graded COO (M)-module. In order to lighten the notation a bit, we will denote, in the rest of this discussion, the vector bundle pp,E associated to P by the representation p as B = pp,E. Our purpose now is to construct an isomorphism 'J : nk (M; B) ---+ n~or,p(P; E) between the graded COO (M)-module nk(M;B) of all (smooth) B-valued k-forms and the graded Coo(M)_ module n~or,p (P; E). The actual construction starts with the choice of a trivializing atlas

U = {(Ua, 'lj;a) I a E I} for P and the associated trivializing atlas {(Ua, :;fa) I a E I} for B [IV.2.l]. Now if a E nk(M; B) is a B-valued k-form on M, then it is represented by local k-forms a~ as al 7r -l(Ua) = I:ia~ @~i (8.2). If'lj;ba is a tra~sition function for P, then the corresponding transition function 'lj;ba for B is given by 'lj;ba = po 'lj;ba [IV.2.1]. It follows that the local k-forms a~ are related on overlaps Ua n Ub by (8.3):

a~lm = 2:a~ p('Ij;ba(m))i j

.

On Ua x G we now define the E-valued k-form aa by

aal(m,g) = p(g-l) 2: a~lm ® ei

(l0.7)

0

Obviously aa is horizontal and aal(m,gh) of type p. Moreover,

.

= p(h-l) aa I(m,g), i.e., aa 0

abl(m,1/>ba(m).g) = p(g-l) 0 p('Ij;ba(m)-l) 0

is horizontal and

(2: a~lm ® ej) j

=

p(g-l) 0 p( 'lj;ba(m)-l) 0

(2: a~ 1m p('lj;ba( m) )i j ® ej)

= p(g-l) p( 'lj;ba(m)-l) (2: a~ 1m ® p( 'lj;ba(m)) (ei) ) 0

0

i

= p(g-l)o (2:a~lm ®ei) = aal(m,g)' i

Since ('Ij;b 0 'Ij;;;l )(m, g) = (m, 'lj;ba(m)· g), we thus have shown that ('Ij;b 0 'Ij;;;l )*ab = aa. In other words, the E-valued k-forms 'Ij;~aa and W;ab coincide on overlaps Ua nUb. It follows that there exists a well defined global E-valued k-form a = 'J(a) on P such that al 7r -l(Ua) = 'Ij;~aa. Moreover, this E-valued k-form is horizontal and of type p, i.e.,

=

a E n~or)P; E). A priori the construction of a depends upon the trivializing atlas U. A way to make it manifestly independent of such a choice would be to take for U the whole fiber bundle structure. However, since any chart added to U is compatible with the elements ofU, this will not change a. And thus, even though the construction of a from depends upon an atlas, the result does not.

'J(a)

a

Chapter VII. Connections

398

10.8 Proposition. The map') : nk(M; pp,E) isomorphism of graded COO (M)-modules.

----t

n~or,p(P; E), 0: ~ ')(0:)

== a

is an

Proof Replacing 0: E nk(M; Pp,E) by a sum of two elements, or multiplying it by a function on M will replace a by the corresponding sum or multiply the result by the given function. Hence the map') is a morphism of graded COO(M)-modules. An E-valued k-form a on P is zero if and only if all local k-forms aa = ('ljI;1 )*a are zero, which is the case if and only if all o:~ are zero (l0.7). But this happens if and only if 0: is zero, showing that') is injective. To prove surjectivity, let a be an E-valued k-form on P which is horizontal and of type p. Since it is horizontal, there exist E-valued functions f~ ... ik on Ua x G such that aa == (( 'ljI;1 )*a) I(m,g) = 2:i1... ik dX il 1\ ... 1\ dX ik ® f~ ... ik (m, g) [10.2]. Since a is of type p, we must have fi~ ... ik(m,g) = p(g-1)ft .. ik(m, e). Decomposing fi~ ... ik(m,g) with respect to a basis (ej) of E as f~ ... ik (m, g) = 2: j f~':'.ik (m, g) . ej, we can define the local k-forms o:~ on Ua by

O:~lm =

L

dxil

1\ ... 1\

dX ik . f~,.J.ik (m, e) .

i1···ik

Since p(g) is even, we thus have by constructionaal(m,g) = p(g-1) 0 2: j O:~lm' Comparing this with the construction of'), we see that if the local k-forms 2: j o:~ ® f.j glue together to form a global k-form 0:, then the a we started with is the one constructed from 0:, i.e., ') is surjective. To prove that the local k-forms 2: j o:~ ® f.j glue together, we note that, by construction, we have ('ljIb 0 7jJ;1 )*ab = aa. Since ('ljIb 0 'ljI;1)( m, g) = (m, 'ljIba(m) . g), this implies that the local functions f~ '" ik must satisfy the compatibility condition

ft .. ik (m, 'ljIba(m) . g)

= f::;' ... ik (m, g) .

Together with the type p condition, this implies that we have

and thus o:~ 1m = 2:i o:~ 1m p( 'ljIba (m)) i j . By [8.7] this means that the O:a glue together to IQEDI form a global pp,E-valued k form on M.

10.9 Remark. As before we use the abbreviation B = pp,E. For O-forms the identification') : nO(M; B) ----t n~or,p(P; E), which is an identification between r(B) and COO(P; E), can be stated in a more intrinsic way using the description [5.11] of the associated bundle. Let f : P ----t E be a function of type p (it is automatically horizontal), i.e., f (pg) = p(g-1) f (p), and let s = ,)-1 (J) E f( B) be the associated section of B. Then we have or all pEP the equality (10.10)

7rx

(p, f(p))

= S(7r(p)) .

§ 10. Principal fiber bundles versus vector bundles

399

This formula can also be used to construct the correspondence f f-+ S as follows. If the functionfis horizontal and oftypep, then (pg, f(pg)) = (pg, p(g-l)f(p)) = (p, f(p))·g, i.e., the map p ~ 7rx (p, f (p)) is constant on the fibers 7r- 1 (m). We thus can define the section s: M ----t B by s(m) = 7r x (P,f(p)) for an arbitrary p E 7r-l(m). Conversely, if s : M ----t B is a section, we claim that there is a unique function f : P ----t E of type p satisfying (10.10). Uniqueness of f follows from the fact that if 7rx (p, e) = 7rx (p, e') then e = e' (the pre-images of 7rx are G-orbits in P x E). Existence follows from the surjectivity of7rx . Since 7rx (p,f(p)) = 7rx (pg,p(g-l)f(p)) it follows that f satisfies f(pg) = p(g-l )f(p), i.e., f is of type p.

10.11 Examples. • Let 7r : P ----t M be a principal fiber bundle with structure group G. Denoting by Conn(P) the set of all connection I-forms on P, we know it is not empty [10.5]. The difference w - Wo of two elements W o , W E Conn(P) being horizontal and of type Ad [10.5], we can define a map Conn(P) ----t nJ,.or Ad (P; g) by W ~ W - Wo for a given fixed Wo E Conn(P). This map is obviously injec'tive; it is also surjective because if 0" is a horizontal g-valued I-form of type Ad, then w = Wo + 0" satisfies the conditions of a connection I-form. We conclude that the space of all connection I-forms (the space of all principal connections) on P is an affine space modeled on the graded CCXJ(M)-module nLr Ad(P; g). Since this space is isomorphic to nl(M; pAd,g), the difference w - Wo of two ~onnection I-forms can be seen as a pAd,g-valued I-form. We could have shown this result directly by combining [2.13] and [3.11]: w - Wo is determined by local g-valued I-forms a - r~ which, according to (2.15), are related on overlaps by

r

Here we have deduced this result from the more general identification given in [10.8]. The presence of the term 1/J~b e Me in (2.15), which contains derivatives of the transition function 1/Jab prevents the individual connection I-forms to be interpreted as sections of some associated bundle. Since the space ofFVF connections on a fiber bundle only depends upon the structure group [5.5], we conclude that the set of all FVF connections on a fiber bundle 7r : B ----t M is an affine space modeled on a graded CCXJ(M)-module of sections of a vector bundle (determined by B) . • Coming back to a principal fiber bundle 7r : P ----t M with structure group G and a connection I-form w, we have seen that the curvature 2-form n = Dw is horizontal and of type Ad [10.5]. We thus can see n as a pAd,g-valued 2-form. And, in a similar way as for the difference of two connection I-forms, this result could have been deduced directly from (4.11) . • Twisting our point of view, we now start with a vector bundle 7r : B ----t M with typical fiber E. Combining [5.3] with the first part of [7.14] then shows that we have EndR(B) = SBAd,EndR(E). Now suppose that \7 is a covariant derivative on B. Then it can also be seen as an FVF connection on B or as a connection I-form w on SB [6.18]. When we see it as a connection I-form, we have the associated curvature 2-form

400

Chapter VII. Connections

n, which in turn corresponds to an SBAd,EndR(E)_valued 2-form, i.e., an EndR(B)valued 2-form via the identification 'J: n2(M; EndR(B)) --t n~or,Ad(SB; E). If we now compare [4.8] with this construction (in particular with (10.7)) and with the local description of the curvature tensor R of the covariant derivative \7 as given in [9.10], we see that 'J(R) = n, i.e., the incarnation of the curvature 2-form n on the structure bundle SB as an EndR(B)-valued 2-form is exactly the curvature tensor R of the covariant derivative. 10.12 Proposition. Let 1[" P : P --t M be a principal fiber bundle with structure group G, let P : G --t Aut(E) be a representation and let B == pp,E be the vector bundle associated to P by the representation p. Let furthermore w be a connection I-form on P and let a E n~or,p(P; E) be a horizontal E-valued k-form on P of type p. Then the exterior covariant derivative Da is also horizontal and oftype p and is given by the formula (1O.l3)

Da

= da + (Tepow) f.o.a.

Moreover, if'Jk : nk(M; B) --t n~or,p(P; E) is the identification between B-valued k-forms on M and horizontal E-valued k-forms on P of type p and if \7 is the covariant derivative on B associated to the connection w, then we have for (J E nk(M; B) the equality (10.14)

i.e., the identification 'J intertwines the covariant derivative \7 and the exterior covariant derivative D. Proof To prove (10.13) we proceed as in the proof of [4.3]: we show that we have equality when evaluating on k + 1 smooth homogeneous vector fields X o , ... , X k that are either horizontal or vertical, and using only vertical vector fields of the form yP, Y E Bg. If all Xi are horizontal, contraction ofthe left gives by definition ofthe exterior covariant derivative the value t(Xo, ... , Xk)da. Contraction of the -wedge product (TeP 0 w) f.o. a with X o, ... , X k yields a sum of terms, each involving the contraction ofw with some Xi, which is zero. We conclude that the equality holds when evaluating on k + 1 horizontal vector fields. Whenever one of the Xi is vertical, the left hand side of (10.13) is zero by definition. If at least two are vertical, contraction of (TeP 0 w) f.o. a with X o , ... , X k yields a sum of terms, each involving the contraction of a with k vector fields among X o, ... , X k . Since among these k at least one is vertical, the result is zero. Still assuming that at least two among the Xi are vertical, contraction of da with X o, ... , X k gives a sum of two terms, the first being a single sum, the second being a double sum (V.7.4). Each summand in the first term contains the contraction of a with k elements among X o, ... , X k and thus is zero. Each summand in the second term contains the contraction of a with k - 1 elements among X o, ... , X k as well as contraction with the commutator of the remaining two. If

§ 10. Principal fiber bundles versus vector bundles

401

the remaining two are not both vertical, at least one among the k -1 is vertical and thus the full contraction is zero. If the remaining two are both vertical, i.e., of the form x P and yP for x, y E Bg, then their commutator [xP, yPj = [x, yjP is also vertical, and thus again the full contraction is zero. We conclude that if at least two among the Xi are vertical, then contraction of the right hand side with Xo, . .. , X k yields zero, and thus in that case too we have equality. Remains the case with only one vertical vector field. By skew-symmetry we may assume that it is the first: Xo = yP for some y E Bg and Xi is horizontal for i :::: 1. If X is any horizontal vector field and (3 any (suitable) R-form, then

t(X)((Tepow) ~ (3) = (t(X)(Tepo w)) ~ (3 - (Tepow) ~ t(X)(3 = -(Tepow) ~ t(X)(3 , where the minus sign after the first equality comes from the fact that TeP 0 w is a I-form, and where the second equality comes from the fact that X is horizontal. Since we have t(yP , Xl, ... , X k ) = t(yp) 0 t(X I ) 0 • . • 0 t(X k ), we obtain the equality t(yP,XI"",Xk)((Tepow)~o:)

= (_l)k (Tept(yP)w) ·t(Xl, ... ,Xk)o: = (_l)k TeP(Y) . t(X l , .. . , Xk)o: ,

where the right hand side should be interpreted as the action of TeP(Y) E EndR(E) on the E-valued function t(Xl"'" Xk)o:. As said before, contraction of do: with the vector fields yP, Xl, ... , X k yields a sum of two terms, the first a single summation and the second a double summation. In the single summation, only the term in which 0: is not contracted with yP remains (all other being zero) and gives yP (t( Xl, ... , X k) 0:). Whenever, in the double summation, the vector field yP does not appear in the commutator, contraction with 0: yields zero, and thus only a single summation remains and we find:

(_l)k. t(yP, Xl'"'' Xk)do: = yP (t(X l , ... , Xk)O:) )+ I: (E(Xp)IE(Xj)) + (-1) O
L

= .c(yP)(t(Xl , ... , Xk)o:)

I:

- L O<j'5.k

I: = (_l)"Sps;k

(E(Xp)IE(Y))

t(Xl, ... ,Xj - l ,

(-l)o
[y , Xj], X)+l,"" Xk)o:

(E(Xp)IE(Y))

t(Xl"'" Xk)(.c(yP)o:) ,

where the last equality follows from [Y.7.l7]. Since 0: is of type p, we have the equality <1>;0: = p(g-l) 00:. Taking the derivative of this equation in the direction of y at g = e

Chapter VII. Connections

402

as in the proof of [4.3] we obtain on the left L(yP)a. On the right hand side we have to compute Ye (p(g-l) 0 a). Taking into account that TeInv = -id we obtain Ye(p(g-l) 0 a) = -(TeP(Y)) 0 a, and thus L(yP)a = -(TeP(Y)) 0 a. Substituting this in the previous result, we obtain

Combining this with the contraction of (TeP 0 w) t.-. a with yP, Xl, ... ,Xk, we see that contraction of the right hand side of (10.13) with yP, Xl, ... ,Xk yields zero, and thus we have equality also in this last case with just one vertical vector field, i.e., we have proven (10.13). To show (10.14), we choose a localtrivialization (U, 1/;) of P and we denote by (U, 1/;) the associated local trivialization of B == pp,E. For ease of notation we introduce the functions Ip : G ---t EndR(E), g ~ p(g-l) and hd : G ---t EndR(g), g ~ Ad(g-l). Given the B-valued k-form (J and the local trivialization (U, ;j;), we have an E-valued k-form (J:;;; on U (8.5). According to (10.7), the local expression (1/;-1) * (Jk(J) of'J k(J as E-valued k-form on U x G is given by the formula

Using (10.13) and the local expression (3.12) of w, we compute:

+ (Tepow) t.-. 'Jk(J) = d(Ip (J:;;;) + (Tepo (8MC + hd r)) t.-. (Ip (J:;;;) = (dIp) t.-. (J:;;; + Ip d(J:;;; + ((Tepo 8MC) Ip) t.-. (J:;;; + ((TeP hd r) I p) t.-. (J:;;; ,

(1/;-l)*(D(JkrJ)) = (1/;-l)*(d(Jk(J) 0

0

0

0

0

0

0

0

where the reader should be aware that all composition operators 0 are in fact alternative notations for wedge products, either (j or t.-.. We now claim that we have the equalities and

Using this, [10.7], and [9.7-i] we obtain:

Since the trivializing chart (U, 1/;) is arbitrary, we obtain the global result (10.14). Moreover, this also shows that Da is horizontal and of type P whenever a is because: 'J k is bijective, so there exist (J such that a = 'Jk ((J), and thus Da = 'J k+1 (\7 (J) is in the image Of'J k +1 and thus also horizontal and of type p. To finish we now prove the tw 0 claims. Contracting dIp with a left-invariant vector field yields:

x

§ 10. Principal fiber bundles versus vector bundles

403

Now IpoLg : G ----+ EndR(E) is the map h ~ p(h-1)op(g-1), and thus, taking into account that Te1nv( x) = -x, we find

t(xg)dlp = -(TeP(x)) op(g-l) = -(TeP(t(xg)8MC)) o Ip(g) = -t(xg)(Tep o 8Mc)oIp). Since x Egis arbitrary, we have shown the first of our claims. To prove the second one, we start by studying the map f : G ----+ EndR(E) defined as

f(h) Evaluating

Te! on x

=

p(g-lhg) 0 p(g-l)

=

p(g-l) 0 p(h) .

E 9 we thus find the equality

(TeP(Ad(g-l)x)) 0 p(g-l) Substituting for x the value x

= t(y(m,g»)f for

=

p(g-l) o TeP(x) .

some y(m,g) E T(m,g) (U x G) we find

t(Y(m,g»)((Tepo hd 0 r) oIp) = (TeP(Ad(g-l)(x))) 0 p(g-l) = p(g-l) 0 TeP(x) =

t(Y(m,g»)(lpo(Tepor)).

Since Y(m,g) is arbitrary, we have also shown the second claim.

10.15 Remark. If a is horizontal and oftype p, we used in the proof of [10.12] an indirect way to prove that Da is again horizontal and of type p. By definition of D it is obvious that Da is horizontal; this is true even if a itself is not horizontal. However, a direct proof that Da is again oftype p involves the exterior derivative of a generalized pull-back. Correctly defining this notion involves a lot of technical problems that were avoided in our indirect proof.

10.16 Corollary. On a principal fiber bundle 1[' : P ----+ M with connection I-form wand curvature 2-form 0, the Bianchi identities are equivalent to the statement that DO = O.

Proof In [4.4] we have seen that the Bianchi identities imply DO = O. From (10.13) we learn that DO = dO + (Te Ad 0 w) t.-.O = dO + (ad R 0 w) t.-. O. In order to show the converse, it thus is sufficient to show that (ad R 0 w) t.-. 0 = - [ 0 A,. w]. Writing w = 2:i Wi ® Vi and 0 = 2: j OJ ® Vj with respect to a basis (Vi) of g, we compute

ij

ij

ij

=-

L oj

1\

wi ® (-1) (E(Vi)IE(Vj» [Vj, Vi]

= - [0 A,. w]

,

ij

where the third equality follows from the fact that wand 0 are even and thus that we havec(w i ) = C(Vi) = c(Oi). The fourth equality follows from [1.6.9], [1.5.9] and graded IQEDI skew-symmetry of the bracket [_, -1.

Chapter VII. Connections

404

10.17 Corollary. Let 1[" : B ----+ M be a vector bundle with typical fiber E and \7 a covariant derivative on it. If R E EndR(B) is the curvature tensor of \7 then \7Ad R = o.

Proof The covariant derivative \7 on B is equivalent to a connection I-form w on the structure bundle SB [6.18]. The representation Ad defines the associated vector bundle SBAd,EndR(E) == EndR(B) and an induced covariant derivative \7Ad on EndR(B) [5.3], [7.14]. According to [10.11], J(R) = 0, and then, combining [10.12] with [4.4] we IQEDI compute J(\7 Ad R) = DO = O. Since J is an isomorphism, the result follows.

10.18 Corollary. Let 1[" : P ----+ M be a principal fiber bundle with structure group G and let w be a connection 110rm on P with associated curvature 2-form O. Let furthermore p: G ----+ Aut(E) be a representation and let a E O~or,p(P; E) be a horizontal E-valued k-form on P of type p. Then D 2 a is given by the formula (compare with [9.10])

Proof Using [10.12] we compute:

D 2a = D(da + (Tepow) t.-.a) = d(da + (Tepow) t.-. a) + (Tepow) t.-. (da + (Tepow) t.-. a) = d((Tepow) t.-.a) + (Tepow) t.-.da + (Tepow) t.-. ((Tepow) t.-.a)

= (TeP dw) t.-. a + ((TeP w) fci,(TeP w)) 0

0

0

1\

a ,

where for the fourth equality we used [V.7.9-iii] and the fact that TeP is constant and thus d(Tepodw) = Tepodw. Since TeP is an EndR(E)-valued I-form, we can apply [1.4] to get (TeP 0 w) fci,(TeP 0 w) = ~ [TeP 0 w f,\ TeP 0 w]. But TeP is a morphism of A-Lie algebras and thus [TeP 0 w f,\ TeP 0 w ] = TeP 0 [ w f,\ w ]. Plugging this in the final equality for D2 a above gives us D2 a = (TeP 0 (dw + ~ [w f,\ w ])) t.-. a, which is the desired result by [4.3].

IQEDI

References

Some other books/papers on supermanifolds (in square brackets the page(s) where the work is cited) [BBH-R] C. Bartocci, U. Bruzzo, and D. Herwindez-Ruiperez, The Geometry of Supermanifolds, MIA 71, Kluwer Acad. Publ., Dordrecht, 1991. [Ber] F.A. Berezin, Introduction to superanalysis, Mathematical Physics and Applied Mathematics vol. 19, Reidel, Dordrecht, 1987. [DW] B. DeWitt, Supermanifolds, Cambridge UP, Cambridge, 1984, [p.271]. [Fre] P.G.O. Freund, Introduction to Supersymmetry, Cambridge UP, Cambridge, 1986. [Ko] B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential geometric methods in mathematical physics (K. Bleuler & A. Reetz, eds.), Proceedings Conference, Bonn 1975. LNM 570, Springer-Verlag, Berlin, 1977, pp. 177-306. [Le] D.A. Leites, Introduction to the theory of supermanifolds, Russian Math. Surveys 35 (1980), 1-64. [Ma] Y. I. Manin, Gauge field theory and complex geometry, Grundlehren der Mathematischen Wissenschaften, 289, Springer-Verlag, Berlin, 1997. General references (in square brackets the page(s) where the work is cited) [Ba] [Bo] [Br] [E.Ca] [H.Ca]

M. Batchelor, The structure of supermanifolds, Trans. Amer. Math. Soc. 253 (1979),329-338, [po 196]. N. Bourbaki, Groupes et algebres de Lie, Ch 4-6, Elements de mathematique, Fascicule XXXIV, Hermann, Paris, 1968; Masson, Paris, 1981, [po 23]. K.S. Brown, Cohomology of Groups, GTM 87, Springer-Verlag, Berlin, New York, 1982, [po 325]. E. Cartan, La topologie des espaces representatifs de groupes de Lie, CEuvres I (part 2), Gauthier-Villars, Paris, 1952, pp. 1307-1330, [po 325]. H. Cartan, Notions d'algebre dijJhentielle; application aux groupes de Lie et aux varietes ou opere un groupe de Lie, CBRM Colloque de Topologie, Bruxelles 5-8 juin 1950, George Thone, Liege, 1950, pp. 15-27, [po 248]. 405

406

[Du] [Eh]

[GHV]

[God] [Gor] [Hi]

[HS]

[Hu] [Iw] [Lie] [MS-V] [Pij]

[Qu] [Ra] [Sc] [So]

[Spa] [Spi] [Sta] [Ste] [Tu1]

References

J. Dugundji, Topology, Allyn and Bacon, Inc, Boston, 1966, [po 93]. C. Ehresmann, Les connexions infinitesimales dans un espace fibre differentiable, CBRM Colloque de Topologie, Bruxelles 5-8 juin 1950, George Thone, Liege, 1950, pp. 29-55, [po 344]. W. Greub, S. Halperin, & R. Vanstone, Connections, Curvature, and Cohomology, vol. I, Academic Press, Inc., San Diego, London, etc., 1972, [pp. 145, 195]. C. Godbillon, Elements de topologie algebrique, Hermann, Paris, 1971, [po 327]. v.v. Gorbatsevich, The construction of a simply connected Lie group with a given Lie algebra, Russian Math. Surveys 41 (1986), 207-208, [po 325]. F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, Berlin, Heidelberg, New York, 1978, [po 145]. M. W. Hirsch & Stephen Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, Inc., San Diego, London, etc., 1974, [p.233]. D. Husemoller, Fibre Bundles, GTM 20, Springer, Berlin, New York, 1994, [p.145]. K. Iwasawa, On some types of topological groups, Annals of Mathematics 50 (1949),507-558, [po 327]. S. Lie, Theorie der Transformationsgruppen Vol III, Teubner, Leipzig, 1893; Reprinted, Chelsea Publishing Company, New York, NY, 1970, [po 325]. J. Monterde & A.O. Sanchez-Valenzuela, Existence and uniqueness of solutions to superdifferential equations, J. Geom. Phys. 10 (1993), 315-343, [po 334]. H.GJ. Pijls, The Yang-Mills equations, Proceedings Seminar 1981-1982 Mathematical Structures in field theories (E.M. de Jager & H.GJ. Pijls, eds.), CWI Syllabus 2, CWI, Amsterdam, 1984, pp. 119-182, [po xii]. D. Quillen, Superconnections and the Chern character, Topology 24 (1985), 89-95, [po 393]. R.M. Range, Holomorphic functions and integral representations in several complex variables, GTM 108, Springer-Verlag, Berlin, New York, 1986, [po 95]. L. Schwartz, Les Tenseurs, Hermann, Paris, 1975, [po xii]. J.-M. Souriau, Structure des Systemes Dynamiques, Dunod, Paris, 1969; English translation, Structure of Dynamical Systems, A Symplectic View of Physics, PM 149, Birkhauser, Boston, Basel, 1997, [po 332]. E.H. Spanier, Algebraic Topology, Springer-Verlag, Berlin, New York, 1966, [po 312]. M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. I, Publish or Perish, Berkeley, 1970, [pp. xii, 123, 135]. J.D. Stasheff, Continuous Cohomology of Groups and Classifying Spaces, Bull. Amer. Math. Soc. 84 (1978), 513-530, [po 325]. N. Steenrod, The Topology of Fibre bundles, Princeton University Press, Princeton,NJ, 1951, [po 145]. G.M. Tuynman, An elementary proof of Lie's third theorem, Publications IRMA 34 (1994), X.1-XA, Internal publication of the Department of Mathematics of

References

[Tu2] [TW]

lyE]

[Wa]

407

the University ofLille, [po 325]. G.M. Tuynman, Functions of class C k without derivatives, Publicacions Matematiques 41 (1997),417-435, [po 97]. G.M. Tuynman & W.AJJ. Wiegerinck, Central Extensions in Physics, 1. Geom. Phys. 4 (1987), 207-258, [po 325]. W.T. van Est, Une demonstration de E. Cartan du troisieme theoreme de Lie, Actions hamiltoniennes de groupes - Troisieme theoreme de Lie, Seminaire Sud-Rhodanien de Geometrie VIII (may 1986) - Travaux en cours 27, Hermann, Paris, 1988, pp. 83-96, [po 325]. F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, GTM 94, Springer-Verlag, Berlin, New-York, 1983, [pp. xii, 300, 300].

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Index of Notation

M(piq, A), 72

Standard sets C: complex numbers N: natural numbers R: real numbers 6k: permutation group Z: integers Z2 = Z/2Z = {O, I}

Fraktur characters 2l,3 ([,62 9'\,22

6k. 22 'I',13

Standard names Ad, 283 ad, 30,284 Det, 58, 77 dim, 61 exp,278 gDet,78 gtr,75 im, 17

Operations related to the creation of A-vector spaces Aut, 11 End, 9 Hom, 9 Hom s \ 24 Map, 8 Maps\ 24 N E ,81 R E ,84

Inv, 268 lac, 116 ker, 17 Lift, 377 rank, 58, 122 Span, 6 supp, 134, 159 tr,76

14 14 0, 18,20 EBiE['

I1iE['

Q!/ E =A,22 ®lE=E,22 k

®i=l' 22 /\,26

Special sets A, 4, 57,114 AC, 188 2l,3 Gl(piq, A), 72

/\,27

NE=A,25 NE=E,25 N,25 EO = {O}, 15

M(m x n, A), 58 409

Index of Notation

410

Symbols attached to A-vector spaces Ew,o,d, 1CX5

E*,9 *E,9 E~, 101 Ebo:, 32, 102 P.l..,193 Operations related to the creation of maps

N,40 I1iEI' 37 I:iEI,35 x,37, 134

Symbols attached to maps 1*, 183,255 *¢, 14 ¢*, 14 Partial derivatives 8i , 114 8ai l m ,209 8x~lm, 209 8 X i, 114 8f.j, 114 81 j8x i , 114 8 f/ 8~j, 114 Tangent something T function, 211 T A-manifold, 204 T' function, 282 Tpoint function, 213 Tpoint A-manifold, 204 Symbols with a fixed meaning ~,11

t( J, 11, 182 LL>,12 \7,366 0, 12 -1-, 61

UJ,3

[ _ f;.. -1,337

[_,_]'28,210 ,(}, 337, 389 t.-., 337, 389 A,40 B, 57, 58, 82, 105, 126, 129 BM,262 CD, 94

C k ,94 Coo, 97 C oo C)o:,97 r, 148

r- 1 ,

l75 (C in]), 1m cfj,272 ([,62 58 c,3,4 e,287 G, 88,106,137 Hk,263 In, 58, 119 L g ,268 £,253 eMs, 68 NIs ,68

of,

Ms,68 m,266 mL,2 mR,2 7r a,221,222

9'\,22 R g ,268 8MC,339

'I', 13 V, 200 Symbols attached to others x M : fundamental vector field, 299 a + ib: complex conjugation, 188 x: left-invariant vector field, 271 t: transpose, 71

Index

2l-graded A-module, 6 commutative ring, 3 commutativity, 3 ring, 3 2l-grading, 3 A-Lie algebra, 270 associated to an A-Lie group, 271 A-Lie group, 142,266 A-Lie subgroup, 292 normal, 296 proper, 300 A-manifold, 128 A-module, 2 A-vector space, 85 action effective, 143 left/right, 142, 266 additive, k-, 2 adjoint representation, 31, 32, 284, 325 Adjoint representation, 283,284 affine connection, 374 algebra 2l-graded commutative, 28 2l-graded Lie, 28 of parity a, 31 associative, 28 of parity a, 32 associated bundle, 151, 364 atlas, 124 adapted to a subbundle, 160

trivializing, 145 automorphism of a module, 11 basis, 46 dual, 67 ordered, 59 orthonormal, 192 Batchelor's theorem, 196 Berezinian, 41, 55, 78 Bianchi identities, 355, 357 bilinear map, 8 bimodule,2 body of A, 57 of a (proto) A-manifold, 126 of a linear map, 82 of a matrix, 58 of a module, 82 of a smooth function, 105, 129 border point, 228 bracket of a Lie algebra, 28 bundle associated, 151, 364 fiber, 145 principal, 155, 301 frame, 361 of morphisms, 170 pull-back, 151 structure, 361 trivial, 147 vector, 156 411

412

Cartan's structure equations, 355, 357-359 center of a Lie algebra, 325 central extension of a Lie algebra, 325 of an A-Lie group, 325 chain rule, 116 chart of an A-manifold, 124 Christoffel symbols, 370 closed differential form, 262 cocycle of a Lie algebra, 325, 332 of an A-Lie group, 325 cohomology de Rham, 263, 327 of a Lie algebra, 325, 332 of an A-Lie group, 325, 332 commutativity, 2l-graded, 3 commuting flows, 236 compatible chart of a fiber bundle, 144 compatible chart of an A-manifold, 124 complex conjugation, 188 connection, 342, 345, 351 FVF, 345 I-form, 351 affine, 374 Ehresmann, 342 flat, 342 integrable, 342 linear, 374 principal, 351 contraction elementary, 74, 274 operator, 12 contraction, elementary, 74, 75 convergence, 286 coordinate (even/odd), 104 cotangent bundle, 248 counter example, see example, counter covariant derivative, 366 exterior, 354 induced, 368 covering, 311-312 Coxeter group, 23 curvature 2-form, 355 curvature tensor, 394

Index

decomposition, 2 degree-a part of a vector bundle, 167 derivation (right/left), 29 determinant, 41, 61, 77, 80, 111 graded, 41, 55, 78-80, Ill, 118,285 DeWitt topology, 93 de Rham cohomology, 263, 327 diffeomorphic, 128 diffeomorphism, 128 differential form, 248 with values in a vector bundle, 383 with values in an A-vector space, 323 dimension differential, 207 evenlgradedlodd/total, 61 of a (proto) A-manifold, 124 direct product of bundles, 153 of A-manifolds, 133 direct sum, 7, 14 of bundles, 164 dual basis, 67 bundle, 170 of a module, 9 ofa morphism, 14 effective action, 143 Ehresmann connection, 342 elementary contraction, 74, 75, 274 embedding, 214 endomorphism, 9 enlarging the structure group, 154 equivalence of bases, 83 of A-Lie subgroups, 293 equivariant map, 305 Euler vector field, 261 evaluation operator, 12 even, 3,4, 56 exact differential form, 262 example, 3, 4, 21, 28, 56, 57, 59, 71, 74,97,110,118,125,132,133, 137, 143, 144, 152, 231, 280,

Index

283, 287, 297, 332, 339, 367, 396,399 counter, 7, 10,60,63,65,66,76,81, 84,87, 103, 123, 130, 148, 163, 191, 192, 216, 221, 224, 244, 300,344 exponential map, 278 of matrices, 287 exterior algebra, 27 covariant derivative, 354 derivative, 249 power, 25 of a vector bundle, 167 family of Lie algebra morphisms, 282 of A-Lie group morphisms, 282 f.g.p, 47-54, 59,172,180,181,196,250 fiber bundle, 145 map, 147 principal, 155, 301 structure, 145 over a point, 145 typical, 145 finite dimensional, 59 type, 46 finitely generated, 46, 48, 50, 51, 196 flat connection, 342 flow, 228 commuting, 236 global, 234 local,228 foliation, 243, 341 form, differential k-, 248 frame, 361 bundle, 361 free 2l-graded A-module, 16 Frobenius' theorem, 243, 244, 342, 355 fundamental vector field, 299 FVF connection, 345 gauge, 351

413

transformation, 351 generator, 16, 46 graded, 56 determinant, 41, 55, 78-80, 111, 118, 285 subspace, 59, 86 trace, 75, 76, 80, 285 transpose, 71 graph, 134, 149 homogeneous, 3, 4 homomorphism, 9 horizontal k-form,395 lift, 342 map, 342 part of a tangent vector, 342 section, 342 submanifold, 342 tangent vector, 342 ideal, 296 identification, 11 immersion, 214 implicit function theorem, 122 independent elements, 46 initial condition, 228 integrable connection, 342 subbundle, 243, 244, 342 vector field, 228 integral manifold, 244 interchanging map, 22 interval, 228 invariance of dimension, 121 invariant k-form, 320 left/right, 320 vector field, 315 inverse function theorem, 121 invertible homomorphism, 11 involutive subbundle, 243 isomorphic fiber bundles, 147 modules, 11

Index

414

isomorphism, 11 of fiber bundles, 147 of Lie algebras, 270 of vector bundles, 157 of A-Lie groups, 267 isotropy subgroup, 304 Jacobi identity, 28, 30-32, 270-272, 284, 326, 332, 356 Jacobian, 116 k-additive, 2 k-form,248 M-dependent,257 with values in a vector bundle, 383 with values in anA-vector space, 323 k-linear map (leftlright), 8 Kronecker delta, 58 leaf, 244, 246, 294 left multiplication, 2 translation, 268 left -invariant vector field, 270 Lie derivative, 253 lift, 377 linear connection, 374 linear map (left/right), 8 local flow, 228 locally finite, 134 Maurer-Cartan I-form, 339, 357 metric on a free graded A-module, 192 on a vector bundle, 194 modeled, an A-manifold on an A-vector space, 124 module, 2-6 momentum map, 332 morphism, 9 of Lie algebras, 30, 270 of vector bundles, 157 of A-Lie groups, 142, 267 nilpotent vector, 81

normal A-Lie subgroup, 296 notational shorthand, 119 odd, 56 ordered basis, 59 orthogonal complement, 193 orthonormal basis, 192 parallel transport, 344, 364 parity, 3, 4 of a linear map, 8 of a section, 158 reversal, 102 shift operation, 32, 102 partition of unity, 95, 128, 135, 136, 159, 167, 175, 194, 195, 197, 199, 263 plateau function, 136, 160, l76, 205, 250, 327 principal connection, 351 fiber bundle, 155, 301 product of bundles, 153 projective, 47 proper A-Lie subgroup, 300 proto A-manifold, 124 pseudo effective action, 143 pseudo metric pseudo metric on a free graded Amodule, 188 on a vector bundle, 194 pull-back bundle, 151 map, 183 of a differential form, 255 E-valued, 324 generalized, 257, 319 of a section, 183 push forward of a section, l78 push forward of a vector field, 218 quotient, 16 bundle, 164 rank, 73

Index

415

of a function, 122 of a matrix, 58 of a vector bundle, 156 reducing the structure group, 154 regular value, 214 related vector fields, 218 representation of a Lie algebra, 30 of an A-Lie group, 267 restriction of a bundle to a sub manifold,

146 right multiplication, 2 translation, 268 right-invariant vector field, 270 ring, 2l-graded, 3 commutative, 3 second countable topology, 128, 245, 247 section, 148 shorthand, notational, 119 signature, 24 simply connected, 312 skew-symmetric, 24 skew-symmetrization operator, 40 smooth A-structure, 124 functions, 94-101 linear map, 85 map between A-manifolds, 128 system, 96 maximal, 97 tree, 96 strong bundle map, 147 structure bundle, 361 constants of a Lie algebra, 272, 273, 280, 297, 328, 340 equations of an A-Lie group, 340 group, 145 subalgebra of a Lie algebra, 28, 293 subbundle, 160 integrable, 243, 244, 342 involutive,243 sub manifold, 130

sub module, 2 2l-graded, 4,6 generated by, 6 sum of, 6 subspace graded, 59, 86 of an A-vector space, 86 sum of submodules, 6 supplement, 15, 63 of a bundle, 165 support of a function, 134 of a section, 159 symmetric,2l-graded, 24 symplectic geometry, 332 tangent bundle, 204 map, 212 generalized, 225, 257 to a subbundle, 244 tensor product, 18 of vector bundles, 166 topology DeWitt, 93 on an A-vector space, 93 second countable, 128, 245, 247 trace, 76 graded, 75, 76,80,285 transition function, 145 transitive action, 267 transpose, 71 graded, 71 of a morphism, 13 transposition operator, 13 trilinear map, 8 trivial bundle, 147,223,224 vector bundle, 158, 223, 224 tri vializing atlas, 145 chart, 144 sections, set of, 160 type p k-form, 396 typical fiber, 145

416

Index

vector bundle, 156 vector field, 2(1) Euler, 261 fundamental, 299 integrable, 228 invariant, 315 left/right-invariant, 270 vertical

wave, 228 wedge product, 26 for vector bundle valued forms, 336, 386 symbol, 26 Whitney sum, 164 without odd dimensions, 106, 136 Yang -Baxter equation, 23

subbundle, 342 tangent vectors, 342

zero section, 158