Algebraic foundations of non-commutative differential geometry and quantum groups

"If E H*,Va,b

E

H: f(ab) =

L

f/(a)ff'(b).

1=1

Thereby one establishes the Z2-graded coalgebra {H*, .1*, c*} over K. If the associative superalgebra {H, p., TJ} is graded-commutative, then the super-coalgebra {H*, .1*, c*} is graded-cocommutative. (3.4.3) Hence the dual H* := HomK(H,K) of a finite-dimensional Z2graded bialgebra {H, p., TJ, .1, c} over K is again some finite-dimensional Z2graded bialgebra {H*, p.* , TJ*, .1*, c*} over the field K. (3.4.3.1) If especially H is a Z2-graded Hopf algebra over K with the antipode a, then H* is again such an object with the antipode a*, which is defined such that

"If E H*,a

E

H: a*(J)(a):= foa(a).

(3.4.3.2) With the trivial Z2-grading of H =: H O, HI := {OJ, one obtains the duality of finite-dimensional bialgebras over K. (3.4.3.3) The opposite structure maps of H are denoted by p'0PP := Il 0 r, .1OPTJ := r 0 .1, and similarly for H*, with the Z2-graded flip r. Obviously the dual of {H, IlOPP, 17, .1opp, c} is {H*, (p.*)OPTJ, TJ*, (.1*)OPP, c*}. The latter is equipped with the antipode a*, if {H, p., TJ, .1, c} allows for the antipode a. (3.4.3.4) The antipode a of {H, p., 77, .1, c} is bijective, because H is finitedimensional. Therefore both {H, J!., TJ, .1opp, c} and {H, p.OPP, TJ, .1, c} are Z2graded Hopf algebras over K with the antipode a-I, Then one easily establishes the following dual Z2-graded Hopf algebras over K:

{H, p., TJ, .1OPTJ, c} --.{H*, (p.*)OPP, TJ*, .1*, c*}, dual

{H, p'0PP, TJ, .1, c} ~{H*, p.*, TJ*, (.1*)OPP, c*}, both duals H* with the antipode (a*)-1 = (a- 1)*. Moreover a : {H, p., 77, .1, c} +-----+ {H, p'0PP, TJ, .1oPP, c} is some isomorphism of Z2-graded Hopf algebras over K, because of the commutative diagrams:

a 0 p. 0 r = p. 0 T(a, a), T(a, a) 0.1 = r

0 .1 0

a.

3.5 Examples of Z2-Graded Hopf Algebras

93

3.5 Examples of Z2-Graded Hopf Algebras (3.5.1) Consider the alternating algebra A(E) of an R-bimodule E. The corresponding structure maps explicitly read, such that

'rip, q E N, 'Vr, s E R : r ® s --+ rs, p.

A(E) ® A(E) 3 (Xl A .. · AXp) ® (YI A··· A Yq) --+xI A··· AYq E A(E), p.

r ® (YI A··· A Yq) --+r(YI A··· A Yq), P.

(Xl A ... A x p ) ® s --+ S(XI A ... A x p ), p.

and T/ is just the canonical embedding of R into A(E). (3.5.1.1) The canonical projection c : A(E) --+ R is some homomorphism of unital associative R-superalgebras, with respect to the natural Z2-grading due to the No-grading of A(E). (3.5.1.2) Consider the R-linear map 6 : E 3 X --+ X ® eR

+ eR ® X E A(E)®A(E),

into the skew-symmetric tensor product of unital associative superalgebras over R. One immediately finds, that 'rIx E E : 6(x)6(x) = O. Hence one establishes a unique homomorphism of unital associative superalgebras ..1, due to the universal property of alternating algebras over R, such that the following diagram becomes commutative.

E

_ _ _8_m_b_8_d_dl_·n..... 9

LA(E)@A(E)

• A (E)

JA

(3.5.1.3) Obviously the unital associative superalgebra A(E)®A(E) is again graded-commutative, just as A(E) itself. (3.5.1.4) The family {A(E),j.t, T/, ..1,c} is some Z2-graded Hopf algebra over R, which is graded-commutative-cocommutative, with the unique antipode 0', such that "Ix E E : o'(x) = -x.

94

3. Coalgebras and Z2-Graded Hopf Algebras

(3.5.1.5) For every f E HomR(E,F), AU) defined by the commutative diagram below is some homomorphism of Z2-graded Hopf algebras over R. E - T( E)

3

X,

@ •.. @

f

l

xp

A ... AXp

• x1

A (f)

p 3

@ kz 1

A(E)

j

T (f)

F - T(F)

E

p

f(i)

>

A k=1

f (Xk )

E

A (F)

(3.5.2) An element a of a coalgebra {A, .1, e} over R is called group-like, if and only if .1(a) = a I8i a. An element b E A is called primitive with respect to the group-like element a, if and only if .1(b) = a I8i b + b I8i a. (3.5.2.1) Let {A,JL, TI, .1,e} be a Z2-graded Hopf algebra over R. Denote the set of group-like elements of {A, .1, e} by G. Then obviously

eA

E

G:= {a

E

A;.1(a) = a I8i a} ~ AO, Va,b

E

G: ab

E

G, e(a) = eR.

Now let a be the unique antipode of {A, JL, TI, .1, e}. Then Va E G: aa(a)

= a(a)a = eA.

Hence the set G becomes some group, with the group operation induced by JL. The elements, which are primitive with respect to eA, are called primitive. The linear subspace of these elements is graded, such that

LA := {a

E

A; .1(a) = eA I8i a + a I8i eA} = (LA n All) EB (LA n AI).

Va, bELA: [a, b]

:=

ab -

L

(-l)xYbYa X

E

LA, e(a)

= O.

x,iiE Z 2

Hence the Lie superalgebra LA over R, consisting of the primitive elements of A, is established.

(3.5.3) Let E(L) denote the universal enveloping superalgebra of a Lie superalgebra L, over a field K of characteristic char K l' 2. (3.5.3.1) The homomorphism of Lie superalgebras over K:

LEBL

3

{x,y} ---+xl8ie+el8iy

E E(L)~E(L),

with the unit e of E(L), fulfills the defining property of universal enveloping superalgebras. Hence there is an isomorphism of unital associative Ksuperalgebras:

E(L EB L) ~ E(L)~E(L).

3.5 Examples of Z2-Graded Hopf Algebras

95

(3.5.3.2) The universal property of E(L) is used in order to define the homomorphism Ll of unital associative superalgebras over K, according to the following diagram. x®e + e ®x

1\

E

E(L) ®E(L)

homomorphism of Lie-superalgebras

E(L)

(3.5.3.3) A homomorphism c: : E(L) --+ K of unital associative superalgebras is defined, such that "Ix E L : c:(x) = 0, c:(e) = eK, with the unit eK of

K. (3.5.3.4) Hence one obtains the Z2-graded Hopf algebra {E(L),fL,7],Ll,c:} over K, with the unique antipode a of E(L) defined as the so-called principal graded anti-automorphism of E(L), such that "Ix

E

L : a(x) = -x, a 0 a = id E(L).

Here fL,7] denote the structure maps of the unital associative superalgebra E(L).

(3.5.3.5) The Z2-graded coalgebra {E( L), Ll, c:} is graded-cocommutative. (3.5.3.6) The Lie superalgebra of primitive elements of E(L) is L itself. For an appropriate proof one needs the Z2-graded version of the PoincareBirkhoff-Witt theorem. Obviously the only group-like element of E(L) is O. (3.5.4) Let G be any group, and denote by H := K(G) the vector space over a field K, which is generated by the basis G. (3.5.4.1) H becomes a unital associative algebra over K with the structure maps

fLH : H ® H

3

L

cklgk

® gl

--+

k,IEljin

K 3

C --+ C def

ea

E

L

cklgkgl E

H,

k,IEljin

H,

with the unit ea =: eH of H. Here the index set: I +---+ G.

[fin

denotes an arbitrary finite subset of

3. Coalgebras and Z2-Graded Hopf Algebras

96

(3.5.4.2) With the structure maps defined by the subsequent diagrams one establishes the cocommutative coalgebra {H, ..1 H, C H} over K.

G

>

free over K

\j)

H

free over K

9

G \j)

9

L

g@gEH@H

def

.1

H

EH

K3eK

~ def

(3.5.4.3) Moreover {H,J-LH,T/H,..1H,cH} is some Hopf algebra over K with the unique antipode (7H, which is constructed by the diagram below, and (7H O(7H = id H.

G ----=------:-:-----.. H free over K \j) 9

L

def

3.6 Smash Product of Z2-Graded Hopf Algebras (3.6.1.1) Consider two Z2-graded bialgebras E and H over a commutative ring R, with the trivial grading of H =: HO, and let J-L : H ® E ---+ E be an even R-linear map, which fulfills the following conditions.

(i) VXEE,V/,gEH:J-L(eH®x)=x, J-L(Jg®x)=J-L(J®J-L(g~x)). Hence E becomes an H-Ieft module over R.

(ii) The pair {J-L, H} measures E to E in the sense, that with the notation: L(f)

H

3

1---+ L If ® If'

E

H ® H, "Ix, y E E, "If

E

H:

LiH 1=1 L(f)

J-L(J ® xy) =

L

J-L(Jf ® X)J-L(Jf' ® y), J-L(J ® eE) = cH(J)eE.

1=1

(iii) J-L is a homomorphism of Z2-graded coalgebras over R.

3.6 Smash Product of Z2-Graded Hopf Algebras

97

(3.6.1.2) The twisted tensor product of Z2-graded coalgebras H®E is constructed over R, the twist being ineffective due to the trivial Z2-grading of H. (3.6.1.3) The so-called smash (or crossed) product of H that Vx,y E E,V/,g E H:

~E

is defined, such

L(f)

U ~ x)(g ~ y) =

L Ut' g) ~ XJ.lU! ~ y), 1=1

which especially implies that (eH ~x)(gl8ieE) = g~x. One thereby obtains some Z2-graded bialgebra over R, which is denoted by H ~i-' E, with the unit

eH

~eE·

(3.6.1.4) If both E and Hare Z2-graded Hopf algebras over R, then H 18ii-' E is also such an object with the antipode: L(f)

H ~i-' E 3 / ~ x

7 "

L (THUD ~ J.l((THUf') ~ (TE(X)) E H

18ii-'

E.

1=1

(3.6.2) Consider the graded-cocommutative Hopf algebras H := K(G) and E := E(L) over a field K of char K i- 2, the former with the trivial grading of H =: HrJ, which were defined previously. (3.6.2.1) The graded tensor product of unital associative superalgebras, and the graded tensor product of Z2-graded coalgebras, are constructed over K. Denote F := H®E. Vh 1 , h2 E H, "lab a2 E E : (hI ~ a1)(h 2 ~ a2) = (h 1 h2) ~ (ala2)' Vg E G : eF(g ~ eE) = eK, L1 F(g ~ eE) = (g I8i eE) ~ (g I8i eE),

"Ix E L : eF(g ~ x) = 0,

L1F(9 ~ x) = (g I8i x) ~ (g

~

eE)

+ (g ~ eE) ~ (g ~ x).

Hence one obtains the Z2-graded Hopf algebra {F,J.lF,T/F,L1 F ,eF} over K. Obviously the Z2-graded coalgebra {F, L1 F , eF} is graded-cocommutative. Here (TF := T((TH' (TE) is the unique antipode of F, and (TF O(TF = id F, with the principal graded anti-automorphism (TE of E. One finds, that Vg E G, x E L : (TF(g ~ x)

= g-1

~ (-x), (TF(g ~ eE)

= g-1 ~ eE.

98

3. Coalgebras and Z2-Graded Hopf Algebras

(3.6.2.2) The group of group-like elements of F is: {g ® eEj 9 E G} +---+ G, indicating an isomorphism in the sense of groups. The Lie superalgebra of primitive elements of F is: {eH ® Xj x E L} +---+ L, with a natural isomorphism of Lie superalgebras over K. Here again one uses the Z2-graded version of the Poincare-Birkhoff-Witt theorem. (3.6.3) This graded tensor product of Z2-graded Hopf algebras over K is generalized with respect to a homomorphism of groups rrL : G --+ Aut(L), into the group of isomorphisms of the Lie superalgebra Lover K onto itself. Due to the universal property of E := E(L), for any 9 E G,rrdg) is lifted to an isomorphism rr(g) : E +---+ E of unital associative superalgebras over K. Hence rrL is lifted to some homomorphism of groups rr : G --+ Aut(E), into the group of isomorphisms of Z2-graded Hopf algebras over the field K, char K =1= 2. (3.6.3.1) The universal property of tensor products allows for the definition of an appropriate structure map

as vector space over K, such that

L

Vh =

ckgk,h' E H,Va,a' E E:

kEl'in

F1r ® F1r 3 (h ® a) ® (h' ® a')

--+(h ® a) IJ,,,

01r

(h' ® a'):=

L

Ck(gkh') ® (arr(gk)(a')) E F1r ,

kEl'in

thereby constructing an associative superalgebra F1r with the unit eH ® eE, which is denoted by H ®1r E.

(3.6.3.2) More explicitly, the following inclusions are shown in the diagrams below. ~

~

• (eH@a;aEE(L)} subalgebra

• (h@eE ;h E K(G)} subalgebra

~ . • E(L) Isomorphism

~ . • K(G) Isomorphism

3.6 Smash Product of Z2-Graded Hopf Algebras

99

(3.6.3.3)

H: (eH ®a) 011' (h®eE) = h®a, Vg E G: (g ® eE) 011' (eH ® a) 011' (g-l ® eE) = eH ® 71'(g) (a). Here one uses, that Vg E G: 71'(g)(eE) = eE, 71'(eH) = id E. Obviously, in case of 1m 71' := {id E} one recovers the skew-symmetric tensor Va E E,h E

product H~E from above. (3.6.3.4) The comultiplication to fulfill

L111'

and counit

£11'

Vg E G, x E L :.111'(g ® x) = (g ® x) ® (g ® eE)

on F1I' are defined in order

+ (g ® eE) ® (g ® x),

.111'(g ® eE) = (g ® eE) ® (g ® eE), £1I'(g ® eE) = eK, £1I'(g ® x) = O. The Z2-graded coalgebra {F1I" .111',£11'} over K is graded-cocommutative. (3.6.3.5) The Z2-graded coalgebras {F1I".111"£1I'} and {F,.1 F ,£F} coincide. For instance, Vg E G, \:!Zi, Z2 E Z2, VX1 E LZI, X2 E L Z 2 :

.1F(g ® X1X2) = .111'(g ® X1X2) = (g ® X1X2) ® (g E eE) + (g ® eE) ® (g ® X1X2) + (g ® xd ® (g ® X2) + (_lYIZ2(g ® X2) ® (g ® Xl), £F(g ® X1 X2)

= £1I'(g ® X1X2) = o.

(3.6.3.6) Hence one obtains the Z2-graded Hopf algebra {F1I" /1011" T/1I" .111" £11'} over K with the unique antipode a1l" such that Vg E G, a E E : a1l'(g ® a) = g-l ® (7I'(g-1)

° aE(a)).

Especially

V9 E G, x E L : a 11' (g ® e E) Vg E G : 71'(g)

= 9 -1 ® e E,

a 11' (e H ® x)

= eH

® (- x).

° aE = aE ° 71'(g) , and therefore a1l' ° a1l' = id F1I"

(3.6.3.7) Again the Z2-graded version of the Poincare-Birkhoff-Witt theorem is used. The group of group-like elements of F1I' is {g ® eEj 9 E G} ~ G. The Lie superalgebra of primitive elements is {eH ® Xj x E L} ~ L. These bijections are isomorphisms, the former in the sense of groups, and the latter in the sense of Lie superalgebras over K, respectively. (3.6.3.8) This Z2-graded Hopf algebra K(G) ®11' E(L) is some special case of the above defined smash product H ®I£ E, defining /10 such that Va E E(L),g E G: /1o(g ® a)

= 71'(g)(a).

100

3. Coalgebras and Z2-Graded Hopf Algebras

3.7 Graded Star Operations on Z2-Graded Hopf Algebras (3.7.1) Let the commutative ring R of coefficients, of an associative superalgebra {H,j.L,17} with the unit 17(eR) =: eH, be equipped with an involutive isomorphism of rings: R '" T +-----+ fER, for instance complex conjugation in case R := C. The bijection: H '" a +-----+ a· E H is called graded star operation on H, if and only if the following conditions are fulfilled. Va, bE H, "IT, s E R : (ra

+ sb)· =

"Ix, fi E Z2 : a E HZ, bE Hi!

==::}

fa· + sb·, a··:= (a·)· = a; a· E H:t, (ab)* = (-l)X ll b·a·.

ObviouslyeiI = eH·

(3.7.1.1) This graded star operation is then lifted to some graded star operation on the skew-symmetric tensor product of associative superalgebras over R, such that Vp EN, %, ... , zp E Z2: HZl ® ... ® HZp ",a1 ® ... ® ap --+

•

(-1 )L l$k
ZkZI

ap• ® ... ® a1.E" H®··· ®H.

(3.7.1.2) The structure mapping j.L is compatible with this graded star operation and its lifting in the sense, that: * 0 j.L = j.L 0 *. (3.7.2) Consider a Z2-graded coalgebra {H, L1, e:} over the commutative ring R, the latter again being equipped with an involution. An idempotent even antilinear bijection * on H is called graded star operation on H, if and only if the structure mappings L1 and e: are compatible with it in the sense, that: L1 0

*= *

0

L1, Va E H : e:(a·) = e:(a).

Here again the bijection * is lifted to the twisted tensor product H if9H of coalgebras over R, namely to TO T( *, *), inserting the Z2-graded twist T, and more generally to H if9 ... ~H with finitely many copies of H.

(3.7.2.1) Inserting the natural R-linear bijections:

one finds the commuting diagram: * 0012 0 T(L1, id H)

0

L1 =

023 0

T(id H, L1) 0 L1 0 *.

3.8 Z2-Graded Lie Coalgebras and Lie Bialgebras

101

(3.7.2.2) The twisted tensor product of two Z2-graded coalgebras over R is thereby equipped with an appropriate graded star operation, such that VX,Y E Z2, Va E HX,b E Hii : (a iZI b)* = (-l)XYb* iZla* E H®H.

(3.7.3) An idempotent even antilinear bijection * on a Z2-graded bialgebra Hover R is called graded star operation on H, if and only if it is compatible with the four structure mappings in the above described sense. (3.7.3.1) In the special case of a Z2-graded Hopf algebra H one demands, that a graded star operation be compatible with the antipode a in the sense, that: a 0 * = * 0 a. (3.7.3.2) In the special case of R :3 r +-----+ f := r E R, for instance real coefficients, and of aoa = id H, the antipode a is some graded star operation onH.

3.8 Z2-Graded Lie Coalgebras and Lie Bialgebras (3.8.1) Let L = LO EB L 1 be a vectorspace over a field K of char K =I- 2, and L(a)

8 : L :3 a

--+

L a; iZI a;' E L iZI L 1=1

a K -linear map, which is even with respect to the natural grading of tensor products. L is called Z2-graded Lie coalgebra, or Lie super-coalgebra, over K, if and only if the following conditions hold. (i) 8 is skew-symmetric in the sense of: r 0 8 = -8.

(ii) The so-called co-Jacobi identity holds: 0: 0 T(8, id) 08 - T(id, 8) 0 8 = T(id, r) 00: 0 T(8, id) 08.

Here id == id L, r denotes the Z2-graded flip on L iZI L, and 0: merely changes the succession of tensor products: (L iZI L) iZI L +-----+ L iZI (L iZI L), both being identified with L iZI L iZI L henceforth.

(3.8.2) Consider a Lie superalgebra, which is also a Z2-graded Lie coalgebra over K. L is called Z2-graded Lie bialgebra, or Lie super-bialgebra, over K, if and only if the above denoted structure map 8 is compatible with the super-commutator in the sense, that V homogeneous a, bEL:

102

3. Coalgebras and Z2-Graded Hopf Algebras L(a)

[a,b]

7

~)al ® [al',b]

+ (-l)X;'Y[al,b] ®al')

1=1 L(b)

+ ~)[a, blJ ® bl' + (-l)XY;bl ® [a, bl']) , 1=1

inserting explicitly 8 (a) and 8 (b), and denoting by

a, b, ai', bl, respectively.

x, y, xl' 'Yl the degrees of

(3.8.2.1) Via the universal enveloping superalgebra E(L) of L, using the super-commutator of E(L)~E(L), one may rewrite the above compatibility condition as the following l-cocycle property:

L

3

[a, b] ---+[8(a), 8(b)]

e

+ [8(a), 8(b)]

E

L®L

c

E(L)~E(L),

inserting the diagonal map, which is defined with the unit e of E(L):

L

3

6

'

a---+a ® e + e ® a E E(L)®E(L). de!

(3.8.3) The dual vector space L* := HomK(L,K) of a Z2-graded Lie coalgebra Lover K, with the natural grading of K-linear forms, becomes some Lie superalgebra over K with the following super-commutator. For a E L, f and 9 E L*, inserting explicitly L1(a), and denoting by xl and z the degrees of al and g, respectively, L(a)

[j,g](a):=

L

f(aj)g(al')(-l)ZX;.

1=1

This definition may be conveniently rewritten, such that Va E L,Vf,g E L*: ([f,g]la) = (J®gI8(a)).

(3.8.3.1) Here one uses the notation, that

(Ii ® ... ® fnlal ® ... ® an) for homogeneous

!k

E

=

L* and ak

E

(_l)El~l
(3.8.3.2) The duality of Jacobi and co-Jacobi identity is shown easily, using that Va E L, Vf,g,h E L*:

([[f,g], h] la) = (J ® 9 ® hIT(8, id) o8(a)), ([f, [g, h]] la) = (J ® 9 ® hIT(id, 8) o8(a)).

3.8 Z2-Graded Lie Coalgebras and Lie Bialgebras

103

(3.8.4) Conversely, consider a finite-dimensional Lie superalgebra L over a field K. Then the dual vector space L* becomes some Z2-graded Lie coalgebra over K, such that Va,b E L,Vf E L*: (8(J)la0b) = (J1[a,b]).

(3.8.5) Hence one establishes the dual vector space L * of a finite-dimensional Z2-graded Lie bialgebra over a field K as some object of the same category. (3.8.6) A K-linear map ¢ : L ----.. L' of Z2-graded Lie coalgebras over K is called homomorphism of such objects, if and only if: T(¢,¢) 08 = 8' 0 ¢, denoting by 8 and 8' the involved structure maps. Thereby one establishes an according category. (3.8.7) Moreover one uses the category of Z2-graded Lie bialgebras over K, its morphisms being expected to be compatible both with the supercommutators and the co-structure mappings. (3.8.8) The corresponding non-graded statements are obtained, taking as odd vector subspace of L the trivial one. (3.8.9) The complex Lie algebra 8l(2, C) == A 1 becomes some Lie bialgebra with the structure maps: [x+ x-l [h , x±] = ±2x±' " The dual vectorspace

= h·

h ----..

° x± ----..e x± 0 h - h 0 x±.

e'

Ai, with the dual basis {g, f±} such that

(g/h) = 1, (gJx±) = 0, (J±/h) = 0, (J±lx±) = 1, (J±lx'f) = 0, is some complex Lie bialgebra, the structure maps of which may be denoted as those of A 1 for convenience.

[g,f±]

= -f±, [j+,rl = OJ

9 ----.. f+ 0

e

r -r

0 f+, f± ----.. ±2(g 0 f± - f± 0 g).

e

(3.8.10) The complex Lie superalgebra B(O, 1) is also some Z2-graded Lie bialgebra, with the structure mappings looking like those of A 1 , merely inserting the anti-commutator of x±. In order to construct the complex Z2graded Lie bialgebra B(O, 1)*, one may use the dual bases {h, x±, (x±)2} and {g, f±, (J±)2}, hand 9 being even, x± and f± odd. (glh) = (J±lx±) = 1, ((J±)21(x±)2) = =t=2, (glx±) = ... = 0. One then calculates easily the following structure mappings.

104

3. Coalgebras and Z2-Graded Hopf Algebras

f±

e

+ (I±)2 ® f~ -

±2(g ® f± - f± ® g)

f~ ® (I±)2,

+ ~ ((1+)2 ® (r)2 - (r)2 ® (1+)2) .

9 -----. r ® f+ - f+ ® r

e

2

3.9 Quasitriangular Z2-Graded Lie Bialgebras (3.9.1) Consider a Lie superalgebra L over a field K of char K existence of an even element r E L ® L, such that Va E L:

(i) (ii)

[r + r(r), o(a)] [[rl2,rl3]

= 0,

o(a)

:=

f:. 2. Assume

a ® e + e ® a,

+ [rl2,r23] + [rI3,r23],T(o,id L)(a)]

= 0,

identifying: (L ® L) ® L +---+ L ® (L ® L) +---+ L ® L ® L. Here one usually denotes, with the unit e of the universal enveloping superalgebra E(L),

hereby inserting an explicit notation of r, such that for instance rl2 = r ® e, n

and using the super-commutators of ®E(L),n = 2,3. Then the even Klinear map:

L:;) a~[o(a),r] de!

E

L®L

defines some Z2-graded Lie bialgebra Lover K. This structure map called coboundary of L.

e

is

(3.9.1.1) If r fulfills the so-called classical Yang-Baxter equation (CYBE), such that [rl2, rl3]

+ [rl2, r23] + [rl3, r23]

= 0,

then L is called quasitriangular with respect to the classical R-matrix r. If moreover r + r(r) = 0, then L is called triangular. (3.9.2) The above defined complex Lie bialgebra A l is quasitriangular with respect to the classical R-matrix

3.9 Quasitriangular Z2-Graded Lie Bialgebras

105

(3.9.3) The Z2-graded complex Lie bialgebra B(O, 1) is quasitriangular with respect to the classical R-matrix

(3.9.4) Let the finite-dimensional complex Lie superalgebra L :f:. A(l, 1) be basic classical, or purely even and simple, with the generators hk, x t = Xk, xi; = Yk, k = 1, ... , s. Then the coboundary e : L ----4 L ® L, such that "1'1 k : hk

----4

e

0, x~

----4

e

dk(X~ ® hk - hk ® x~),

establishes L as some Z2-graded Lie bialgebra over C. The integers dk, k = 1, ... , s, are chosen such that -

1-

"1'1k,l: dkrkl = dink =: Bkl, dk:f:. 0, Bkl:= nidi = Blk·

(i)

0, "1'~ k : dk > O. "1'k E T : dk > O. (iii) The greatest common divisor of {dki k = 1, ... , s} is equal to 1. (ii) In the purely even case of T = Here

~ {I, ... ,s} denotes the set of indices k such that x~ are odd.

T

(3.9.5) Consider a simple finite-dimensional complex Lie algebra L of rank m, with the 3m generators hk,Xk = xt,Yk = xi;,k = 1,oo.,m. The Lie bialgebra L is quasitriangular with respect to the classical R-matrix m

r:=

L (B-

p

1) kl hk ® hi

kJ=1

+ 2L

dk j X+(J3j) ® x- (J3j).

j=1

Here the positive and negative root vectors are defined most conveniently by means of the automorphisms Tk, k = 1, ... ,m, of the complex Lie algebra L, as was explicated in Chap. 1. Along the longest word O'kl 0 . . . 0 O'k p of the Weyl group, J31:= O:kllx±(J3d:= X~l' and ~j:

x±(J3j) :=Tk 1 ooo'OTkj_l(X~), , J3j :=O'k 1 ooo,oO'k j _1 (O:kj)' The braid-like relations of these automorphisms serve for the independence of r from the choice of the longest word.

(3.9.5.1) Inserting the usual defining representation p of Am, mEN, on cm+l

one obtains: 1

m+l

r

----4

T(

p,p

kk ®E kk - - ~ ~ E m+1 ) m + 1 LJ m+l m +1 k=1

+2

~ LJ

19~I~m+1

~

LJ

l~k
E kk

m+l

®E II

kl E m+1 ® Elk m+1'

m+1

106

3. Coalgebras and Z2-Graded Hopf Algebras

(3.9.5.2) For sentations:

em, m

L L

r -----+

~

3, and D m, m

4, one obtains the defining repre-

(E kk - E 2m +l- k ,2m+l-k)

1:5k:52m

+2

~

E kl

® Elk

±2

®

L

E kk E kl

® E 2m +l- k ,2m+l-1

1:5k
L

- 2

E kl ® E 2m + 1-

k ,2m+l-I,

l::;k
with the upper sign for

(3.9.5.3) For B m , m

~

em, and the lower one for D m. 2, the defining representation reads:

L L

r -----+2

(E kk - E 2m + 2 - k ,2m+2-k)

® E kk

1:5k:52m+l,k#m+l

+4

E kl

® (Elk _ E 2m + 2 - k ,2m+2-1) .

1:5k
3.10 Duals of Quasitriangular Z2-Graded Lie Bialgebras Consider a Z2-graded Lie bialgebra L over a field K, char K :f; 2, which is quasitriangular with respect to the classical R-matrix r. (3.10.1) Let p : L -----+ Mat(d,K) be a representation of Lon Kd, in the sense of unital associative superalgebras. The Z2-grading on Kd is arranged such that its Cartesian unit vectors E; are even for 1 ::; k ::; m, odd for m + 1 ::; k ::; m + n = d. The classical R-matrix is then represented on the two-fold skew-symmetric tensor product of M at(d, K): d

r -----+ r p =: T(p,p)

L

rijkl

E~j ®E;l E Mat(d,K)®Mat(d,K),

i,j,k,l=l

with the basis matrices (E~j)kl := bik bjl, i, ... , l = 1, ... , d. Sometimes one conveniently uses an isomorphism of unital associative superalgebras over K: Mat(d, K)®Mat(d, K)

f-----+

Mat(d 2 , K),

which is denoted explicitly in Chap. 8.

Duals of Quasitriangular Z2-Graded Lie Bialgebras

107

(3.10.2) The Z2-graded Lie bialgebra A r over the field K is defined by the following super-commutation relations and Lie comultiplication of generators akl, k, l = 1, ... , d. Define k := 0 for 1 ~ k ~ m, and 1 for m + 1 ~ k ~ d; the generators akl are taken as even or odd for k + [ being so. The coboundary e of A r is defined such that: A r 3 akl

e

d

~) _1)(k+3)(3+ i ) akj

® ajl - ajl ® akj EAr ® A r .

j=l For i, ... , l = 1, ... , d, one demands that d

(-1)(i+3)(k+i)aij akl- akl aij =

2: (a ip rpjkl- rijkp apl p=1

+(_1)(i+3)(l+k+i)akp rijpl- (-1)(i+3+l)(k+i)riPkl apj)' (3.10.3) Two Z2-graded Lie bialgebras L 1 and L 2 over K are called dual with respect to a K-bilinear form: L 1 x L2 -- K, if and only if 'Val,b 1 E L 1,a2,b2 E L 2:

defining an according K-bilinear form on tensor products as explained previously. (3.10.4) The above defined Z2-graded Lie bialgebras Land A r are dual with respect to the K-bilinear form:

(3.10.5) The super-commutation relations of A r may be rewritten conveniently in terms of the graded tensor product over an associative unital superalgebra A over K, inserting an injective homomorphism a : A r - - A of such objects. This construction is explained in detail in Chap. 8, in order to rewrite the so-called main commutation relations (MCR). d

Mat(d,A) 3 [bkl;k,l = 1, ... ,d]

--.+

2: bkl ®E;l E A®Mat(d,K). k,I=1

[b ij ; i,j

= 1, ... ,d] ®A hi; k,l = 1, ... ,d] d

--.+

2: i,j,k,I=1

(_1)(i+3)<:kl bij Ckl ®Ey ®E;l,

108

3. Coalgebras and Z2-Graded Hopf Algebras

= (_I)(i+J+k+ilb+(k+i)(jlHl6jp 6l r ab (i9 E~q (i9 E;s, for i, ... ,s = 1, ... ,d. With this product, the K-vector space

Mat(d, A)®AMat(d, A) +-----+ A (i9 Mat(d, K) (i9 Mat(d, K)

becomes an associative superalgebra with the unit eA (i9 fd (i9 fd· (3.10.6) Denoting fA := eA (i9 fd, r A := eA (i9 r, with the unit eA of A, inserting the graded flip T on M at(d, K)®Mat(d, K), T( id A, T) =: TA, such that TA(rA) = eA (i9T(r), and abbreviating [o:(akz)jk,l = 1, ... ,d] =: [a], one obtains ([a] (i9A fA) (fA (i9A [a]) - (fA (i9A [a]) ([a] (i9A fA)

= ([a]

(i9A fA

+ fA (i9A

[a]) r A - r A ([a] (i9A fA

+ fA (i9A

[a]) .

(3.10.6.1) A family of quasitriangular complex Z2-graded Hopf algebras, inserting for its universal R-matrices appropriate representations on Cd (i9C d: R(q) -dd (i9 fd

+ rlnq + 0

(lnq)2), q E C\l- 00,0],

tends on Cd, for q --+ 1, to an according quasitriangular Z2-graded bialgebra with respect to r, to the first order in lnq. Especially the quantum YangBaxter equation tends in this limit to the classical one. Therefore the above super-commutation relations may be called semiclassical main commutation relations. This limit will be discussed in Chap. 8.

(3.10.7) In the case of sl(2, C) one finds the following semiclassical dual, denoting r +-----+

all

== a, aI2 == b, a2I ==

~2 [~0 ~10 -1~ 0~] '

C,

a22 == d.

T +-----+ transposition,

o

0 0 1 ab - ba = -b, bd - db = -b, ac - ca = -c, cd - de = -c, ad = da, be = cb. With the additional relation a = -d one recovers the dual Ai of AI. (3.10.8) In the case of Am, mEN, inserting the usual defining representation on cm+1 such that

~+1i,j, k, l

: rijkl = 6ij 6kl 6i k - _1-6ij6kl

m+l

+ 26il

6jkO(k -l),

one obtains the super-commutation relations [aij, akd = aij 6kl(6j k - 6i k) + akl6ij(6il - 6i k) + 2ail 6j k (O(k -l) - O(k - i» + 2akj 6il (O(j -l) - O(k -l»,

denoting O(n) := 1 for n E N, and 0 for n E -No.

Deformed Tensor Product of Semiclassical MCR-Type Algebras

109

3.11 Deformed Tensor Product of Semiclassical MeR-Type Algebras (3.11.1) Consider a Z2-graded Lie bialgebra Lover K, which is quasitriangular with respect to the classical R-matrix r E L ® L, and let 0:' and (3 be representations of the Lie superalgebra L on KP and Kq, respectively. The corresponding dual Z2-graded Lie bialgebras A(r; 0:') and A(rj (3) are then generated by aij and bkl, 1 ~ i,j ~ p, 1 ~ k, I ~ q, with the previously described grading and semiclassical MCR. The tensor product of these two representations is constructed as usual, inserting the diagonal map 8, which is uniquely extended to the comultiplication on the universal enveloping superalgebra E(L): 6

L3x--+x®e+e®x

KP ® Kq

3

u®v

--+

T(a,{3) --+

~T6(0:',{3)(X):

O:'(x)(u) ® v + u ® (3(x)(v)

E

KP ® Kq.

Denoting ')' ~ T6 (0:', (3), and the generators of the Z2-graded Lie bialgebra A(r; ')') by Cra, 1 ~ r, s ~ pq, one easily finds that

(XIC(i-1)q+k,U-1)q+l)

= (-1)(i+ 3)k(xlaij)8kl + 8ij (xlbkl)'

(3.11.2) On the other hand, denoting r (8(x)laij ® bk1 ) = ([8(x), r]/aij ® bkl) =

t. (1;1 ((xlaim)(r~lamj)(r~lbkl)(

=:

L:f=l r~ ®r~, one calculates that

= -([8(x), r(r)]laij ® bk1 ) _1)(i+3)(k+i)

-(xlamj) (r~laim)(r~lbkl)(_l)k+i) q

+

L ((xlbkn)(r~laij)(r~lbnl)( _1)i+3 n=l

-(xlbnl) (r~ laij) (r~lbkn)( -1 )(i+3)(k+i)) )

=

t. (1; ((xlamj)(r~lbkl)(r~laim)

-(xlaim)(r~ Ibkl) (r~lamj)( -1 ) (i+3+1)(k+i) ) q

+

L ((xlbnl)(r~lbkn)(r~laij)( _1)(i+3)(k+i+l) n=l

110

3. Coalgebras and Z2-Graded Hopf Algebras

(3.11.3) Denoting r(a,{J) := T(o:, {3)(r), and similarly r({J,a), one is thereby heuristically led, remembering that the involved bilinear forms may be degenerate, to the following theorem. A homomorphism of Lie superalgebras over K is established: A(r; 1) 3C(i-l)q+k,(j-l)q+1 --+

(-1)
+ 8ijbkl

E A(r; o:)®A(r; {3),

into the Lie superalgebra of generators aij and bkl, 1 ~ i,j ~ p, 1 ~ k, l ~ q, with respect to the following relations: The generators aij among themselves, and bkl among themselves, fulfill the super-commutation relations of A(r; 0:) and A(rj {3), respectively. Moreover they obey the mixed relations (-1)
= ~ (r(a~{J)a. _ r~a,{J)a '(_1)
+

~(r(~,{J)b (_1)(i+J)(k+i+1) _ r(~,{J)b ) L..... oJnl kn oJkn nl , n=l

(-1)
»)

= ~(r({J~a)b _ r({J'
+ L..... ~ (r({J,a~ a.tm (_1)(k+i)(i+3+1) klmJ

_ r({J,a) a .)

kltm mJ .

m=l

(3.11.3.1) Using again faithful copies into an associative unital superalgebra

A, with the previously introduced abbreviations lA, [a] and [b], and denoting r~a,{J) := eA ®r(a,{J), and similarly r<,f,a), these relations can be short-written as the subsequent mixed semiclassical MCR. ([a] ®A lA)(lA ®A [bJ) - (lA ®A [bJ)([a] ®A lA)

= ([a] ®A lA + lA ®A [bJ)r~,{J»)

- r~a,{J)([a] ®A lA

and similar relations, interchanging [a] with [b], and indicated homomorphism can be abbreviated as:

+ lA ®A 0:

[bJ),

with {3. The above

and may be called deformed tensor product of semiclassical MCR-type algebras.

Deformed Tensor Product of Semiclassical MeR-Type Algebras

111

(3.11.3.2) The proof of this theorem is based on the following almost trivial lemma:

T(-y, 'Y)(r)

+--+

d~,Q) + r~~,t3)

+ r~~,Q) + T~~,I3),

where the lower indices denote embeddings into an appropriate four-fold tensor product of matrices over K. One then calculates that [a] @AIA + I A @A [b] fulfills the semiclassical MeR-type relations with respect to T(-y, 'Y)(r).

4. Formal Power Series with Homogeneous Relations

Since the relations and costructure mappings of quantum algebras, which are investigated in Chap. 8, contain exponentials in the generators, and in particular with respect to the universal R-matrix, one is forced to introduce an appropriate topology on the algebra or coalgebra in question. If the ring of coefficients cannot be equipped with a suitable natural topology, one just uses its discrete topology, thereby ending up with formal power series. The theory of formal power series over an arbitrary set of commuting indeterminates is presented in the volume Algebre. Chapitres 4 a. 7, by N. Bourbaki (1981). Formal power series in finitely many generators without any relations, and their topological tensor product, can be used in order to prove on an algebraic level the Baker-Campbell-Hausdorff formula, as is shown for instance in the monograph on Lie algebras by N. Jacobson (1962). Formal power series with homogeneous relations, and also their topological tensor product, are constructed easily, factorizing with respect to the smallest closed ideal of the set of relations. Inhomogeneous relations, for instance of exponential type, are not compatible with the topology of formal power series. Possibilities to escape this difficulty, which is just typical for quantum algebras, are presented in Chap. 8.

4.1 Polynomials in Finitely Many Commuting Indeterminates (4.1.1) Let X := {Xki k E I} be a finite set, with the index set 1:= {1, ... ,p}, and R a commutative ring. The commutative unital associative algebra of polynomials in the commuting indeterminates Xl, ... ,Xp over R is defined. R[X]

== R[xl, ... ,xp ]

:= T(E)jC, E:= R(X),

C := ideal( {XkX/

- X/Xki

k, lEI}),

with the unit e := {eR' 0, 0, ...} + C. The equivalence classes Xk + C, k E I, are conveniently denoted as

Xk

itself.

114

Formal Power Series with Homogeneous Relations

(4.1.2)

R[X] = R({X~l ... x;P; nl,"" np E No}), with an R-basis of monomials. Here one denotes Vk E I :

x2 := e.

(4.1.3) This algebra of polynomials, which is just the symmetric algebra S(E) over the R-bimodule E, is universal in the following sense. Let w : E - - A be an arbitrary Weyl map into a unital associative algebra A over R, which means that

"Ix, y E E : w(x)w(y) = w(y)w(x). Then :3 unique homomorphism w. : S(E) - - A of unital associative Ralgebras, such that w. 0'" 0 (3 = w, according to the diagram below.

L

E _----lP~_. T(E)

(j)

Weyl map

_ _ _ _K:.:..-

•

jalgabra-homomorPhlsm

S(E)

m..

algebra-homomorphism

(4.1.4) Let M be an R-bimodule. The elements of M[X] :=. M ® R[X] are called polynomials in the commuting indeterminates Xl,'" ,xp , with the coefficients EM.

M = R({mk;k E K}) ~ M[X] = R({mk ®X~l .. ,x;Pjnll ... ,n p E NOj k E K}). (4.1.5) Both R[X] and M[X] are No-graded with respect to the grading, such that Vnl, ... ,np E No : dey X~l ... x;P := nl

+ ... + n p ,

dey e = 0, dey 0 := -00.

(4.1.6) Let A be a unital associative algebra over R, and {ak; k E I} a family of pairwise commuting elements of A. Due to the above universal property, :3 unique homomorphism a : R[X] - - A of unital associative algebras over R, such that Vk E I : a(xk) = ak. The family {ak; k E I} is called generic, if and only if a is injective. One then conveniently denotes V¢ E R[X] : a(¢) =: ¢(al, ... ,ap ).

(4.1.6.1) In particular let A be commutative. Then the elements of 1m a are called polynomial functions in p variables E A.

4.1 Polynomials in Finitely Many

Commuting Indeterminates

115

(4.1.6.2) An insertion of polynomials into polynomials is defined by the special choice of A:= R[Y], Y := {Yt, ... , Yq}, considering the image of a.

R[X]

3

Xk ~ de!

'L..J "

r(k) ymt 1 ml· .. m q

•••

yqmq

E

R[Y].

ml, ... ,m q EN o

(4.1.7) Let X := {Xl, ... ,xp} and Y:= {YI, ... ,Yq} be finite sets, and A a commutative unital associative algebra over R. Take

"l/Jk :=

r~L.mqy~l ... y;;q E R[Y], k = 1, ... ,p,

L ml,···,mqENo

and denote by ~ : R[X] --+ R[Y] the· unique homomorphism of unital associative algebras over R, such that Vf.k : ~(Xk) = "l/Jk. Furthermore take aI, ... ,aq E A, and denote by a : R[Y] --+ A the unique homomorphism of unital associative algebras over R, such that \1ft : a(YI) = al. Thereby one obtains the mapping: p

R[X] x

q

(TI R[Y]) x (TI A)

--+

A,

which may be called composite polynomial in variables

R[X] 3 ¢>:=

--+

" ENO rnl ... n xl L.Jn 1,"" n p P

..•

xp

R[Y] 3 "l/JI, ... , "l/Jp A 3 al,'" ,aq ¢>("l/Jk(al; t = 1, ... , q); k = 1, ... ,p) = a

L

IT ( L

r nl ... np

nl, ... ,npEN o

k=l

E

A.

np

nl

}

0

~(¢»

r~Lmqa~l ... a;;q) n~ EA.

ml, ... ,mqEN o

(4.1.8) The partial derivations 8~~ E DerR(R[X]) of such polynomials are defined, such that

(4.1.9) Consider again the insertion of R[Y] into R[X], with the notation from above, and denote Vf.k : a(xk) =: "l/Jk E R[Y]. One easily calculates the following chain rule. {)

'V¢> E R[X], \1ft : ~ VYI

0

p

{)

{)

k=l

VXk

VYI

a(¢» = Lao ~(¢»~"l/Jk E R[Y].

116

Formal Power Series with Homogeneous Relations

(4.1.10) Let the field of rational numbers Q be a subring of R. Then one finds easily Taylor's formula. For two disjoint sets of indeterminates X := {Xl, ... 'Xl'} and Y := {Yl,"" Yl'}' consider the following polynomial E R[X U Y] with the coefficients r : Dl' No --+ R.

~

rnloO.np(Xl

+ Ylt

1

'"

(xl'

+ Yl')n p

nt •...• npEN o

=

(4.1.10.1) This formula also holds for the images of polynomials in a commutative unital associative algebra over R.

4.2 Power Series in Finitely Many Commuting Indeterminates (4.2.1) Let G be a group, and also a Hausdorff topological space. G is called topological group, if and only if the mappings: GxG

3 {g,h} --+ gh E

G, G

3 9 --+ g-l E

G

are continuous, the former with respect to the product topology. The direct product of a family of topological groups is some topological group, with respect to the product topology. (4.2.1.1) A subgroup F of a topological group G becomes a topological group be means of the subspace topology; if F is open, then it is also closed in Gj the closure of F is some subgroup of G. (4.2.1.2) Let N be an invariant subgroup of a topological group G. The quotient topology on the factor group GIN is defined in the following manner. A subset V of GIN is called open, if and only if 11'-1 (V) is an open subset of G. In this case the canonical projection 11' : G 3 9 --+ gN E GIN is both open and continuous. Let 4> : GIN --+ T be a map into a topological space T; if 4> 0 11' is continuous, then 4> itself is continuous. (4.2.1.3) This quotient topology on GIN is Hausdorff, if and only if the subset N of G is closed; in this case GIN is some topological group. If especially N is an open subset of G, then the quotient topology is discrete.

Power Series in Finitely Many Commuting Indeterminates

117

(4.2.1.4) Let Gk, k = 1,2, be topological groups, and consider a homomorphism of groups ¢ : G 1 ---+ G 2 . ¢ is continuous, if and only if it is continuous at the unit e1 of G 1 ; in this case its kernel keT ¢ is some closed subset of G 1 , and the three group homomorphisms:

G1

---+

GdkeT ¢

---+

1m ¢

---+

G2

are then continuous. Two topological groups G1, G2 are called isomorphic, if and only if there is an isomorphism of groups: G 1 +--+ G 2 , which is also a homeomorphism. (4.2.1.5) Assume existence of some countable basis {Vn ; n E N} of neighbourhoods of the unit. Let {9k; kEN} be a sequence of elements of G. It is called convergent to 9 E G, if and only if "In EN: 3m EN: 'Vk ~ m : 9k9- 1 E Vn ; in this case one writes 9 = limk_oo 9k· It is called Cauchy sequence, if and only if "In EN: 3m EN: 'Vk,l ~ m: 9k91 1 E Vn . Obviously every convergent sequence is of Cauchy type.

(4.2.1.6) G is called complete, if and only if every Cauchy sequence converges to an element of G. In this case G/ N is complete too, for every closed invariant subgroup N of G. (4.2.1.7) For every topological Abelian group G, there is a (modulo isomorphism) unique complete topological Abelian group the so-called compleand tion of G, such that G is isomorphic to some topological subgroup of G lies dense in = G.

a,

a,

a, a

(4.2.1.8) Let Gk, k = 1,2, be topological Abelian groups, F1 a subgroup of Gl, and assume G2 to be complete. Then every continuous homomorphism of groups: F1 ---+ G2 can be extended uniquely to a continuous homomorphism of groups: F 1 ---+ G 2 • (4.2.1.9) The topological ring R[X], with the h -adic topology defined below, is not complete with respect to the addition of polynomials. (4.2.2) The commutative unital associative algebra of polynomials R[X] over a commutative ring R, in the indeterminates Xl,' •• , X p EX, is equipped with the so-called I 1-adic topology. Here 11 denotes the ideal of polynomials without constant term,

h

:= {

L nl, ... ,npENo

Tnl ...

npx~l .•. x;P;

T n1 ... np

E R, TO... O =

o}.

118

Formal Power Series with Homogeneous Relations

(4.2.2.1) "1m EN: Vm(O) :=

={

1m := sum( {h··· 1m; h,···,1m E II})

rnl ... npx~l···x;p; nl+ ... +np<m~rnl...np=O},

L n ......npENa

\10(0)

:=

R[X], V1 (0) = It, Vm(O) ~ Vm+1(O),

n

Vm(O) = {O}.

mENa

For every m E No, Vm(O) is an ideal of R[X]. (4.2.2.2) There is an obvious isomorphism of unital associative algebras: R[X]/ II +-----+ R. (4.2.2.3) "1m E No, VI E R[X]: Vm(f):= {f+g;g E Vm(O)}, Vo(f) = R[X]. (4.2.2.4) With this countable basis of neighbourhoods {Vm(f)j mE No} of IE R[X], the topology Tl is constructed. VI E R[X] : WI := {V ~ R[X]; 3m EN: Vm(f) ~ V}. The set U/ER[Xl WI fulfills the axioms for the neighbourhoods of an according topology. Therefore Tl

:=

{V

~

R[X]; VIE V : V E WI} 3 0

is the unique topology on R[X] with exactly these neighbourhoods. By its very construction, VI E R[X], "1m E No: Vm(f) E TI. (4.2.2.5) The ring R[X] is topological, because the following mappings are continuous with respect to the product topology. 2

IT R[X] 3 {f, g} 1+ 9 E R[X], R[X] 3 I IT R[X] {f,g} ----4lg R[X]. ----4

+-----+

-IE R[X],

2

3

E

(4.2.2.6) Obviously the topology

Tl

is of Hausdorff type.

(4.2.3) Let {fk j kEN} be a sequence of polynomials E R[X]. This sequence is called convergent to I E R[X], if and only if "In EN: 3m EN: Vk

Ik - IE Vn(O); limk-+oo Ik.

~

m:

in this case one writes I = It is called Cauchy sequence, if and only if "In EN: 3m EN: Vk, l

~

m : !k

-

II E Vn(O).

Of course every convergent sequence is of Cauchy type.

Power Series in Finitely Many Commuting Indeterminates

119

(4.2.4) The commutative unital associative algebra R[[X]] == R[[xt, ... ,xp ]] of formal power series in the indeterminates Xt, . .. ,xp E X over R is defined.

R[[X]]

L

:= {:=

Tnt ... npxft .. ·x;P :

nt •...• npENo

fINO 3 {nt, ... , n p }

->T nt ... np

E

R}.

An R-linear combination of such power series is defined naturally, and the product of formal power series is defined by an appropriate rearrangement. Let (k), k = 1,2, be such power series, with the coefficients T~~~ ..np, respectively. V8t, 82 E R:

(1)(2):=

L

L

mEN o nt'···''''pENo nt

+... +rn.p=tn

(4.2.4.1) R[X] is considered as some subalgebra of R[[X]] in the natural sense. (4.2.4.2) For an R-bimodule M, the elements of M[[X]] := M ® R[[X]] are called formal power series in the indeterminates Xt, ... ,xp , with the coefficients E M. (4.2.5) The following family of ideals of R[[X]] is used as some countable basis of neighbourhoods of O. '1m E N:

V m(O)

L

:= {

Tnt ...npxft ... X;Pj

nt .....npENo

nt

+ ... + n p < m

~ Tntoo.np = o},

With the resulting Hausdorff topology one establishes the topological ring

R[[X]]. (4.2.6) Convergence of sequences, and Cauchy sequences of formal power series, are defined with respect to their addition.

120

Formal Power Series with Homogeneous Relations

(4.2.7) The topological ring R[[X]] is complete with respect to the addition of formal power series. Obviously R[X] lies dense in R[[X]]. Moreover the subspace topology of R[X], which is induced from R[[X]], coincides with the topology Tl of polynomials defined previously. Therefore R[[X]] is the unique completion of R[X]. "1m E No: Vm(O) = Vm(O), the closure of Vm(O) in R[[X]]. (4.2.8) The R-bimodule R[[X]] is the following direct product of bimodules over R. p

R[[X]] = II~, Vi

E 1:=

II No : ~:= R.

iEI

The product topology of R[[X]], inserting the discrete topology of R, coincides with the topology, which is induced by the basis of zero-neighbourhoods defined above, all of which are ideals. (4.2.9) Let A be a commutative unital associative algebra over R, equipped with a Hausdorff topology, which is compatible with the addition. Let A be complete with respect to the addition. Moreover assume the existence of some countable basis of zero-neighbourhoods, all of which are ideals of A; then A is some topological ring. (4.2.9.1) Let a : R[[X]] ~A be a homomorphism of unital associative algebras over R, which is continuous. Then Vfk : limn......co(a(Xk)}n = O. (4.2.9.2) Let {all"" a p } be a family of elements of A, such that Vfk : limn_co a~ = O. Then 3 unique homomorphism a : R[[X]] ~ A of unital associative algebras over R, such that a is continuous, and Vfk : a(xk) = ak. One conveniently denotes V¢ E R[[X]] : a(¢) =: ¢(all"" ap ). (4.2.9.2.1) There is a unique homomorphism a : R[X] ~A of unital associative algebras over R, such that Vfk : O(Xk) = ak. Since a is uniformly continuous, it is uniquely extended to some continuous map a : R[[X]] ~ A. Moreover a is some homomorphism of unital associative algebras over R, which is uniformly continuous. (4.2.10) A formal power series ¢ E R[[X]] fulfills the condition limn_co ¢n = 0, if and only if its constant term vanishes, such that ¢ E V 1 (0). (4.2.11) An insertion of formal power series E R[[Y]] into a power series ¢ E R[[X]] is constructed. Denote X := {Xl"", x p}, let 'fIJi, ... , 'l/Jp E R[[Y]], and assume that Vfk : limn......co'l/Jr = O. Then 3 unique continuous homomorphism of unital associative algebras a : R[[X]] ~ R[[Y]], such that Vfk: a(xk) = 'l/Jk.

Power Series in Finitely Many Commuting Indeterminates

121

(4.2.12) One easily calculates, that

L

(eR - t)-l =

t n E R[[t]].

nENo

(4.2.13) The formal power series cP E R[[X]] is invertible, if and only if TO + cPl' such that cPl E Vl(O), and such that 3so E R: TOSO = eR·

cP =

(4.2.13.1) Let cP = TO + cPl with the above assumptions. Then 3"p E Vl(O) : cP = To(eR -"p). Moreover 3 unique continuous homomorphism a: : R[[t]] -+ R[[X]] of unital associative algebras, such that a:(t) = "p. Hence

a:(eR - t)a:((eR - t)-l) = eR, a:(eR - t)

=

eR - t/J = socP.

(4.2.14) For any ill' .. ,ip E No, the partial derivation:

R[[X]] 3 cP:=

T

x n1 ... xnp nl···np 1 p

nl ... ·.npENo

&

~Xk := n(n -1)··· (n -i uX k {}o

n

~ 0 xk

uX k

:= xkn£or n E N 0,

+ l)x~-1

&

n

~xk:=

for i,n E Nand n ~ i, or N 0 3 n < i ,

O£

uX k

is an R-linear uniformly continuous mapping. (4.2.15) Let X := {Xl, ... ,X p} and Y := {Yl,"" Yq} be disjoint sets. The mapping:

R[[X]][[Y]]

L

3

( L

mlo ....mqEN o f---+

~ L...J

nl,

T~7.~:~;nq)X~l ... x~p ) y~l .•. y;nq

npENo

T(m1 mq)x n1 ... xnpy m1 .•. y mq E nl n p 1 p 1 q

R[[X U Y]]

nl ..... mqENo

is an R-linear bijection, which is uniformly continuous from the left to the right hand side.

122

Formal Power Series with Homogeneous Relations

(4.2.16) This R-linear bijection is used in order to write down Taylor's formula. Let X:= {Xl, ... ,Xp}, Y:= {Yl, ,Yp}, X n Y = 0, Z:= {Zl, ... ,zp}, and

L::

:=

rn1 ...npzfl

z;p

E

R[[Z]].

nl, ... ,npENo

The image of with respect to an insertion of formal power series, such that Vik: R[[Z]] 3 zk ---+Xk + Yk E R[[X U Y]], is denoted naturally:

R[[Z]]

3

L::

---+

=:

rn1 ... np(Xl

+ Ylt 1 ... (xp + Yp)n

p

(x + y) E R[[X U Y]].

With this notation, using the above R-linear bijection, one obtains:

y~l ... Y~ E R[[X]] [[Y]].

(4.2.16.1) If especially the field Q is some subring of R, then one obtains Taylor's formula, which is written here with an obvious abuse of notation.

(X + y)

=

(4.2.17) Consider the unique continuous homomorphism a: R[[X]] ---+A of unital associative algebras over R, such that Vik : a(xk) = ak, for ak E A such that Vi k : lim n .- oo ai: = o.

(4.2.17.1) A is an R[[X]]-left module over R, with the module-multiplication: 3 {, a} ---+ a(
R[[X]] x A

V,1/J E R[[X]] : D(1/J)

= a(
(4.2.17.2) For every family {b k E A; k = 1, ... ,p}, :3 unique continuous Rderivation D of R[[X]] into A, such that Vik : D(Xk) = bk. Moreover D is uniformly continuous, and

Power Series in Finitely Many Commuting Indeterminates

123

(4.2.17.3) Let L1 E DerR(A) be continuous. Then the above theorem immediately implies, that

8

p

L: 0:( -8
'V¢ E R[[X]] : L1 oo:(¢) =

Xk

k=l

(4.2.17.4) "'Iil, a~l is the unique continuous R-derivation of R[[X]] into itself,

such that

unit

for k = l for k::j:. l .

"'Iik:Xk--+ { 0

(4.2.18) Every R-derivation D of R[[X]] into itself is uniformly continuous, because "1m E No : 1m Dlv m+l(O) ~ V m(O). (4.2.18.1) Let d : R[X]

'Vj,g

E

--+

R[[X]] be an R-linear map, and assume that

R[X] : dUg) = (df)g

+ j(dg).

Then 3 unique R-derivation D of R[[X]] into itself, such that DIR[Xl = d. (4.2.18.2) The family {a~k; k = 1, ... ,p} is some basis of the R[[X]]-left module DerR(R[[X]]).

(4.2.19) If R is an integral domain, then R[X] and R[[X]] are integral domains too. (4.2.20) Let the field Q be some subring of R, and consider R[[t]]. (4.2.20.1) The exponential function is defined as exp(t) == et :=

tn

L: I ' 00

n=O

e(t):= et

-

1,

l(t):=

n.

00

tn

n=l

n

L:( _l)n+l_.

(4.2.20.2) exp(t

+ s)

= exp(t) exp(s) E

R[[t, s]].

8 atl(t)

00

= L:(-t)n = (1 +t)-l. n=O

124

Formal Power Series with Homogeneous Relations

(4.2.20.3) In the sense of inserting formal power series into formal power series defined above one finds, that

a

a

_

atloe(t) = 1, ateol(t)=(1+eol(t»)(1+t) 1, hence

10 e(t) = e 0 l(t)

= t.

(4.2.20.4) e(t + s) = e(t) + e(s)

+ e(t)e(s),

l(t)

+ l(s) = l(t + s + ts).

(4.2.21) Let again Q be some subring of R, and denote X := {Xl, ... ' X p }. The ideal V 1(O) of R[[X]] is an open and therefore also closed subset of R[[X]]. With the notation V 1 (O) =: 11 it is considered as topological subgroup of R[[X]], with respect to the addition. The subset V 1 (1) := {1 + ¢; ¢ E V 1 (O)} of R[[XlI is also open and therefore closed; as topological subgroup of R[[X]] , with respect to the multiplication, it is denoted by L 1. (4.2.21.1) Let ¢,1/1 E III and consider the unique continuous homomorphisms of unital associative R-algebras a1 : R[[t]]

----+ R[[X]],

a2: R[[t, s]] -. R[[X]], a: R[[t]] -. R[[X]],

such that a1(t) = a2(t) = ¢, a2(s) = 1/1, a(t) = ¢+1/1. Denote the-exponential function of ¢ by exp(¢) := a1(exp(t)), exp(1/1):= a2(exp(s», exp(¢ + 1/1) := a(exp(t». Then from the partial sums one concludes, that exp(¢ + 1/1) = exp(¢)exp(1/1). Here one uses that a2(exp(t + s)) = exp(¢ + 1/1).

(4.2.21.2) V¢

E 11:

exp(¢) -1 = e(¢) := a1(e(t», l(¢) := a1(l(t».

Vm EN: ¢ E Vm(O) ==> e(¢) E Vm(O) ¢:=}

exp(¢) E V m(1) := {1 + ¢; ¢ E V m(O)}.

Therefore the group homomorphism exp : 11

(4.2.21.3) V¢,1/1 E 11 : l(¢) + l(1/1) = l(¢ + 1/1 + ¢1/1).

----+

L 1 is continuous.

Power Series with Homogeneous Relations

125

(4.2.21.4)

= loe(¢) = ¢.

V¢ E 11 : eol(¢)

Hence one establishes the homeomorphisms e : 11 ~ 11, e- 1 = l.

(4.2.21.5) One therefore obtains the homeomorphic group isomorphism exp : 11 ~ Ll, and denotes the logarithm In := exp-1. (4.2.21.6) V¢, t/J E L 1 : In(¢) V¢ El l

:

10(1

+ In(tP) =

+ ¢) =

In(¢tP).

l(¢).

(4.2.21.1) Let L1 E DeTR(R[[X]]) be continuous. V¢ Ell: L1 0 exp(¢) = exp(¢)L1(¢). V¢ ELl: L1(¢) = ¢L1(ln¢).

4.3 Power Series in Finitely Many Indeterminates with Homogeneous Relations (4.3.1) Let X := {Xk; k E l} be a finite set, with the index set I := {I, ... ,p}, and denote by F == R(X) := T(R(X)) :=

E9 T"(R(X)) rENo

the free algebra over X, consisting of polynomials in these generators, with the coefficients from a commutative ring R, and with the unit e == eF.

R(X) = R({Xkl ,,,xkn;kl, ... ,kn

E

l;n

E

N} U {e}).

(4.3.2) The direct product of R-bimodules F == R((X)):=

II T"(R(X)) rENo

is an R-bimodule, which can be viewed as the following set of families of mappings.

R((X)) = { {TO E R,

=: TO

{IT

+L nEN

I 3 {k 1 ,

L

,

kn } -Tkl ...kn E R;n EN} }

Tkl knXkl'· 'Xkn}.

kl .....knEI

Obviously F is an R-submodule of F.

126

Formal Power Series with Homogeneous Relations

(4.3.2.1) With the product of these formal power series in p algebraically independent indeterminates defined below, one obtains an associative algebra F over R, with the subalgebra F of F, and the unit

e == ep =

eF := { eR, {

IT

I --+{O} --+ Rj n EN} } .

+

r(l) r(2) Xk' .. XI E kt ... k n It ...I", t "'

F.

qEN\{l} m.nEN;m+n=q kt .....kn.lt .....I"'EI

(4.3.2.2) With the countable basis of zero-neighbourhoods \10(0) := F, VmEN:

Vm(O):=

{L L

rkt ... knXk t ·· 'Xk n ;

"In

< m:

rkt ... k n =

O}

nEN kt .....knEI

:J Vm+l(O),

each of them being an ideal of F, and topological ring of Hausdorff type.

nmEN

Vm(O) = {O}, F becomes some

(4.3.2.3) By means of an analogous basis of zero-neighbourhoods {V m(O); mE No}, which are ideals of F, one constructs the topological ring F, which also is of Hausdorff type. F is the unique completion of F with respect to the addition. Obviously "1m E No : V m(O) = Vm(O), the closure of Vm(O) in F, is both closed and open in F. Vo(O) = F. (4.3.2.4) This topology of F is just the product topology of the Cartesian product

R (II R), 1:= x

UnEN(nn

I),

I

inserting the discrete topology of R. (4.3.3) Let S be a subset of F, and denote by ] the closure of the ideal J of S in F. Since the closure of an ideal in a topological ring is again an ideal of this ring, one can define the factor algebra F/] =: R«X,S)). With the usual quotient topology, such that the projection "if : F --+ F /] is both continuous and open, one obtains some topological group F /] with respect to the additionj it is of Hausdorff type, because] is closed in F by definition.

Power Series with Homogeneous Relations

127

Moreover, since the set of ideals {V~(O) := 7f(Vm(O)); mE No} is some basis of zero-neighbourhoods for this quotient topology, the ring F j] is topological. Here -8

-8

'<1m E No: V m+l(O) ~ V m(O) ~ 7l'(Vm(O)), -8

-

-

--

and V m(O) is both open and closed in FjJ. Obviously FjJ is complete with respect to the addition. (4.3.3.1) If, : F j] ---+ T is such a map into a topological space T, that , o7f is continuous, then, itself is continuous. (4.3.3.2) Relations S are called polynomial, if and only if S ~ F; in this case the closures in F, of the ideal Jfin of Sin F, and of the ideal J of S in F, coincide. (4.3.3.3) Polynomial relations S are called homogeneous of degree r E N, if and only if S ~ T'(R(X)). The relations S are called homogeneous, if and only if S is the disjoint union of homogeneous relations of pairwise distinct degrees. In this case the factorization is compatible with the natural N ograding of the tensor algebra F, which was used in order to construct the topology of F. Therefore relations S are assumed to be homogeneous, if one aimes at the factorization with respect to their closed ideal. Otherwise the resulting factor algebra over R might collapse. (4.3.4) Let the relations S be homogeneous, and assume that they allow for an appropriate rearrangement of monomials, such that FjJfin = R ({x~t .. ·x;P

+ Jfin;nl, ... ,np E No}),

Vf.k:

x2 := e.

Then the R-linear bijection:

L

R[[X]] 3

rnt ... npx~l ... x;p

nt .....npENo +-----+

L

rnl ... npx~l .. ·x;P +] E

R«X,S))

nt .....npENo

is some homeomorphism. In the special case of commuting indeterminates, Le., S := {XkX/ - X/Xk; k, lEI}, this homeomorphism is an isomorphism of unital associative algebras over R. (4.3.5) Let an associative algebra A over R with the unit eA be equipped with a Hausdorff topology, which is compatible with the addition, and let A be complete with respect to the addition. Moreover assume existence of a countable basis {Im; mEN} of zero-neighbourhoods in A, all of which are ideals of A; then A is some topological ring.

128

Formal Power Series with Homogeneous Relations

(4.3.5.1) Consider a continuous homomorphism a: R«X)) --A of unital associative R-algebras. Then

vm EN: 3n EN: "In 2: Ti, Vk 1, ... , k n E I: a(xk

J

) •••

a(xkJ E 1m .

(4.3.5.2) Let {at, ... , a p} be a family of elements of A, such that

vm EN: 3n EN: "In 2: Ti, Vkl, ... ,kn E I

: ak J

•••

akn E 1m .

Then 3 unique continuous homomorphism a : R( (X)) - - A of unital associative R-algebras, such that Vk E I : a(xk) = ak; moreover a is uniformly continuous. One then conveniently denotes V¢ E R«X)) : a(¢) =: ¢(a1,' .. ,ap ).

(4.3.5.3) For a family {1Pk + J; k = 1, ... ,p} of formal power series tPk E V 1(0) c R«Y)), Y := {Yt,oo',Yq}, 3 unique continuous homomorphism a : R( (X)) - - R( (Y, T)) of unital associative R-algebras, such that Vk E I : a(xk) = tPk + J; a is uniformly continuous. Here J denotes the closure in R( (Y)) of the ideal J of a homogeneous subset T of R(Y). (4.3.6) A formal power series which homogeneous relations ¢ E R«X, S)) is invertible, if and only if 3 ¢1 E V 1(0), and TO, So E R, such that TOSO = eR and

a(x) = (/J :=

J

of homogeneous relations S in R«X)).

e«(/J) := a(e(x)), exp(¢):= a(exp(x)) = 1 + e(¢), l(¢):= a(l(x». V¢ E 11 : eo l«(/J) = l 0 e(¢) = ¢. V(/J, t[J El l , "1m E No : (/J - t[J E {X + J; X E V m(O)} =: 1m = } e(¢) - e(t[J) , l(¢ -l(t[J) El m ' Hence one establishes the homeomorphism exp : 11 4----+ L 1 := {I +¢; ¢ and defines the logarithm as inverse of the exponential function, In := exp-1. V¢ El l

:

In(1 + (/J) = l(4J).

E

Id,

Tensor Product of Formal Power Series with Homogeneous Relations

129

(4.3.8.1) Especially let R be the field K := R or C. Then the homeomorphism exp = In- l is naturally extended to some homeomorphism: K((X,S)) ~ K((X,S))\Il, which is again denoted by exp = In-I.

+ ¢l E K( (X)) : exp(¢ + ]) := e CO exp(¢l + ]), In(do + ¢l) = Indo + In(1 + dOI¢t}.

Veo, do E K, do ::/= 0, V¢l E VI (0), ¢ := Co

Here one assumes] ::/= K((X)), which equivalently means eo E C is uniquely determined by ¢ + J.

e ¢], such'that

4.4 Tensor Product of Formal Power Series with Homogeneous Relations (4.4.1) Let J i be an ideal of an algebra Ai over the commutative ring R, for i = 1, 2. Then one finds an isomorphism of R-algebras: Al A2 Al ~A2 J ~ J 3 [ad ~ [a2] ~ [al ~ a2] E (AI ~ J ) + (JI ~ A )' I 2 2 2

(4.4.2) Let Fi be the free R-algebra over a set Xi, and J i the ideal in Fi of relations Si ~ F i , for i = 1,2. Denote by F the free R-algebra over Xl U X 2, with the sets Xl and X 2 being disjoint, and by J the ideal in F of the set of relations Sl U S2 U {XIX2 - X2XI; Xi E Xi, for i = 1, 2}

~

F,

suppressing the natural injection of Fi into F. Then one finds an isomorphism of unital associative R-algebras: FI F2 J ~ J 3 I

2

[/d ~ [12] ~ lith]

F E J'

(4.4.3) Consider topological rings of Hausdorff type Fd J i =: R( (Xi, Si,))' i = 1,2, Xl := {Xl, ... ,Xp}, X 2 := {Yl, ... ,Yq}, Xl nX2 = 0, with the closure J i of the ideal J i in F i of homogeneous relations Si ~ F i . The tensor product FI

F2

I

2

J ~ J ~

F I ~ F2 J ' J:= (FI ~ J 2 )

--

+ (JI ~ F2 ),

of unital associative R-algebras is equipped with the Hausdorff topology, which is due to the countable basis of zero-neighbourhoods W~(O) := sum( {¢l ~ ¢2

+ J; ¢i E V

mi

(0), for i = 1,2;

ml,m2 E NO;ml +m2 = m E No}) 2 W~+l(O),

130

Formal Power Series with Homogeneous Relations

all of which are ideals of (F1 Q9 F2 )/ J = wtf (0). In case of no relations, such that J i = {O}, i = 1,2, these zero-neighbourhoods are denoted by Wm(O), and then nmENo Wm(O) = {O}.

(4.4.3.1) The unique completion of this topological ring FdJl again some topological space of Hausdorff type.

Q9

F2 1J2 is

(4.4.3.2) Denote by D the ideal of the homogeneous relations 8 := 8 1 U 8 2 U {XkYl - Ylxk; k = 1, ... ,Pi l = 1, ... , q} in R( (X 1ux2) ). Then one obtains some homeomorphism of the above defined completion onto the formal power series in the indeterminates Xl, ... , Yq, with relations 8:

+ J1) Q9 (2 + J2) +-----+ 12 + DE R«X1 UX2,8».

R«XlJ 81» Q9 R«X2' 82» 3(1

This homeomorphism is then used to establish this completion as some unital associative R-algebra, which is some topological ring of Hausdorff type.

(4.4.3.3) Since the polynomials in Xl, ... ,Yq with homogeneous relations 8 lie dense in R«X 1 U X 2, 8», the image of { + J; E F1 Q9 F2} lies dense in the completion Fd J 1 Q9 F2 / J 2 . (4.4.4) Let aI, a2, a be the unique continuous homomorphisms: --+ R( (Xl UX2, 8» of unital associative R-algebras, with the notations from above, such that

R[[t]]

a1(t) = 1

+ D,

a2(t) = 2

+ D,

a(t) = 1

+ 2 + D,

considering 1 E R«X1, 8 1», 2 E R«X2, 8 2» as elements of -8

R«X1 UX2,8», and assume both 1 and 2 E VI (0). exp(t) ~ exp(l) de!

+D

+-----+

(exp(d

exp(t) ~exp(2) + D

+-----+ eR Q9

exp(t) ~exp(l + 2)

+D =

de!

de!

exp(d Hence one finds, that +-----+

Q9

exp(2)

+ Jd Q9 eR,

(exp(2)

+ J 2 ),

exp(dexp(2)

+ J.

+D

Tensor Product of Formal Power Series with Homogeneous Relations

131

(4.4.5) Consider the free Lie algebra L(X) over a set X with the coefficients R == K, over a field K of char K = O.

E

L(X)

K -lin span(X U {multo commutatars of elements EX}).

:=

Then F := T(K(X» is the universal enveloping algebra of L(X). Denote by .:1 : F - F ® F the unique homomorphism of unital associative algebras, such that

suppressing the embedding of eK into F.

(4.4.5.1) The composite mapping: F - F®F-F®F, inserting.:1, is uniformly continuous. Therefore :1 unique continuous extension .:1: F-F®F. This extension .:1 of.:1 is uniformly continuous, and .:1 is also some homomorphism of unital associative K-algebras. Moreover the closure L(X) of L(X) in F is some subalgebra of the commutator algebra FL, and

L(X)

= {4> E K((X»;~(4» = eK ®4>+4>®eK}.

(4.4.5.2) V4>,1/J E L(X): exp(4))exp(1/J)

~exp(4))exp(1/J)

®exp(4))exp(1/J),

.1

and therefore In(exp(4»exp(1/J» E L(X).

(4.4.5.3) V4>,1/J E L(X) exp(4)) exp(1/J)

c

V 1 (0):

= exp(4) + 1/J + ~[4>, 1/J] + /2([[4>, 1/J], 1/J] + [4>, [4>, 1/J]]) + X),

with an element X E L(X) n V 4(0).

(4.4.5.4) The Jacobi identity shows, that

L(X) =

EB U(X), rEN

Lr(X)

:=

K -lin span({[... [[Xk 1 ' Xk 2 ], xk31···, xkrl; k t , ... , k r = 1, ... ,p}) C ?(K(X».

Consider the K-linear map

T(K(X» K,(eK)

3

Xk 1

•••

:= 0, "Irk:

K,:

Xk r dJ[' .. [[Xkl' Xk 2 ], xk 31· .. ,Xkr] E L(X),

K,(Xk)

:=

Xk.

132

Formal Power Series with Homogeneous Relations

One shows by an induction with respect to r EN, that Vr EN: r(X) = {4> E ?(K(X)); ~(4)) = r4>}. Here one uses, that ~IL(X) is some derivation of L(X). (4.4.5.5) Consider formal power series E K(({s,t})). Since ( l)m+1

In(exp(s) exp(t» = ~ ~ sP1tql LJ LJ mpl!ql!'" P !q ! mEN E(m) m m

sPmt qm

'"

E L(X),

this series can be rearranged by means of the unique continuous extension ---+ L(X) of ~. An insertion of formal power series then yields the Baker-Campbell-Hausdorff formula. "14>, t/J E VI (0):

It : K ((X))

(l)m+l ( m L , -,... I , L(Pk mEN E(m) mpl·ql· Pm·qm· k=l

In(exp(4» exp(t/J» = L

+ qk)

)-1

[... [4>,4>] ... ,4>],t/J] ... ,t/J] ... ,4>] ... ,4>],t/J] ... ,t/J],

"-"-"' PI

E(m) (4.4.5.6) V4>,t/J

E L(X):

--......--...

:= {PI, ... , qm E

exp(4))t/Jexp(-4>)

~--......--...

ql

E L(X).

_

"14>, t/J E V dO) : exp(4»t/J exp(-4» = t/J + [4>, t/Jl

qm

Pm

No; \lfk : Pk

+ qk > O}.

Hence one finds, that 00

+L m=2

1

-, [4>, ... [4>, t/J] ...]. m. ---......-.m

(4.4.5.1) These series of multiple commutators can be factorized with respect to the closure"] in F of an ideal J of homogeneous relations S c F.

5. Z2-Graded Lie-Cartan Pairs

Lie-Cartan pairs serve for a suitably unified algebraic description of the covariant exterior derivation of differential forms with respect to a connection on a finite-dimensional real differentiable manifold, of the covariant derivation on real or complex vector bundles, and of cohomologies of Lie algebras. For instance as a prototype, consider the pair {L := T(M), A := Coo (M)} of smooth vector fields and scalar fields on an rn-dimensional real differentiable manifold M, and the R-bilinear mappings:

Lx A

3

{X,f}-+Lxf de!

E

A, A x L

3

{J,X}-+fX de!

E

L,

inserting the Lie derivation Lx of scalar fields with respect to a vector field X, and the module-multiplication of the A-left module Lover R. On the other hand, the E-valued Chevalley cohomologies of a Lie algebra L over a field K provide an easy application of the Lie-Cartan pair {L, K} with the K-bilinear mapping: Lx K - + {O}, and K x L - + L defined as the module-multiplication of L; here E denotes a vector space over K. In case of an atlas consisting of two or more charts, the Coo(M)-left module T(M) of smooth vector fields on an rn-dimensional real differentiable manifold M is projective, but not free over the commutative unital associative algebra of scalar fields Coo(M). Correspondingly one constructs projectivefinite right modules over an associative unital algebra, thereby generalizing the concept of real or complex vector bundles. An appropriate generalization to Lie superalgebras and graded-commutative unital associative superalgebras leads to the notion of Z2-graded LieCartan pairs. These provide an algebraic scheme for the calculus of real Z2graded differential forms, as one step towards a theory of supermanifolds. On the other hand one obtains an elegant approach to the theory of cohomologies of Lie superalgebras. The concept of Lie-Cartan pairs was developed by D. Kastler and R. Stora (1985), their Z2-graded generalization is due to A. Jadczyk and D. Kastler (1987). An according differential geometry on Grassmann algebras, including the so-called Berezin integral, is presented in a review by R. Coquereaux, A. Jadczyk, and D. Kastler (1991). The matrix differential geometry, which was constructed by M. DuboisViolette, R. Kerner, and J. Madore (1990), can be obtained from an ap-

134

5. Z2-Graded Lie-Cartan Pairs

propriate Matn(C)-connection ofthe Lie-Cartan pair {Derc(Matn(C», C}. Generalizing this framework to Matn(C)-valued smooth functions on an mdimensional real connected differentiable manifold, the above authors developed models of gauge theory. An algebraic approach to the notions of covariant derivation, its curvature, and connection was performed by R. Matthes (1991), starting from an associative algebra over a field and the Lie algebra of its derivations. The differential geometry on real or complex fibre bundles is then reformulated within this algebraic framework. For the classical theory of differentiable manifolds the reader is referred to the corresponding volume by J. Dieudonne (1972), and to the monograph by S. Lang (1972). Cohomologies of Lie algebras over a field are treated on the abstract level of homological algebra, which is due to H. Caftan and S. Eilenberg (1956), by N. Jacobson (1962). Cohomologies of Lie superalgebras were studied by D. A. Leites (1976).

5.1 Tensor Products over Graded-Commutative Algebras

eo

(5.1.1) Let E = EB E 1 be an A-left module over R, with a unital associative superalgebra A over a commutative ring R. Assume A to be gradedcommutative, and moreover, that Vz1 , Z2 E Z2, Va E

Ail, X

E EZ~ : ax E EZdz~.

Then E is called graded A-left module over R. With the R-bilinear map:

E xA

3

{x,a}

dJ L

(_l)ZlZ~aZlxZ~ =: xa E E,

zl,z~EZ~

E becomes also an A-right module over R. Then E is called graded Abimodule over R. One immediately finds, that Va, bE A, "Ix E E: a(xb) = (ax)b.

(5.1.2) Let 8 be a homogenous derivation of A, 8 E Derh(A), Le., Vfi E Z2,a E AiI,b E A: 8(ab)

= 8(a)b+ (-1)lI a8(b), 1m 81AV Z

~

AlI+ Z ,

with the degree z of 8 being fixed. An endomorphism d of E is then called 8-derivation of E, if and only if d E Endh(E) fulfills the property that Vfi E Z2,a E Ail, x E E: d(ax)

= 8(a)x + (-l)lI ad(y). Z

Tensor Products over Graded-Commutative Algebras

135

(5.1.3) One then immediately finds the following lemma. Let d be a bderivation, and d' a b'-derivation of E. Then

[d, d'] := dod' - (_l)ZZ' d' 0 d E End~+Z' (E) is some [b, b'l-derivation of E, with the super-commutator

[b, b'] := bob' - (_l)ZZ' b' 0 b E Der~+z' (A), and with the degrees z of band d, z' of b' and d', respectively. (5.1.4.1) Let E, F be graded A-bimodules over R. Their tensor product over A is defined as the following factor module over R. E®A F:= (E®F)

N1

/N1,

:= sum({(xa) ® y - x ® (ay);x E E,y E F,a E A}).

(5.1.4.2) The diagram below, inserting a E A, allows for the definition of an A-left module E ® F over R, such that "Ix E E,y E F,a E A: a(x®y)

=

(ax) ®y.

ExF3{X,y} - - - - - - - - _ " x®y

E

E ®F

R-Iinear R-bilinear

def

I

(ax)®y EE®F

(5.1.4.3) Obviously this A-left module E ® F over R is graded with respect to the natural Z2-grading of E ® F. Hence E ® F is some graded A-bimodule over R. (5.1.4.4) Since the above factorization is compatible with this A-left module structure of E ® F, there is an A-left module E ®A F over R such that, with respect to this factor module, Va E A, x E E, y E F: a(x ® y

+ N1) = (ax) ® y + N1.

(5.1.4.5) Since the above R-submodule is Z2-graded,

N1

EB (N1 n (E ® F)Z),

=

ZE Z 2

one can define, that Vz E Z2, V t

t + N1

E

(E ®A F)Z <===> 3t' de!

+ N1

E

E E ®A F:

(E ® F)Z : t' - t

E

N1.

136

5. Z2-Graded Lie-Cartan Pairs

Thereby one establishes the graded A-bimodule E 0A F over R. For homogeneous elements one finds, that \tZi', Z2, Y E Z2, VXl E EZl, X2 E F~, a E AY :

(-l)ZlYa(Xl 0 X2

+ N~)

+ N~ (ax2) + N~

= (xla) 0 X2

= Xl 0

=

(_1)Y Z 2 (Xl 0 X2

+ N~)a.

(5.1.5) More generally, let Ell' .. ,Ep be graded A-bimodules over R. One then factorizes with respect to the R-submodule N~ := sum( {Xl 0 ... 0 (axi) 0 ... 0 x p

- (-l)y(z,+"+Zj-tl xl 0··' 0 (axj) 0··· 0 xp; Vfk:XkEE~\ZkEZ2;

l~i<j~p; aEAY,yEZ2}),

and obtains the graded A-bimodule over R

(5.1.6) Let E, F be graded A-bimodules over R. With the natural Z2-grading of the R-bimodule HomR(E,F), one defines Vz E Z2: Hom~(E,F):=

{¢ E Homh(E, F); Vy E Z2,a E AY,x E E: ¢(ax) = (-lYYa¢(x)} ,

as an R-submodule of HomR(E, F). Then

HomA(E,F):=

E9 Hom~(E,F). ZE Z 2

With the definition, that Va E A,x E E,¢ E HomA(E,F): (a¢)(x):= a¢(x),

one then establishes the graded A-bimodule HomA(E,F) over R. (5.1. 7) Let E l , ... , E p , F be graded A-bimodules over R. Consider the graded A-bimodules E 1 0A'" 0A Ep and HomA(E 1 0A'" 0A Ep,F) over R, in order to establish the universal property of the tensor product over A, according to the subsequent diagram. p

-------..>@Ek k=1

"p ,,' I '------=-----;:--;';-;----~. F~ R-multilinear

Tensor Products over Graded-Commutative Algebras

LA(E l , ... ,Ep;F)

:=

{A

IT Ek

p :

---+

137

Fj

k=l

'v'Z,Zl,""Zp E Z2,'v'Xl E Ef1, ... ,Xp E E;p,'v'a E AZ,'v"{j: \ ( Xl,'" ,ax . , •.. ,Xp) -_ Ap J

(_l)z(zJ+oo+zp) Ap \ ( Xl,'"

,Xp) a } .

Here obviously the natural Z2-grading of Ap corresponds to that of the uniquely induced map A; E HomA(E l ®A'" ®A Ep,F). (5.1.8) Let expecially E l = ... = E p =: E, and consider the following graded-alternating forms.

A~(E,F):=

{A

p

L~(E,F):= LA(~;F);

E

p time3

'v'Zl, ... , zp

E Z2, 'v'Xl E EZl, ... , x p E EZp, 'v'P:= [ .1. ..

\ p (x.J 1 , ••• , x·Jp )

A

~

Jl ... Jp

= \ p (Xl , .•. , Xp ) (_l)T + L:{ l~kil} p

A

] E Pp

:

Zik Zil } .

With an appropriate R-submodule, namely

N~ := sum( {Xl ® ... ® Xp . . z· z· _ XJ1 ® ... ® XJp (-1) T p +~ L...{ l~kJd Jk Jj. , 'v'fk: Xk E EZk,Zk E Z2; P:=

[.l ~ ] Jp

Jl

E p p }),

one establishes the double-factorization with respect to N~ and N~, Le., the R-linear bijections:

((

~E) /N~) /N~A

+---+

(0 E) /(N~ + N~)

+---+ ( (

0 E) /N~) /N~A'

such that p

'v't E

0

E: (t + NAP) + N~A

with the R-submodules

+---+

t + N~ + N~

+---+

(t + N~) + N~A'

138

5. Z2-Graded Lie-Cartan Pairs N~A := sum( {t

+ N~; t E N~}),

N~A := sum( {t

+ N~; t E N~}).

The graded tensor product over A is then defined as A

P (P) 0 E /(N~ +N~).

0AE:=

The implicit definition, such that Va E A, VXI,"" x p E E : a(xi ® ... ® x p

+ N~ + N~)

= (axI) ® X2 ® ... ® x p

+ N~ + N~,

yields the graded A-bimodules ®~E, and accordingly, HomA (®~E,F).

(5.1.8.1) One then establishes the universal property of the graded tensor product over A, according to the next diagram, with Ap E A~ (E, F), and an according unique A-linear graded-alternating form ).p E H omA ( ®~ E, F). p

XE --~> ®E

p

p

p _ _~>

A

® E _ _~), ® E A

A

(5.1.9) Let E be a graded A-bimodule over R. The tensor algebra of E over A is constructed. For p = 2,3, ... , p

T~(E)

:=

0 AE , Tl(E):= E, ~(E):= A, TA(E):= E9 T~(E). pENo

(5.1.9.1) For convenience, denote Vp VXI,""

xp

E

E : Xl ® ... ® x p

~

+ N~

2, =: Xl ®A ... ®A X p E T~(E).

(5.1.9.2) The tensor product over A is associative, modulo an A-linear bijection, as is shown in the following diagrams.

Tensor Products over Graded-Commutative Algebras

139

P..

A-linear One then similarly contructs a. : E 0A (E 0A E) ~ (E 0A E) 0A E, and finds that a. 0 p. = id, p. 0 a. = id. (5.1.9.3) The product of TA(E) is defined, omitting the canonical embeddings, such that "Ip, q EN, "IXl, ... , xp, Yl, ... , Yq E E, Va, bE A:

(Xl 0A 0A Xp)(Yl 0A ... 0A Yq) = Xl 0A 0A Xp 0A Yl 0A ... 0A Yq, ab is defined from A, aYl and xlb are defined from E, a(Yl 0A (Xl

0A

0A Yq) = (ayd 0A Y2 0A ... 0A Yq, 0A Xp)b = Xl 0A ... 0A Xp-l 0A (xpb).

(5.1.9.4) Hence TA(E) is an associative algebra over R with the unit eA. Moreover TA(E) is some graded A-bimodule over R, with the usual Z2grading of T(E) used for an appropriate grading of TA(E). Especially TA(E) is some graded A-algebra over R, because one finds, that

(5.1.9.5) The graded A-bimodule B over R is called graded A-algebra over R, if and only if B is a unital associative algebra over R, and moreover

140

5. Z2-Graded Lie-Cartan Pairs

(5.1.9.5.1~ Let B, B' be graded A-algebras over R. An A-linear mapping = 0 + E HomA(B, B') is called graded algebra-homomorphism over A, if and only if 0 (eB) = eB', and

Vi, fi E Z2, 'v'b l E B Z , b2 E B : ii(b 1b2 ) = (_1)Y ZIii(bI)ii(b2 ). In this case, if especially is even, then it is a homomorphism in the sense of unital associative algebras over R. (5.1.9.6) The mappings k E HomA(Ek, Fk), k = 1, ... ,p, induce an Alinear map

due to the universal property of tensor products, such that 'v'XI E E 1 , ... , x p E E p:

T~(ll"" L =

p)

(0

P

Xk)

A k=l

(_1)2:19t(X~k). A k=l

Z1,oo.,zp,iiEZ2

(5.1.9.7) Accordingly an A-linear mapping E HomA(E, F) induces TA( E Hom~ (E, F):

TA()(XI ®A ... ®A Xp)TA()(YI ®A .,. ®A Yq)

= (-1)qY(ZI+oo,+zp)TA()(XI ®A .. , ®A x p ®A YI ®A ... ®A Yq)· With the definition, that 'v'a E ~(E) : TA()(a) := a E ~(F), one establishes the covariant functor T A , according to the diagram below. E

____ e_m_b_e_d_d_in-=g

• T (E) A

covariant

functor

F _ ____e_m_b_e_dd_in-"g'-----

--. T IF) •

A I'

Tensor Products over Graded-Commutative Algebras

141

(5.1.9.8) The universal property of the tensor algebra of E over A is shown in the subsequent diagram. ---------~>

E

7A(E)

B Here E and B are graded A-bimodules over R, and moreover B is some graded A-algebra over R. Then ¢ E HomA(E,B) is extended to a unique graded algebra-homomorphism ¢. over A, which is explicitly contructed by means of the universal property of tensor products. (5.1.9.8.1) For p = 2,3,4, ... , :J unique ¢~ E HomA(T~(E),B), such that

~

(0

P

Xk) =

L (-1)2::::: zdp-k)Y¢Y(XI)'"

¢ii(x p).

iiEZ~

A k=l

(5.1.9.8.2) The direct sum of ¢~, Zl, ... , zp E Z2, P E No, ¢l := ¢, Va E A : ¢~(a) := aeB, yields the desired homomorphism ¢ •. (5.1.9.9) The alternating algebra of E over A is constructed, using the graded tensor product over A.

EB A~(E), A~(E) := A, A~(E) := E,

AA(E):=

pENo •

p

Vp;::: 2: A~(E):= 0AE.

Then the product of AA(E) is defined, such that VXI, ... , x p, YI, ... , Yq E E, Va, b E A:

(Xl

@

= Xl

00.

@ .

@Xp + N~ 00

+ N~)(YI @. @ Yq + N1 + N~) @ Xp @ YI @ ... @ Yq + N~+q + N~+q, 00

(Xl @,oo@Xp+N~ +N~)b=XI @oo'@Xp_l @(xpb) +N~ +N~, a(Yl

@ ... @

Yq

+ N1 + N~)

=

(aYd

@

Y2

@ ... @

Yq

+ N1 + N~,

with the elements of A~ (E) being inserted in the natural manner. Thereby AA(E) becomes some graded A-algebra over R, with the following universal property: For any ¢ E HomA(E, B), which is graded-alternating in the sense defined below, there exists a unique graded A-algebra homomorphism ¢ over R, such that the following diagram is commutative.

142

5. Z2-Graded Lie-Cartan Pairs E----~. TA

(E) -----~ AA (E)

B

Here ¢ = ¢o

+ ¢I

is assumed to be graded-alternating in the sense, that

V1Jl,Y2,Zl,Z2 E Z2,'v'Xl E EZl,X2 E EZ2:

(5.1.9.10) Obviously both EBpENo N~ and EBpENo N~ are ideals of the tensor algebra T(E). One trivially finds an R-linear bijection:

with the definitions: N~

+ N1 + N~ + N~

=

{o}.

(5.1.9.11) Obviously the alternating algebra of E over A is spanned by the following monomials.

+ N~ + N~ := TP(E)/(N~ + N~).

Xl t\A ... t\A X p := Xl @ ..• @ X p

E A~(E)

(5.1.9.12) Every even A-linear mapping ¢ E Hom~(E, F) induces AA(¢) E HomA(AA(E), AA(F)), due to the subsequent diagram.

E - - - - _ . ~(E) - - - - _ . AA(E)

l~ F

[J;.(# ~.

~(F)

jAA(# • AA(F)

Here again one uses the Z2-grading of AA (E) by homogeneous representatives, because both N~ and N~ are Z2-graded R-submodules of TP(E), for p ~ 2.

5.2 Projective-Finite Modules

143

5.2 Projective-Finite Modules (5.2.1) Consider an A-right module E over R, with the unital associative algebra A over R. An := EBn A, as an R-bimodule. Furthermore with the definitions, that 'v'{al, . .. ,an } E An,'v'b,c E A: {al, ... ,an}b:={akb;k=I, ... ,n}, c{al, ... ,an }:={ ca k;k=I, ... ,n}, An becomes an A-A-bimodule over R, Le., by definition, (i) An is an A-left module over R, and also an A-right module over R;

(5.2.2) The map 'Y E HomA(An,E)

~

HomR(A n , E)

is called generating n-map for E, if and only if'Y is surjective; in this case,

A E HomA(E,A n ) ~ HomR(E,A n ) is called lift for 'Y, if and only if 'Y bijective and A injective.

°A =

id E. If A lifts 'Y, then 'Y 11m

A

is

(5.2.3) The canonical projections and embeddings of An are denoted, such that \11k:

Obviously,

~

L..J 1rk

° 1r k =

°d An , Uflk l VI'

t

: 1r

k ° 1r1 = { id a A, k...J.kl = l .

k=l

'

T

(5.2.4) Assume, that there is some lift Afor an appropriate generating n-map 'Y for E.

'Y01rk E HomA(A,E) ~ HomR(A,E), k c := 1r ° A E HomA(E,A) ~ HomR(E,A).

\11k:

Ck:=

k

n

L k=l

n

Ck

k

° 1r = 'Y,

L 1rk ° c k=l

{ek, e k ; k =

n

k

= A,

L

ek

° e k = 'YO A = id E.

k=l

Here the family 1, ... ,n} is called dual n-basis, or coordinatization of E. Especially {1rk,1rkjk = 1, ... ,n} is some coordinatization of An. An A-right module E over R is called projective-finite, if and only if there is some dual n-basis of E, for some n E N.

144

5. Z2-Graded Lie-Cartan Pairs

(5.2.5) One conveniently uses the natural R-linear bijection: E :3 x

+----+

x : A :3 a

----+

xa E E, x E HomA(A, E).

Obviously, v¢ E HomA(A, E) : ¢ = (¢(eAW. Especially one finds: A :3 a

+----+

a: A

:3

b ----+ ab E A,

a E EndA(A) ~ EndR(A).

In the latter case, "10: E EndA(A) : 0: = (o:(eA))·' Here A is considered as an A-right module over R. (5.2.6) n

Vf.k,Vx E E: x k := c k ox, X = L€k ox k . k=l n

Vf.k, V'IjJ E HomA(E, A) : 'ljJk := 'IjJ 0 Ck, 'IjJ = L

'ljJk

ck .

0

k=l

(5.2.7) Let E2 be an R 2 -right module, and F an R 1-R2 -bimodule, Le.,

Vy E F, Tl E Rl' T2 E R 2 : (TlY)T2 =

Tl (YT2)'

In this case, defining

one obtains an R1-left module

HomR~(E2, F).

Here R 1 and R 2 are rings.

(5.2.8) Obviously HomA(E, A) is an R-submodule of HomR(E, A). Moreover, defining V¢ E HomA(E,A),a E A: a¢:=

a o¢,

which equivalently means, that "Ix E E : (a¢)(x) becomes an A-left module over R.

=

a¢(x), HomA(E, A)

(5.2.9) V¢ E HomA(E,A),x E E,a E A: (¢ox)(a) = ¢(x)a E A, ¢ox = (¢(x)t E EndA(A).

5.2 Projective-Finite Modules

145

(5.2.10) The A-right module E over R is assumed to be projective-finite. Denoting

one finds the following direct sum decomposition of the A-right module An over R, according to the next diagram.

1m>' = 1m p, An = 1m p EB 1m(id An - p).

1 E

~.,--->

1m p

>

An

] E

1_ _]

(5.2.10.1) A right (left) module over a ring is called projective, if and only if it is some direct summand of a free right (left) module over this ring. Obviously the A-right module E is projective, because it is isomorphic with the direct summand 1m p of An. The projective-finite A-right module E over R is also called a finitely generated projective right module over A. (5.2.11.1) Consider the ring M atn(R) of matrices [rik; i, k = 1, ... , n] with the components rik E R. With module-multiplications, such that V[rik; i, k = 1, ... , n] E Matn(R), r E R:

r[rik; i, k = 1, ... ,n] := [rrik; i, k = 1, ,n] = [rikr;i,k = 1, ... ,n] =: [rik;i,k = 1, ,n]r, Matn(R) is an R-bimodule, hence an associative algebra over R, with the unit

In

:=

[e R o

0 ] . Then

Matn(R) ®A, as the tensor product of algebras

eR

over R, is an associative algebra over R, with the unit In ® eA. (5.2.11.2) On the other hand, consider the ring Matn(A) of matrices [aik : i, k = 1, ... , n] with the components aik E A, and define modulemultiplications, such that V[aik; i, k = 1, ,n] E M at n (A), r E R:

r[aik;i,k = 1, ... ,n]:= [raik;i,k = 1, ,n] = [aikr; i, k = 1, ... , n] =: [aik; i, k = 1, , n]r,

146

5. Z2-Graded Lie-Cartan Pairs

in order to obtain an associative algebra Matn(A) over R, with the unit

eA

l

o

0 ]

. Then the universal property of the tensor product allows for

eA

an establishment of an isomorphism in the sense of unital associative algebras over R:

Matn(R) ® A

:3 [rikj

i, k = 1, ... , n] ® a

+-----+

[rika;i,k = l, ... ,n] E Matn(A).

(5.2.11.3) Let E EndA(An) module An over R, and define

'vii, k : EndA(A)

:3 ~ :=

~

i

1r 0



EndR(An), with respect to the A-right

0

1rk : A

:3

a ~ ~(eA)a E A.

Here A is considered as an A-right module over R. Then, in this sense,

EndA(A):=

{ E

EndR(A)jV'a,b E A: (ab) = (a)b},

with the unit id A, is an associative algebra over R. Hence one establishes an isomorphism in the sense of unital associative algebras over R:

which can be extended to endomorphisms over R:

(5.2.11.4) An A-endomorphism of the projective-finite A-right module E over R can be written as matrix, the components of which are Aendomorphisms of A, according to the diagram below, the bijections of which are isomorphisms in the sense of unital associative algebras over R.

I

-

To every E EndA(E) belongs an R-linear map: n

HomA(A,E} :3x = LCk ox k k=l n

~ ox =: iJ =

LCI oyl E HomA(A, E), 1=1

1

5.3 Lie-Cartan Pairs

147

with components such that n

vrl: yl

=

Lei

k O¢>Oek ox .

k=l

5.3 Lie-Cartan Pairs (5.3.1) Let A be an Abelian unital associative algebra, and L a Lie algebra, both over the commutative ring R. The Lie-Cartan pair {L, A} is established by the R-bilinear maps: L x A 3 {k,a}

----t

ka E A, A x L 3 {a,k}

----t

ak E L,

which obey the following conditions. Va, bE A, Vk, l E L: [k, l]a = k(la) -l(ka), k(ab) = (ka)b + a(kb), a(kb) = (ak)b, (ab)k = a(bk), eAk = k, [k, all = ark, l] which then imply, that

+ (ka)l,

OLa = kO A = OA, (-k)a = k( -a) = -(ka), OAk = aOL = OL, (-a)k = a( -k) = -(ak), keA = OA, [ak, l] = ark, l] - (la)k. These conditions expecially imply, that the following two left modules over R are constructed. (5.3.1.1) An A-left module Lover R is established, Le., L is an A-left module, with the elements of the ring A used as coefficients, and Va E A, k E L, r E R : r(ak)

= (ra)k = a(rk).

(5.3.1.2) An L-Ieft module A over R is established. (5.3.1.3) Consider the derivation algebra of A, Le., the subalgebra DerR(A) of the commutator algebra (EndR(A))L. Define an R-linear map >. : L ----t (EndR(A))L' such that

Vk

E

L: A

3

a ----t ka EA. A(k)

Then 1m >. S;;; DerR(A), and>. is some homomorphism in the sense of Lie algebras over R, Le., Va, bE A, Vk, l E L : >'(k)(ab) = (>'(k)(a))b + a('x(k)(b)), >'([k, l]) = ['x(k), 'x(l)] = 'x(k) 0 'x(l) - 'x(l)'x(k). Here the commutator algebra (EndR(A))L is defined, such that

V¢>, 'l/J E EndR(A) : [¢>, 'l/J] := ¢> 0 'l/J - 'l/J 0 ¢>.

5. Z2-Graded Lie-Cartan Pairs

148

(5.3.2) Let {L, A} be a Lie-Cartan pair over R, and consider an A-left module E over R, which explicitly means that (i) E is an R-bimodule;

(ii) E is an A-left module, which may be considered as an A-bimodule, defining \:Ix E E, a E A : xa := ax; \:Ix E E, a E A, r E R: r(ax)

(iii)

An R-linear mapping p: L only if

---+

= (ra)x = a(rx).

(EndR(E»L is called an E-connection, if and

\:Ik E L, a E A, x E E : p(k)(ax) = a(p(k)(x»)

+ (ka)x.

An E-connection p is called local, if and only if \:Ik E L,a E A,x E E: p(ak)(x) = a(p(k)(x». (5.3.2.1) The curvature of an E-connection p is defined as an R-bilinear mapping '" : L x L

3

{k, l}

---+

de!

p(k)

0

p(l) - p(l)

0

p(k) - p([k, l])

E

(EndR(E»L.

An E-connection p is called flat, if and only if '" = 0, which equivalently means that p is some representation of Lon E, Le., a homomorphism of Lie algebras over R. (5.3.2.2) Usually an A-left module E over R is induced by some representation p. of A on E, Le., a homomorphism of unital associative algebras p. : A

---+

EndR(E), \:la, bE A : p.(ab)

= p.(a) 0

p.(b).

One then defines the bilinear mapping: AxE

3

{a,x}

---+

ax:= p.(a)(x)

E

E.

(5.3.2.3) Let Pl, P2 be E-connections, and rl, r2 E R. Then rl

+ r2 = eR

~

rlPl

+ r2P2

is an E - connection.

(5.3.3) The Lie-Cartan pair {L, A} is called (i) degenerate, if and only if \:Ik E L, a E A : ka = 0; it is called

(ii) injective, if and only if the implication r\:la E A : ka = 0 ~ k = 01 holds, which equivalently means, that the homomorphism of Lie algebras .A is injective.

5.3 Lie-Cartan Pairs

149

(5.3.3.1) Let L',A' be subalgebras of L,A. The family {L',A'} is called subpair of {L, A}, if and only if Vk E L', a E A' : ka E A', ak E L'; obviously in this case {L', A'} is some Lie-Cartan pair. Especially the subpair {L, ReA} is called the depletion of {L, A}. (5.3.3.2) Consider the Lie-Cartan pair {Der(A), A}, thereby short-writing Der(A) == DerR(A), with two R-bilinear maps defined such that

Va,b

E

A, E Der(A): (a)(b)

:=

a¢(b), ¢a:= ¢(a).

Obviously this Lie-Cartan pair is injective. Moreover every subpair {L, A} of it is injective. Here the subpair {L,A} of {Der(A), A} is established by the assumption, that Va E A, ¢ E L : a¢ E L; L is an appropriate subalgebra of

Der(A). (5.3.3.2.1) Conversely let the Lie-Cartan pair {L,A} be injective, Le., A injective, as indicated in the following diagram. L

• • 1m A -. Der(A) -- (EndR(A)h Lie algebraisomorphism

The Lie-Cartan pair {1m A, A}, which is established by the products, such that V¢ = A(k) E 1m A, a E A : ¢a := ka, a¢:= A(ak),

is some subpair of {Der(A),A}.

(5.3.3.3) Let the Lie-Cartan pair {L,A} be degenerate. Then its defining properties establish an A-left module L, with the commutative ring A providing the coefficients. Moreover an E-connection p yields, for k E L, some homomorphism p(k) in the sense of A-left modules. (5.3.4) Important special cases of E-connections, with respect to the LieCartan pair {L,A}, are established by the choices E:= A, E:= L, with the corresponding left module-multiplications provided by the product of A, and the Lie-Cartan bilinear map: A x L --+ L, respectively.

150

5. Z2-Graded Lie-Cartan Pairs

(5.3.5) In order to generalize the calculus of differential forms, consider an E-connection p, with respect to the Lie-Cartan pair {L, A}. One defines the direct sums of R-bimodules A*(L,E) :=

EB AP(L,E), AO(L,E):= E, A1(L,E):= HomR(L,E), pENo

A:4(L,E):=

EB A~(L,E),

A~(L,E):= E,

pEN o

with the A-linear mappings as elements of A~(L,E) : = {AI E HomR(L,E);Vk E L,a E A: Al(ak)

= aAl(k)}

= HomA(L, E),

and with the alternating R-multilinear, and especially alternating A-multilinear mappings, Vp ~ 2, as elements of AP(L, E) := { Ap :

IT

L

---+

Ej VkI.' .. , kp E L:

f=31 ~ i < j ~ p: ki = kj 1===} Ap(k1, ... , kp) =

o},

A~(L,E):= {A p E AP(L,E);Vk 1, ... ,kp E L,V1 ~ j ~p,Va E A:

Ap(k1, ... ,akj, ... ,kp) = aAp(k1, ... ,kp)}.

Obviously these alternating mappings are skew-symmetric, Le., Vp

~ 2,VA p E AP(L,E),Vk1, ... ,kp E L,V Ap(k 1, ... ,kp)

with the sign

Tp

[.l... ~ ]

=:

)1 .. ')p

P

E Pp

:

= (-1)"'P Ap(k jll ... kip),

of the permutation P of p elements.

(5.3.5.1) The mappings i(k), p(k), Oo(k), Op(k), Do, p, Dp, which shall be defined now, are R-endomorphisms of A*(L,E), Vk E L, as one proves easily. Vp E N,A p E AP(L,E),x E E,Vk,ko,k 1, ... ,kp E L: (i(k )(Ap+l) )(k 1 , •.. ,kp) := Ap+l (k, k 1 , ... , kp), i(k)(x) := 0, i(k)(Al) := Al(k), (p(k)(Ap))(kI."" kp) := p(k)(Ap(k 1, ... , kp)),

as an extension of the E-connection p,

5.3 Lie-Cartan Pairs

151

P

(Bo(k)(>'p))(k 1 , ... , kp) := -

L >'p(k

1 , ... ,

kj - 1 , [k, kjl, kj +1,'''' kp),

j=1

Bo(k)(x) := 0, (Bo(k)(>'I))(k 1 ) = ->'I([k,k 1 ]), Bp(k) := Bo(k) + p(k), Bp(k)(x) = p(k)(x), (bO(>'p)) (k o, k 1 ,···, kp) :=

L

(_l)i+j >'p([ki , kjl, ko,···, ~i"'" ~j"'" kp),

O:S;i<j:S;p

(bO(>'I))(k o, k1 ) = ->'I([ko, kd), bo(X) := 0, p

(jJ(>'p) )(ko, k1 , ••• ,kp) :=

L (-l)j p(kj )(>'p(ko, ... , ~j' ... ,kp)),

j=o (jJ(>'I))(ko, k1 ) = p(kO)(>'I(k 1 )) - P(kI)(>'I(ko)), jJ(x)(k) := p(k)(x), bp := bo + jJ, bp(X) = jJ(x). Here for instance the notation of ~j means, that this argument is omitted. (5.3.5.2) Suppressing the canonical embeddings one finds, that 'tip E N,k E L: I1P(L, E)

--+ i(k)

I1p-l(L, E), I1p-l(L, E)

--+ p(k),8 o (k),8 p (k)

I1p-l(L, E),

(5.3.5.3) With the R-bilinear mapping: A x 11*(L, E) --+ 11*(L, E), such that'tla E A,p EN, >'p E I1P(L, E), x E E, 'tikI,"" kp E L:

(a>'p)(k 1 , ... , kp) := a>'p(k 1 , ... , kp}, ax being defined in the sense of the A-left module E over R, 11*(L, E) becomes an A-left module over R. (5.3.5.4) An endomorphism ..1(a) of 11*(L, E), in the sense of A-left modules, is defined. 'tip E N,>.p E I1P(L,E),x E E,a E A,'tIk,ko,k 1 , ... ,kp E L: p

(..1(a)(>'p))(ko, ... , kp) :=

L( -l)j(kja)>'p(ko, ... , ~j"'"

kp),

j=O

(..1(a)(>'I))(ko,kI)

= (k oa)>'I(kI) -

'tIpEN,'tIaEA:I1P(L,E)

(k 1a)>'I(ko), (..1(a)(x))(k):= (ka)x.

~ I1p+l(L,E).

A-hnear

152

5. Z2-Graded Lie-Cartan Pairs

(5.3.5.5) Obviously Oo(k), Do, and L1(a) are A-endomorphisms of A·(L, E). "Ia,b E A,>' E A·(L,E),k E L:

L1(a)(b>') = b(L1(a)(>.)), Oo(k)(a>.) = a(Oo(k)(>')), Do(a>.) = aDo(>'), p(k)(a>.) = a(p(k)(>')) + (ka)>', Op(k)(a>.) = a(Op(k)(>')) + (ka)>., p(a>.) = ap(>.) + L1(a)(>.), Dp(a>.) = aDp(>') + L1(a)(>.). (5.3.5.6) Furthermore R-endomorphisms K,(k, l), K(k), and defined. "Ip E N, >'p E AP(L, E), x E E, "Ik, l, ko, k1 ,. 00' kp+2 E L :

Pi,

of A·(L, E) are

as an extension of the curvature, P

(K(k)(>'p))(ko, ... , kp) :=

L( -1)j(K,(k,kj )(>'p))(ko, 00"

~j"'" kp),

j=O

(K(k)(>'l))(ko, kd = K,(k, kO)(>'l(kd) - K,(k, kd(>'1(ko)), (K(k)(x))(l) := K,(k, l)(x) , (Pi,(>'p))(kl,. 00' kp+2) := =

2~!)

L 1. .. P+2] [ .

.

(-1f'P(K,(kjllkj,)(>'p))(kh,oo.,kjp+2)' =:PEP p +2

)1·· ')p+2 (Pi,(>'1))(kl, k2, ka) = K,(kl, k2)(>'1(ka)) + K,(k2' ka)(>'1(k1)) + K,(ka, k1)(>'1(k2)), Pi,(x)(k, l) := K,(k, l)(x). The definition of Pi, is valid for an arbitrary commuting ring R, with 2(~!} understood to drop out. "Ip EN, k E L:

AP-1 (L E) ,

AP(L, E)

--+ K(k,/)

--+ i«k)

AP-1 (L E) "

AP+1(L E) AP(L E) "

--+

AP+2(L E).

'k

'

(5.3.5.7) "Ip E N, >'P E AP(L, E), "Ik, l, kl, . .. ,kp E L : (i(k) i(k)

0

0

i(l)(>'p+2))(k 1, . .. ,kp) = >'p+2(l, k, kl, ... ,kp), i(l)(>'2) = >'2(l, k), i(k) 0 i(l) IEEBA1(L,E) = 0,

i(k) 0 i(k) = 0, i(k) 0 i(l) + i(l) 0 i(k) = 0, Oo(k) 0 Oo(l) - Oo(l) oOo(k) = Oo([k, lJ), p(k) 0 p(l) - p(l) 0 p(k) = K,(k, l) + p([k, lJ),

5.3 Lie-Cartan Pairs

153

extending the curvature to A$(L, E),

Op(k) 0 Op(l) - Op(l) 0 Op(k) = (Jp([k, l]) + K-(k, l), Do 0 Do = 0, po i(k) + i(k) 0 P= p(k), Do 0 i(k) + i(k) 0 Do = Oo(k), Dp 0 i(k) + i(k) ODp = Op(k), i(k) 0 Oo(l) - Oo(l) 0 i(k) = i([k, l]), i(k) 0 p(l) = p(l) 0 i(k), i(k) 0 Op(l) - Op(l) 0 i(k) = i([k, l]), p(k) 0 Do = Do 0 p(k), Oo(k) 0 Do = Do oOo(k), Dp 0 Dp = Pi" Op(k) ODp - Dp 0 Op(k) = Pi,(k) = i(k) 0 Pi, - Pi, 0 i(k). (5.3.5.8) If the E-connection p is Bat, one especially obtains Vk, 1 E L :

K-(k, l) = 0, Pi,(k) = 0, Pi, = 0, p(k) 0 p(l) - p(l) 0 p(k) = p([k, l]), Op(k) 0 Op(l) - Op(l) 0 Op(k) = Op([k, l]), Dp 0 Dp = 0, Op(k) ODp = Dp 0 Op(k) = Dp 0 i(k) ODp. (5.3.5.9) Vk E L : 1m i(k)!AA(L,E) U 1m Op(k)IAA(L,E) ~ AA(L, E).

(5.3.5.10) With the definition, that V¢J E EndR(E),a E A,x E E: (a¢J)(x) := a¢J(x),

EndR(E) obviously becomes an A-left module over R, as E itself. (5.3.5.11) If the connection p is local, i.e., if p : L (EndR(E))L is Alinear, then one especially finds, that V>. E AA(L, E) : Dp(-?-) E AA(L, E). (5.3.6.1) Let E be equipped with an R-bilinear map:

Ex E

3

{x,y} -

xy

E

E,

thereby establishing an algebra E over R. The so-called wedge product is defined as an R-bilinear map: A$(L, E) x A$(L, E) - A$(L, E), such that

Vp, q E No : AP(L, E) x Aq(L, E)

3

{>.P' JLq} -

>'P 1\ JLq

E

Ap+q(L, E).

Vx,y E E,Vp,q E N,V>.p E AP(L,E),JLq E Aq(L,E),Vk1, ... ,kp+q E L:

154

5. Z2-Graded Lie-Cartan Pairs

x /l.y:= xy, (x /I. Ap)(k l , (A p /I. y)(k l , (Ap /I. /1.q)( k l , :=

(AI

-i-t p.q. /I.

, k p) := xAp(k l , , kp), , k p) := Ap(kl, , kp)Y, , kp+q)

L

p

(-1r Ap(kj1 ,···, kjp)/1.q(k jp +1'···' k jp +q),

1. ... P .+ q ] =:PEP p + q [ )1·· ')p+q

/1.I)(k l , k2 ) = Al (k l )/1.l(k2) - Al (k2)/1.l(k l ).

Hence 11*(L, E) is some No-graded algebra over R. (5.3.6.1.1) Especially let the product of E be A-bilinear, Le.,

"Ix, y E E, Va, bE A : (ax)(by) = (ab)(xy). In this case the wedge product is A-bilinear too, Le., VA,/1. E 11*(L, E), Va,b E A : (aA)

/I.

(b/1.) = (ab)(A

/I.

/1.).

In this case 11 A(L, E) is some subalgebra of 11* (L, E). (5.3.6.2) If E is an associative algebra over R, then 11*(L, E) is also associative, Le., Vp, q, r EN, "lAp E 11 P (L,E),/1.q E 11q(L, E), Vr E 11r(L,E),Vkl, ... ,kp+q+r E L:

(A p /I. /1.q

/I.

vr)(kl, ... , kp+q+r ) =

-At p.q.r.

L [

(-1rp

1,..p+q+r] " =:PEP p + q + r )1·· ·)p+q+r

Ap(kj1 ,· .. ,kjp)/1.q(kjp+l'·· .,kjp+q)Vr(kjp+q+t'·· .,kjp+q+r )· In particular, if E is unital, then of course 11*(L, E) is unital too. (5.3.6.3) If the algebra E over R is commutative, then one finds that Vp,q E No, "lAp E 11 P (L,E),/1.q E 11 q(L,E): Ap /l./1.q = (-1)pq/1.q /l.A p . (5.3.6.4) If E is a Lie algebra over R, then 11*(L, E) is some Lie superalgebra over R, with an appropriate Z2-grading.

Vp E Z2 : (11*(L, E))P:=

E9

11 P (L, E).

pEprlNo

Vp,q,r E No, "lAp E 11P (L,E),/1.q E 11q(L,E),vr E 11 r (L,E) :

Ap

/I.

/1.q

+ (-1 )pq/1.q /I. Ap

= 0,

Ap /I. (/1.q /I. vr ) = (Ap /I. /1.q) /I. Vr + (-1)pq/1.q /I. (A p /I. vr ).

5.4 Real Differential Forms

155

(5.3.6.5) The following above defined R-endomorphisms of A*(L, E) are super-derivations of the superalgebra A· (L, E), with the Z2-grading as above, such that Vp,q E No, "1>'1' E Ap(L,E),/-lq E Aq(L,E),Vk E L:

i(k)(>'p 1\ /-lq) = (i(k)(>.p)) 1\ /-lq + (-1)1'>,1' 1\ (i(k)(/-lq)), Oo(k)( >'1' 1\ /-lq) = (00 (k) (>.1')) 1\ /-lq + >'1' 1\ (Oo( k )(/-lq)), <50 (>'1' 1\ /-lq) = (<50 (>.1')) 1\ J.1.q + (-1)1'>'1' 1\ (<50 (J.1.q)). (5.3.6.6) Especially assume, that Vk E L, "Ix, y E E:

p(k)(xy) = (p(k)(x))y + x(p(k)(y)). In this case, Vp, q E No, "1>'1' E Ap(L, E), J.1.q E Aq(L, E), Vk E L :

p(k) (>.1' 1\ J.1.q) = (p(k)(>.p)) 1\ J.1.q + >'1' 1\ (p(k)(J.1.q)), Op(k)(>.p 1\ J.1.q) = (Op(k)(>.p)) 1\ J.1.q + >'1' 1\ (Op(k)(J.1.q)). Furthermore in this case, Vp, q EN, "1>'1' E Ap(L, E), J.1.q E Aq(L, E) :

p(>'p 1\ J.1.q) = (p(>'p)) 1\ J.1.q + (-1)1'>,1' 1\ (p(J.1.q)), <5 p(>'p 1\ J.1.q) = (<5 p(>'p)) 1\ /-lq + (-1)1'>,1' 1\ (<5 p(J.1.q)).

(5.3.6.7) An R-endomorphism ).1 : Aq-1(L, E) --+ Aq(L, E) is established for q E N. "1>'1 E Al(L, E), Vq E N, VJ.1.q E Aq(L, E), Vko, k 1, , kq E L, Vy E E :

().1 (J.1.q))(k o, k 1, .. ·, kq) := (>'1

1\

J.1.q)(ko, k 1,

, kq)

q

=

2) -1)j >'1 (k j )J.1.q(ko, ... , ¥j' ... , kq),

j=o

5.4 Real Differential Forms (5.4.1) Consider an m-dimensional real Coo-manifold M, with the Abelian unital associative algebra COO(M), and the Lie algebra T(M) of Coo-vector fields, both over R. On an arbitrary chart ¢ of an atlas for M, the smooth vector field X is acting with the components {k, k = 1, ... , m.

X: M 2 dam ¢

3 q=

¢-l(x)

--+

{q,

[uJ}

E T(M),

[u] = {k(X) a~k'

with an equivalence class [u] of smooth curves u : ] -1, +1[

--+

M, u(O) = q.

156

5.

Lie-Cartan Pairs

Z~Graded

(5.4.2) The Lie-Cartan pair {T(M), Coo(M)} is established with the following two R-bilinear maps. Vq E dom ¢, VI E Coo(M), X E T(M) : (fX)(q) := {q, I(q)e(x) a~k },

a

(Xf)(q) := Lxl(q) = e(x) axkl(x),

-Ix-

with the natural basis vectors used for the partial derivations of the scalar field Ion the chart ¢. Obviously 1m eCOO(M) := {I}, defining the unit scalar field.

(5.4.3) The real Lie algebra of Lie derivations of smooth scalar fields,

L(M) := {Lx: Coo(M) - - COO (M); X E T(M)}, is isomorphic with the real Lie algebra of COO-vector fields: T(M)

3 X

+-+

Lx

E

L(M),

as some subalgebra of the real Lie algebra DerR(Coo(M» = Der(Coo(M». Moreover the family {L(M), Coo (M)} is some subpair of the Lie-Cartan pair {Der(Coo(M», COO (M)} , defining V/,g E COO(M),VX E T(M): (fLx)(g):= I(Lxg)

= LIX9.

Hence in the sense of the above isomorphism of Lie algebras, the pair {T(M), COO(M)} is some subpair of {Der(Coo(M) , COO (M)}.

(5.4.4) Consider the Lie-Cartan pair {T(M), COO(M)} and an E-connection p, with E := COO(M), using the product of this algebra for the left module-

multiplication. This real-linear map is then acting as p: T(M)

3

X

-+

¢

E

(EndR(COO(M)))L,

"IX E T(M), V/,g E Coo(M) : ¢(fg) = I¢(g) + (Lxf)g. Obviously the COO(M)-connection defined by: T(M) 3 X local and flat.

-+

Lx is both

(5.4.5) Again with respect to the above Lie-Cartan pair, consider the choice E := T(M), using one of the two Lie-Cartan bilinear maps for the definition ofthe COO(M)-left module E over R. Here this real-linear map is acting as p : T(M) 3 X - + ¢ E

(EndR(T(M)))L,

"IX, Y E T(M), VI E COO(M) : ¢(fY) Obviously the T(M)-connection:

T(M)

3

X -- Lx : T(M) del

is flat, but non-local.

3

Y

-+

= I¢(Y) + (Lxf)Y.

[X, Y]

E

T(M)

5.4 Real Differential Forms

157

(5.4.6) Defining, with respect to the Coo(M)-left module of real COO-tensor fields ofrank {s,r} E No x No, E:= T;(M), the Lie derivation of smooth tensor fields with respect to X E T(M) provides an E-connection p: T(M)

3

X

Lx

-+

de!

E

(EndR(E))L,

because one finds, that Vr, s E No,

"IX E T(M),f E Coo(M),t E T:(M): Lx(ft) = fLxt Obviously, for r

~

1 or s

~

+ (Lxf)t.

1, this connection is flat, but non-local.

(5.4.7) On every chart ¢ of an atlas for M, the Coo(dom ¢)-left modules of smooth vector fields, more generally of smooth {r, s }-tensor fields, and especially of differential forms, are free with respect to the bases indicated below.

{a~k jk =

{a:

k1

-+

® ... ®

a:

k•

-+

T(dom ¢)

= TJ(dom ¢),

®dx lt ®···®dx 1r jk 1 , •.. ,lr = 1, ... ,m}

T:(dom ¢),

{dX ll -+

1, ... ,m}

/\ •.. /\

dx 1r ; 1 ~ II < ... < lr ~ mj 1 ~ r ~ m} U {unit}

E(dom ¢),

inserting the unit of Coo (dom ¢). (5.4.8) Every differential form m

t r E Er(M), 1 ~ r ~ m,

E9 Er(M) =: E(M), r=O

can be viewed as an alternating R-multilinear map r

Ar

:

II T(M)

-+

Coo (M),

such that VX 1 , ... ,Xr E T(M), Vq E M: tr(q)(Xk(q)j k

= 1, ... , r) = (Ar(Xl>""

One denotes Eo(M)

= Coo(M),

Er(M)

:=

Xr))(q).

{O} for r

E

Z \ {O, 1, ... , m}.

158

5. Z2-Graded Lie-Cartan Pairs

(5.4.9) Obviously not every element of A*(T(M), COO(M)) is induced by an appropriate differential form t E E(M). For instance consider the convolution form>. E Al(T(Rm, COO (Rm)), such that: T(Rm) 3 X =: f..l aa I - + f x >.

: Rm

3

x -+ de!

( TI(X JRm

Vil : Tl

E

y)f..l(y)dy,

CO'(Rm ),

inserting smooth integral kernels with compact support. (5.4.10) The usual wedge product of differential forms on M can then be written as wedge product of the correspronding special elements of A*(T(M), COO (M)). (5.4.11) Obviously A*(T(M), COO(M)) is some unital associative algebra over R. Moreover the above defined injection jr : Er(M) 3 t r - + jo := id COO (M),

>'r E Ar(T(M), COO(M)) for 0 ~ r

~

m,

is linear with respect to the coefficients E COO(M); hence its image is some submodule of the COO(M)-left module A*(T(M), COO(M)). Obviously Vifr :

1m jr Vr

~

~ ACOO(M) (T(M),

COO(M)).

m + 1 : ACOO(M) (T(M), COO(M))

= {O}.

(5.4.12) The real endomorphisms d,Lx, and ix of E(M), are written explicitly on the domain of one chart. Here d denotes the exterior derivation, and Lx the Lie derivation with respect to the vector field X E T(M), of real differential forms. Let X := f..k a~k E T(M), f E COO(M),

t r .'-for 1 ~ r ~ m, with x := ¢(q) E ran ¢, on one chart ¢, dam ¢ =: M. VI ~ r ~ m:

L

iXtr+l(q) =

f..k(X)TkI1 ...dx)dx I1 /\···1\ dx1r ,

1~1t < ...
(eV kTI1 ...l r VI

~

r

~

m -1 :

+ Tkh ...lr Vit f..k + ... + TI1 .. .lr_1kVlre)(X).

5.4 Real Differential Forms

(VII TI 2 ... 1r +l Lxf(q)

-

Vl 2 Tl1h ... lr+1

= f..k(x)Vkf 0 ¢-l(X),

+ - ... + (-lrVl

ixf:= 0, iXtl(q)

r

159

+ 1 Tl1 .. .lJ(X).

= f..k(X)Tk(X),

df(q) = dx1Vd 0 ¢-l(X), dt m := 0.

(5.4.12.1) These real endomorphisms of E(M k ), k = 1,2, are compatible with the pull back 'l/J. : E(M2) ~ E(Ml) of differential forms, which is induced by a diffeomorphism 'l/J : M 1 +----+ M 2 , as is indicated in the diagram below.

with the according push forward of vector fields, such that X2 := 'l/J.(X1 ). (5.4.12.2) Obviously the unital associative algebra E(M) over R is Z2graded by Ef'(M) :=

EB Er(M),

f

E Z2·

rEf'

Then Lx is some even derivation of E(M), ix and d are odd derivations of E(M), for every smooth vector field X on M. (5.4.12.3) Lx oLy - Ly oLx = L[x,Yj, ix oix = 0, dod = 0, Lx = ix 0 d + d 0 ix, Lx 0 d - do Lx = 0, ix 0 Ly - Ly 0 ix = i[x,Yj, for smooth vector fields X, Y. Therefore the real linear span of the set {d, Lx, ix j X E T(M)} is some subalgebra of the real Lie superalgebra

5. Z2-Graded Lie-Cartan Pairs

160

DerR(E(M»

:=

E9 Derit(E(M» fEZ~

of super-derivations of E(M). (5.4.12.4) The de Rham-cohomologies are defined with respect to the restrictions dIEr(M)' 0 $ r $ m.

. . . -+ Em(M) -+{O} -+ {O} -+ .... d",

d",+1

"Ir E Z: Hr(M) := ker dr/lm dr-I' The above defined pull back t/J., and the Lie derivations Lx with respect to vector fields X E T(M), are complex-morphisms of degree zero, Le., compatible with d, and not changing the rank r of a differential form. (5.4.13) The interior product of the smooth vector field X with the differential form t r + ll 0 $ r $ m -1, is obtained from the endomorphism i(X) of the real vector space A*(T(M), COO(M». "Iq E dam ¢, 1

0

I

a

I

VI $ r $ m -1, "IX:= ~ OXI,XI := ~loxl"" ,Xr := ~r

L

'Vtr+l:=

oxal '

lr T!1."lr+l dxh /I. ... /I.dx +1 E Er+l(M),

l:$h < ...
Ar+l := jr+l(tr+l) :

«i(X)(Ar+I»(XI , ... ,Xr »(q) = (ixtr+l)(q)(X I (q), = Tlh ... dx)~I(x)~~1 (x)

W

:= TJldx l E

EI(M), VAl

:=

, Xr(q» ~~r(x),

jl(Y) :

(i(X)(AI»(q) = (AI(X»(q) = Y(q)(X(q» = TJI(X)~I(x) = ixY(q), "If

E

COO(M) : i(X)(f) = ixf = 0 by definition.

(5.4.14) For 1 $ r $ m - 1, consider the differential form t r E Er(M), and denote

5.4 Real Differential Forms

161

(5.4.14.1)

«60(A r»()(0, ... ,)(r»)(q) m

L

=

(_I)i+j

O~i<j~r

Tlt ... dx)(~f'Vk~~l - ~j'Vk~~1 )(x)

L ll •... ,lr=l

m

(J4r+l()(O,··· ,)(r»)(q) =

L l~ll

L('Vk7"/l ...d x ))

< ...
(-lrp~6(x)~fl (x) .. · ~:r(x),

L

.. . lr] kll .. . =:PEP r + 1 [ JJI .. ·Jr

(Lx; (Ar()(o, ... ,JI j"

.. ,)(r )))(q)

~ ej(x)'i7, C'~~l'l,,.{x )~, (x) ... e'; (X») , with an obvious notation of components of t r , and of )(o"",)(r E T(M), on dom ¢ 3 q. (5.4.14.1.1) Especially, for r = 1 ::; m - 1, one finds, denoting E ran ¢, that

x := ¢(q)

(J42(XO, )(l»)(q) =

L

('VkTl - 'VlTk)(X)~~(x)~{(x),

l~k
«60(AI»()(0,)(d )(q) = -7"/(x) (~~V' kd - ~~V'k~b)(x), I

L( -1)j (Lx; (AI ()(i;») )(q) = (~~V' k( 7"/d) -

dV' k( 7"/~b) )(x),

j=O

with the notation io := 1, il := O. (5.4.14.2) Hence for 1 ::; r ::; m - 1 the exterior derivation can be written as r

+ L( -1)j Lx; (Ar()(o, ... , Jl j , ... , )(r» j=O

= (Op(Ar»)()(O,oo.,)(r),

inserting the COO(M)-connection p : T(M) flat.

3)( - +

Lx, which is local and

162

5. Z2-Graded Lie-Cartan Pairs

(5.4.14.3) For r = m one obtains, that "lAm E Am(T(M),Coo(M)),VXo, ... ,Xm E T(M): m

~(-I)j Lx; (Am(XO, " "

Jl j , ... , X m)) =

-(80 (A m))(XO, " " X m).

j=O

(5.4.14.4) For scalar fields one finds, that "If E Coo(M),X E T(M) :

iI(df) =: J1.1, J1.1(X) = 8p(f)(X) = p(f)(X) = p(X)(f) := Lx/. (5.4.15) Similarly the Lie derivation of differential forms on M can be rewritten, such that VI r m, "IX, Xl,"" X r E T(M), Vt r E Er(M) :

:s :s

Vr

(Xll ... ,Xr ) = LX(Ar(X1 , .•• , X r )) + (tfO(X)(Ar ))(X I , ... ,Xr ) = (tfp(X)(Ar))(X 1 , ... ,Xr ),

(5.4.15.1) "If E COO(M) : Lxf = p(X)(f) = tfp(X)(f).

(5.4.16) Therefore the Cartan calculus of differential forms on M provides an example of the above machinery of Lie-Cartan pairs. Take X E T(M);

\lfr : ixIEr(M)

= j;.!l

0

i(X)

0

jr, ix ICOC(M) = i(X)lcoc(M) = 0;

'v'(fr : Lx IEr(M) = j;l 0 tfp(X) 0 jr; 'v'(f-Ir : dIEr(M) = j;';l o8p 0 jr,

dIEm(M)

= 8p 0 jm = 0,

inserting the above defined local flat COO (M)-connection p, and with an obvious meaning of j;l.

5.5 Cohomologies of Lie-Cartan Pairs

:s

(5.5.1) The diagrams jrH 0 d 0 j;l, jr 0 Lx 0 j;l, O:S r m, jmH := 0, are generalized to an arbitrary Lie-Cartan pair {L, A} over a commutative ring R, with respect to an A-connection p. Here A is also considered as an A-left module over R, using the product of A for the module-multiplication too.

5.5 Cohomologies of Lie-Cartan Pairs

163

(5.5.1.1) The R-endomorphisms d, Lk, ik of A*(L, A) are defined appropriately. Vp E N,A p E AP(L,A),Vk,ko,k 1,

oo.

,kp E L,Va

E

A: (da)(k o):= koa,

(dAp)(k o, k 1, ... , kp) := (80 (A p»(ko, k 1, ... , k p) p

+ L( -1)j kjAp(ko,

00"

~j'"'' kp),

j=O

(LkA p)( kll ... ,kp) := kA p(k 1, ... ,kp) + (Oo(k)(A p))(k ll ... , kp), Lka := ka, ik := i(k), ika = 0, ikA1 = A1(k). (5.5.1.2) Now take the flat local A-connection Po :

L

3

k

--+

Po(k): A

3

a--+ka

E

de!

A.

Then obviously Vk E L, a E A :

d - 8poiEB

>1 AP(L,A) P-

L k = Bpo(k) := Bo(k)

= 0, 0PO := 80

+ Po,

8po (a) = 0,

+ Po(k).

(5.5.1.3) One then obtains the following relations. Vk, l E L :

ik

0

i,

+ i, 0

ik = 0, Lk

0

L, - L,

0

Lk = L[k,I] , dod = 0,

Lk 0 d - d 0 Lk = 0, ik 0 d + d 0 ik = Lk, ik

0

L, - L,

0

ik = i[k,I].

Therefore the real linear span of {d, L k ,ik; k E L} is some subalgebra of the Lie superalgebra

DerR(A*(L,A»:=

E9 Derh(A*(L, A» ZE Z 2

of super-derivations of alternating R-multilinear forms on L, with the values in A. (5.5.1.4) On an m-dimensional real Coo-manifold M, choose A := COO(M), L:= T(M),

°

and restrict the above Lie superalgebra to the images of jr, ~ r ~ m, and apply j;1. Then one obtains the real linear span of {d,Lx,ix;X E T(M)}, in the sense of the foregoing subchapter. (5.5.1.5) The de Rham-cohomologies are generalized to the cohomologies of don A*(L,A).

164

5. Z2-Graded Lie-Cartan Pairs

(5.5.2) In order to investigate cohomologies of Lie algebras, consider the Lie-Cartan pair {L, A} over R, and an E-connection p. One finds, that Vk E L,>" E A*(L,E),a E A: i(k)(a>")

= a(i(k)(>")),

Op(k)(a>..)

= a(Op(k)(>")) + (ka)>..,

8p(a>..)

= a8p(>") + L1(a)(>..).

(5.5.2.1) Let p be flat. The cohomologies H:(L, E) are defined with respect to 8p • 8p 0 8p = 0, Vp E Z : H~(L,E) := ker 8~/lm 8~-1, H~(L, E) = E/{O} .

... ---. {O} ---.{O} ---. E ---. A1(L, E) ---. 6;;2

6;;1

6~

... ---. AP(L, E) ---. AP+l (L, E) ---. .... 6~

(5.5.2.2) Vk, l Op(k)

0

E

L:

Opel) - Opel)

0

Op(k) o8p - 8p oOp(k) i(k)

0

Op(l) - Opel)

0

Op(k)

= 0,

= Op([k, lD, i(k) 0 i(k) = 0, i(k) o8p + 6p i(k) = Op(k), 0

i(k) = i([k, lD.

Therefore the R-linear span of {8 p,i(k),Op(k);k E L} is some subalgebra of the Lie superalgebra (EndR(A*(L, E)))L.

(5.5.2.3) The elements of A*(L, E) are called E-cochains. Especially the elements of ker 8p are called E-cocycles, those of 1m 8p E-coboundaries. Obviously Op(k) turns E-cocycles into E-coboundaries, because

(5.5.2.3.1) Inserting the flat local A-connection Po, choosing E := A, one obtains the A-valued Chevalley cohomologies of L with respect to d. (5.5.2.4) Consider again the choice E := A, hence an L-Ieft module E over E L, a E A : ka := o. Then the Lie-Cartan pair means, that

R. Especially choose Vk

Va, bE A, Vk, l E L : (ab)k = a(bk), eAk = k, [k, all = [ak, l] = ark, l].

The R-linear mappings d, Lk, ik, k E L, are then acting such that

5.5 Cohomologies of Lie-Cartan Pairs

Vp EN, Ap E AP(L, E), Vk, ko, kl>"" kp E L, Va E A : da

L

dAp(ko, kl> ... ,kp} =

165

= 0,

(-l)i+j Ap([ki , kj ], ko, . .. '~i' ... '~j"

..

,kp),

O~i<j~p

dAl (k o, kI) = -AI ([k o, kIl), dA2(ko, kl> k2) = -A2([ko, kIl, k2) + A2([ko, k2], kI) - A2([kl> k2], ko), p

LkAp(kl>"" kp) = -

L Ap(k 1, ... , kj-l> [k, k

j ],

kj+l>"" kp}, Lka = 0,

j=l

LkAl (k 1) = -AI ([k, k1]), LkA2(kl> k2} = -A2([k, kIl, k2) - A2(k 1, [k, k2]), ikAp+1 (kl> ... ,kp) = Ap+1 (k, kl, ... ,kp), ika = 0, ikAl = A1(k), ikA2(kI) = A2(k,kI).

The Lie superalgebra of these super-derivations of A-cochains for L was written down above. Due to the special case of Po = one obtains the cohomologies of L with respect to 0o = d.

°

(5.5.2.5) In the special case of A := R, and demanding that Va E A, k E L : ka := 0, the R-bilinear map: A x L ---+ L is defined as the module-multiplication of L, and E is an R-bimodule. Therefore the defining conditions on Lie-Cartan pairs are essentially reduced to the zero-map: LxA ---+ {o} ---+ A. Obviously every E-connection p is local, because p is an R-linear map.

Vk E L, a E A, x E E : p(k)(ax) = p(ak)(x}. In the case of p := 0, one finds the E-valued Chevalley cohomologies H&(L, E), with respect to the R-endomorphism 0o of A*(L, E}, 0o 0 0o = 0. (5.5.2.5.1) Especially choose E := A := R. Then d = 0o induces the cohomologies H&(L, R}. Using the universal property of the alternating product AP(L}, with the alternating algebra A(L}:=

EB AP(L) pENo

of Lover R, one obtains an isomorphism of (AP(L})* with AP(L, R), p E No, in the sense of R-bimodules, due to the diagram below. P X L

• AP(L)

fLR~f

166

5. Z2-Graded Lie-Cartan Pairs

Here f E AP(L, R) induces uniquely j E (AP(L»)* := HomR(AP(L), R), with the following R-linear bijections. Vp EN: AP(L, R) 3

f

(A 1 (L))* = (

---.+

j

{~}) *

E

(AP(L))*, AO(L) =

---.+

{~}

R =: AO(L, R),

---.+

HomR(L, R) =: A 1 (L, R).

(5.5.2.5.1.1) In the special case of real coefficients, R := R, and dim LEN, one finds real-linear bijections: AP(L, R)

---.+

(AP(L))*

---.+

AP(L), p E No.

Hence one obtains an isomorphism: A(L) +--+ A*(L, R), in the sense of unital associative algebras over R, using carefully an appropriate basis of m

A(L) := EBAP(L) p=o over R. Here one finds, denoting dim L =: m, that Vp > m : AP(L) = {O}. V0

~ p ~ m: dim AP(L) =

m

( ; ) , dim A(L) = 2

.

(5.5.3) The so-called matrix geometry, Le. non-commutative differential geometryover unital associative algebras of complex matrices, which was constructed by M. Dubois-Violette and collaborators (1990), can be formulated within the framework of Lie-Cartan pairs {L, A} over R, and their E-connections, choosing

R := C, A := C, L := Derc(E), L x C -----+{O} -----+ C, de!

using the module-multiplication of the complex Lie algebra L for the complexbilinear map: C x L -----+ L, and inserting the connection p: L -----+ (Endc(E))L by means of derivations of E, such that

Vk E L,x E E: p(k)(x) := k(x)

= kx.

The complex vector space E is chosen as some unital associative algebra, such that derivations on E can be defined. Then A*(L, E) is such an object too, moreover carrying an No-grading and corresponding Z2-grading. The connection p is flat and local. The endomorphism d of E, which is defined such that Vp EN, x E E, k E L: d - 8p I AP (L,E) := 0, (dx)(k) := kx E E, dod = 0,

5.6 Z2-Graded Lie-Cartan Pairs

167

is an odd nilpotent derivation of A*(L, E). More explicitly Vp EN, Vko, k 1 , ... , kp E L, VA p E AP(L, E) :

(dAp)(ko, ... ,kp)

L

(_1)i+j Ap (lk i , k j ], ko,···, ~i"'"

~j"'"

k p)

O~i<j~p

P

+ L( -1)j kjAp(ko, ... , ~j"'"

kp).

j=O (5.5.3.1) These differential forms over E may be restricted to the smallest Z2-graded differential subalgebra of A*(L,E), which contains E itself as subalgebra, in order to obtain an appropriate differential envelope of E. (5.5.3.2) The above indicated version of matrix geometry is obtained by the choice of E := Matn(C). In this case, every derivation of E is an inner one, Le., can be expressed by means of the adjoint representation of the commutator algebra gl(n, C) of Matn(C). Therefore the complex Lie algebra Derc(Matn(C)) is isomorphic to sl(n, C). Moreover the above indicated differential envelope of Matn(C) is A*(Derc(Matn(C)), Matn(C)) itself. (5.5.3.3) Furthermore models of gauge theory were constructed, starting from

E

:=

COO(M)

~

Matn(C),

on a finite-dimensional real connected differentiable manifold M.

5.6 Z2-Graded Lie-Cartan Pairs (5.6.1) Let L be a Lie superalgebra, and A a unital associative superalgebra, both over a commutative ring R. Assume A to be graded-commutative, Le., with vanishing super-commutator, such that Vp, ij E Z2, Va E Ail, bE All : ab = (-1)

pq ba.

The family {L, A} is called Z2-graded Lie-Cartan pair, if and only if the following conditions hold. (i) L is some graded A-left module over R, Le., L is an A-left module over R, and Vp,ijE Z2,Va E AP,k E Lli: ak E Lp+q.

(ii) A homomorphism of Lie superalgebras

168

5. Z2-Graded Lie-Cartan Pairs

into the Lie superalgebra of super-derivations of A, is established, Le., an even R-linear map>. conserving the super-commutator: L 3 k --+ >'(k): A 3 a--+ka E A. de!

(iii) "la, bE A, "Ik

E L: a(kb) = (ak)b.

(iv) "Ip, ij E Z2, "Ia E AP, k E Lq, l E L : [k, all = (-l)pqa[k, l]

+ (ka)l.

(5.6.1.1) These conditions on the pair {L,A} explicitly mean an establishment of R-bilinear mappings: A x L --+ L, L x A --+ A, which obey the following relations. "Ip, ij, r E Z2, "Ia E AP, k E Lil, l ELf', bE A : (ab)k

= a(bk),

eAk

= k,

ak E Lp+q, ka E Ap+q,

k(ab) = (ka)b+ (-l)pqa(kb), [k,l]a = k(la) - (-l)qrl(ka), a(kb) = (ak)b, [k, all = (-l)pqa[k, l]

+ (ka)l.

These relations imply keA

= OA,

[ak,l]

= a[k,l]- (-l)(p+q)r(la)k.

(5.6.2) Let E = EO EB EI be a graded A-bimodule over R. An even R-linear map p: L --+ (EndR(E»L is called E-connection, if and only if "Ip, ij E Z2, "Ia E AP,k E Lil,x E E: p(k)(ax) = (-l)pqa(p(k)(x»

+ (ka)x.

(5.6.2.1) One then immediately finds, that "Ip, ij E Z2, "Ik E LP, x E Eq, a E A: p(k)(xa)

= (p(k)(x»a + (-l)pqx(ka).

(5.6.2.2) An E-connection p is called local, if and only if "Ik E L,a E A,x E E: p(ak)(x) = a(p(k)(x».

(5.6.3) The curvature K, of an E-connection p is defined as an R-bilinear map, such that "Ip, ij E Z2 : LP x Lil 3 {k,l} --+

K,(k, l) := p(k)

0

p(l) - (-l)pq p(l)

0

p(k) - p([k, l]) E (EndR(E) )'{+q.

In particular p is called flat, if and only if its curvature vanishes, Le., if and only if p is a homomorphism of Lie superalgebras over R, which means a representation of L on E.

5.6 Z2-Graded Lie-Cartan Pairs

169

(5.6.4) If Pi, P2 are E-connections, then

'
+ r2 = eR ~ rlPl + r2{J2

is an E - connection.

(5.6.5) The Z2-graded Lie-Cartan pair {L, A} is called degenerate, if and only if '
Lx R

(5.6.7) Obviously A is some graded A-left module over R. The above defined map Ayields some flat local A-connection Po, according to the diagram below. L - - A , - - ' De'R(A)

>

(EndJA)) L

f

I def

L

3

k ---+ Po(k) : A

3

a

---+

ka

E

A.

Po

Here

'
E

L : po(k)

E

DerR(A) := Der~(A) EB Der1(A).

(5.6.8) In case E := L, the adjoint representation of L yields some flat L-connection a.

L

3

k ---+ ad k : L ad

3

1 ---+ [l, k] E L,

with the homomorphism ad in the sense of Lie superalgebras over R:

L --;;;(EndR(L))L, 1m ad ~ DerR(L). L

3

k

---+

a(k) : L

3

l---+[k,l] del

E

L.

Especially, if {L, A} is degenerate, then the L-connection a is local. (5.6.9) The torsion Tp : L x L ---+ L, of an L-connection P, is defined as an R-bilinear mapping. '
V x Lii

3

{k,l}

---+

Tp(k,l):= p(k)(l) - (-l)pqp(l)(k) - [k,l]

such that obviously Tp(k,l) = (-1)pq+1Tp(l,k).

E

V+ q,

170

5. Z2-Graded Lie-Cartan Pairs

(5.6.10) Consider an E-connection p, with the Z2-graded Lie-Cartan pair {L,A} over R. The R-bimodule of R-multilinear maps AI' : IJP L --+ E is denoted by V(L, E), pEN.

EB Lp(L,E) =: L*(L,E).

L1(L,E) = HomR(L,E), LO(L,E):= E,

pENo

(5.6.10.1) Henceforth take R := K 2 Q, with a field K of characteristic zero. For p ~ 2, the vector space Ap(L, E) of graded-alternating mappings AI' : IJP L --+ E is defined, with the permutations PEP I' of p elements.

AI'(L, E)

:= {

AI' E V(L, E);'fZ1, ... , zp E Z2, Vk 1 E LZl, ... , k p E UP, VP:= [

\ (k·J 1 ,

AI'

••• ,

?... ~ ] E

J1 00 .Jp

PI' :

z· z· k·Jp ) = AI' \ (k 1 , ... , kI' )(-1) T P +~ L..,,{I$i;·,j 1i Ji' "

}

counting just those odd elements of L, which are transposed by P. For instance,

For A3 E A3(L, E) one obtains for example, with the above notation of degrees, A3(k 1 , k2, k3)

= A3(k3, k 1 , k2)( _1)Z l z3+ z1z3 = A3(k3, k2, kd( -1) !+ZIZ1+ ZIZ3+

1 3'

Z Z

One then usually denotes

AO(L,E) := E, A1(L,E):= HomK(L, E), A*(L,E):=

EB Ap(L, E). pENo

(5.6.10.2) A~(L,E):= {AI' E Ap((L,E);VZ,Zl,oo.,Zp E Z2,

Vk 1 E LZl,. 00' k p E LZp, Va E Ai:, "11 ~ j ~ p : Ap(k 1 , .. . ,akj , ... ,kp) =

(-ly(zJ+ oo -+ zp)A p (k 1 , • .. kp)a

= (_I)z(lI+zl+oo'+z;-tlaAp(kl'

here y E fi := x - Zl - ... -

zp

E Z2, for Ap ( k 1 ,

A~(L,E):= E, AA(L,E):=

EB A~(L,E). pEN o

' k p )}; ,

kp ) E EX, pEN.

5.6 Z2-Graded Lie-Cartan Pairs

171

Especially, for p = 1,

A1(L,E):= {Al E HomK(L,E)jVi,Zl E Z2,Vk l E LZl,a E A Z Al(ak l ) = (-1)ZZl>'l(kda }.

:

(5.6.10.2.1) Here the Z2-degree ii of >'p is used, such that

AP(L, E) =

E9 (AP(L, E))ii. iiE Z 2

(5.6.10.3) In order to understand these permutations, the graded alternator A p is defined. Vp

~

2, "lAp E F(L, E), VZ1, ... ,zp E Z2,

Vk l E LZ1 •....kpEL·p, VP:= [ .1. .. ~ ] E P p Jl '" Jp

:

(17(P)>'p)(k l , ... ,kp) := (-1) Tp+ L{l~'i,,) Zi, zi,' Ap(k jll ... ,kjp )' Ap :=

1

,

p.

L

17(P).

PEP p

Obviously 17 : P p metric group P p .

--+

EndK(LP(L, E)) is some representation of the sym-

A; = A p, because Vp ~ 2, VP E P p : 17(P)Ap = Ap17(P) = Ap. One then finds, that Vp

~

2:

(5.6.10.4) For pEN, the vector space F(L,E) is Z2-graded due to the definition, that Vii E Z2 : Ap E (F(L, E))ii <==> de!

VZ1, ... , Zp

E Z2, Vk l E LZl, ... , k p E LZp : Ap(k l

, ... ,

kp) E EZ1+···+zp+Y.

LP(L,E) = E9(F(L,E))1/. iiE Z 2

For p = 1, one obtains the usual Z2-grading of HomK(L, E). This intrinsic grading of L*(L, E) must be distinguished carefully from the total grading defined below.

172

5. Z2-Graded Lie-Cartan Pairs

(5.6.11) The K-endomorphisms i(k),p(k),Oo(k),Op(k),60 ,p,6p, of L·(L,E) are defined, inserting the intrinsic grading. Vp EN, Vii, z, Zo, zp E Z2, "lAp E (V(L, E»ii, Vk E Ll, ko E Lzo, , kp E LZp, "Ix E Eii :

(i(k)(Ap+I)(k l , ... , kp) := (_1)(P+i+II)z Ap+i(k, k l , ... , kp), i(k)(x) := 0, i(k)(AI) := (_1)(1+II)z AI(k), (p(k)(Ap»)(k l , ... , kp) := (-1)PZp(k)(A p(k l , ... ,kp», as an extension of the E-connection p, P

(OO(k)(Ap))(k l , ... , kp) :=( _1)1+PZ ~) _1)Z(II+ Z1 +",+zj-d j=l

Ap(kl. ... , kj - l , [k, kjl, kj+i,"" kp), Oo(k)(x):= 0, (OO(k)(Ad)(kI) = (-1)1+Z+ ZII A1 ([k,kd)' Op(k) := Oo(k)

+ p(k),

(6 0 (A p»(ko,k l , ... ,kp) := (_1)i+ j +(Z;+Zj)(zo+··+z;-d+ zj(z;+,+,,+zj-d

L

O:S;i<j:S;p

Ap([ki , kjl, ko,···, ~i"'" ~j"'" kp), 8o(x) := 0, (80(AI»(ko, kI) = -AI([ko, kID, (6 0 (A2»)(k o, kl. k2) = -A2([ko, kil, k2) + (-1)ZlZ2A2([ko,k2],kd - (-1)%O(Zl+ Z2)A2([k l ,k2],ko), p

L( _1)i+ Zj(y+zo+",+zj-d

(p(Ap))(k o, k l , ... , kp) :=

j=O

p(kj)(Ap(ko, ... , ~j"'" kp», p(x)(k) := (-1)YZ p(k)(x), (p(Ad)(ko,kd = (-1)ZOllp(ko)(AI(kd) - (-1)zdy+zo)p(k l )(AI(ko», 8p := 60 + P, 6p(x) = p(x). (5.6.11.1) These endomorphisms show the following properties, for pEN.

L 3 k L 3 k

:-+

linear :-+

linear

i(k) : LP(L, E) :--+ LP-l (L, E); hnear

p(k); Oo(k),Op(k): LP-I(L,E)

60 ,p,6p : LP-I(L,E)

:-+

linear

V(L,E).

:-+

linear

LP-l(L,E);

5.6 Z2-Graded Lie-Cartan Pairs

173

(5.6.11.2) Let fj, i E Z2j then L Z 3 k :-+ i(k)j p(k),Bo(k),Bp(k): (L*(L,E))fi :-+ (L*(L,E))lI+Z j linear

linear

fJo,p,fJp: (L*(L,E))fi ~ (L*(L,E))fi, (L*(L,E))ii:= linear

EB (LP(L,E))ii. pENo

(5.6.11.3) The vector space A*(L, E) is invariant under these K-endomorphisms. (5.6.12) With the convention, that Va E A,p E N,>.p E IJ'(L,E),x E E,Vk 1 , ..• ,kp E L: (aAp)(kl, ... , kp) := a>'p(k 1 , ... , kp), ax being defined in the sense of the graded A-left module E over K, L * (L, E) becomes some graded A-left module over K. Obviously

Va E A,>. E AA(L,E): a>' E AA(L,E). (5.6.12.1) Vx, i E Z2, Va E AX, k E U, V>. E L*(L, E) :

Bo(k) (a>.) = (_1):1:% a( Bo (k )(>.») , fJ o(a>.) = afJo(>'), i(k)(a>')

= (-l)ZZa(i(k)(>')).

(5.6.12.2) Moreover one finds, that Vk E L : 1m Bp(k)IAA(L,E) <; AA(L, E), 1m i(k)IAA(L,E) <; AA(L, E). If the connection p is local, then Vk E L : 1m fJpl AA (L,E) <; AA (L, E).

(5.6.13) An appropriate total grading of L*(L, E), and accordingly of EndK(L*(L, E)), is defined. Vp EN, fj E Z2 : >'p E (IJ'(L, E))lI+P ¢=:}

\1zi", ... , zp E Z2, Vk 1 E LZl, . .. , kp E LZp : \ (k 1, ... , Ap

kp ) E

EZl

+··+zp+lI .

The Lie superalgebra (EndK(A*(L,E)))L is understood with respect to this total grading. (5.6.13.1) Vi E Z2,k E L Z : i(k) E (End K (L*(L,E)))Z+I, p(k),Bo(k),Bp(k) E (EndK(L*(L,E)))Z, fJo,P,op E (EndK(L*(L,E)))I.

174

5. Z2-Graded Lie-Cartan Pairs

(5.6.13.2) The K-endomorphisms K(k,l), I>. of L*(L,E) are defined. \:Ip E N,\:Iy,X,Z,ZI," .,Zp+2 E Z2,\:IAp E (LP(L,E))iJ,x E E, \:Ik E LX, l E L t , k l E LZl, ... ,kp+2 E LZp+~ :

(K(k, l)(A p))(k l , ... ,kp) := (-1 )p(x+z) K(k, l)(Ap(kl, ... ,kp)), (I>.(Ap))(k l

._ L

, ... ,

kp+2)

(_1)1I(z.+Z;)+i+j+I+(Z.+Z;)(zl +--+Z._I)+Z;(ZHI+--+Z;_I)

l$i<j$p+2

K(k i , kj)(Ap(k l , ... , ~i"'" ~j"'" kp+2)),

(I>.(AI))(kl, k2, k3) = (-1)1I(ZdZ~)K(kl' k2)(AI(k 3))

+ (-1 )1I(zl +Z3)+1+Z3Z~ K(k l , k3)(AI (k 2))

+ (_1)1I(z~+z3)+(z~+Z3)zl K(k2, k3)(AI (k l )), I>.(x)(k,l):= (-1)1I(x+z)K(k,l)(x). One immediately finds, that \:Ip E No, \:Ix, Z, Y E Z2:

LX x L Z 3 {k,l}

---+

K(k,l): (LP(L,E))iJ

---+

(£P(L,E))1I+ x+ z,

(£P(L,E))iJ ---::-+(£p+ 2(L,E))iJ. It

Here the intrinsic grading of L*(L,E) is inserted. Obviously I>. is even with respect to the total grading too.

(5.6.13.3) Using the total grading for the super-commutator of K-endomorphisms of A*(L, E) one finds, assuming henceforth char K :f 2, that \:Ik, l E L:

= 0, [Bp(k), Bp(l)] - Bp([k, l]) - K(k, l)IAO(L,E) = 0, I>.IAO(L,E) = 0, fop, i(k)] - Bp(k)IAO(L,E) = 0,

[i(k), i(l)]IAO(L,E) op 0 op -

[i(k), Bp(l)] - i([k, l])!AO(L,E) = 0, [op, Bp(k)]

+ [i(k), I>.lIAo(L,E)

=

0.

(5.6.13.3.1) Inserting the connection p:= 0, one obtains the subalgebra K -lin span({i(k), Bo(k), 00; k E L}) of the Lie superalgebra (EndK(A*(L, E)))L, with the total grading of A*(L, E). (5.6.13.3.2)

\:Ik,l E L: [p(k),p(l)] - p([k,l]) - K(k,l)IAO(L,E) = 0, [Bo(k),p(l)]IAO(L,E) [00,13]

+ pop -

= 0,

I>.IAO(L,E)

= 0,

[p,i(k)] - p(k)IAO(L,E) = 0, [i(k),p(l)]IAO(L,E) = 0.

5.6 Z2-Graded Lie-Cartan Pairs

175

(5.6.14) Especially consider the vector space A*(L,A) over a field K 2 Q, char K = 0, taking E := A, and using the product of A as module-multiplication too. The intrinsic grading 80 and the total grading 8 of A*(L,A) were defined above, such that p

degree >-'p(kl,"" kp) =

L degree k

j

+ 8o>-'p, 80 >-'p + P =: 8>-'p,

j=1

for homogenous elements >-'p E AP(L, A), and k1 , ... , kp E L, pEN. On AO(L, A) := A, the gradings 80 and 8 shall coincide with that of A itself. On Al(L,A) = HomK(L,A),80 is just the usual Z2-grading of homomorphisms.

(5.6.14.1) The A-connection Po defined below is obviously flat and local. L

k ---+ Po(k) : A Po

3

3

a ---+ ka

E

de!

A.

Obviously Vk E L : Po(k) E (EndK(A))L.

(5.6.14.2) The K-endomorphisms d, Lk, and ik of A*(L, A) are defined. E Z2, Vk E LZ,a E AX:

Vx,z

d-

0po

lEa

>1

AP(L,A) := 0, (da)(k):= (_l)XZka,

p-

L k := 0Po(k)IAO(L,A)' ik := i(k)IAO(L,A)' One immediately finds the subalgebra K -lin span({d, Lk, ik; k E L}) of the Lie superalgebra (EndK(A*(L, A)))L' with respect to the total grading 8.

Vk, l

E

= 0, [d, L k] = 0,

L : [ik' izJ

= L[k,I], dod = 0, = Lk, [ik' Ld = i[k,I]'

[Lk, LzJ [d, ik]

Remember here the total degrees and intrinsic degrees, for homogenous k E L:

80d

= 0,

80Lk

= 80ik = degree

k, 8d

= I,

8Lk

= 8ik + I = degree

(5.6.14.3) An appropriate wedge product is defined: 2

II A*(L, A) 3 {>-', Il}

---+

>-./\ Il E A*(L, A),

such that

Vp, q E N, "Ix, il, ZI,"" Zp+q E Z2, Va E AX, bE Aii, V>-'p E (AP(L,A))X,llq E (Aq(L, A))ii, k1 E LZl, ... ,kp+q E LZp+q :

k.

176

5. Z2-Graded Lie-Cartan Pairs

a /\ b:= ab, (a 1\ J.tq)(kl, ... , k q) := (-1)qX aJ.tq(kl, ... , k q),

(Ap 1\ b)(k 1 , .•. ,kp) := (_1)II(Zl +,,+zp) Ap(k 1 , ... ,kp)b,

(Ap

/\

J.tq)(k 1 , ... , k p+q) := (_1)qx

(p;-p.q.~)! (Ap+qvp+q)(kl"'"

kp+q),

vp+q(kl, ... , kp+q) := (_1)I/(Zl+"+ Zp )Ap(k 1 , ... , kp)J.tq(kp+l,"" k p+q), with the intrinsic degrees

degree Ap(kl, degree J.tq(kp+1 ,

x of Ap , Y of J.tq

inserted, Le.,

,kp) = x + Zl + ... + zp, , kp+q) = Y + zp+1 + ... + zp+q'

This product explicitly reads, inserting the graded alternator A p +q ,

with the intrinsic degree x + y of Ap 1\ J.tq. (5.6.14.3.1)

(All\J.tl)(k ll k2) = (-1)X+I/Z1Al(kl)J.tl(k2) - (-1)X+Z1Z1+IIZ1Al(k2)J.tl(kI). (5.6.14.4) This algebra A*(L, A) over K is associative, with the unit eA. Moreover A*(L, A) is graded-commutative with respect to the total grading defined above. With the above notation this explicitly means, that

a

a(A p 1\ J.tq) = aA p + aJ.tq, Ap

1\

J.tq = (_1)(x+p)(lI+q) J.tq /\ Ap ,

a /\ J.tq = (_1)x(l/+q) J.tq /\ a, a 1\ b = (-1)Xl/b 1\ a. (5.6.14.5) The algebra A* (L, A) over K is some bigraded differential algebra with the Z2-graded derivation d, with respect to the total grading a, and the obvious No-grading. Vp,q E No,VA p E (AP(L,A»8,xp,J.tq E (Aq(L,A»8/l q :

d>.p E AP+l(L,A), 8o(d>'p) d(>'p 1\ J.tq) = (d>'p)

1\

J.tq

Obviously the Z2-grading construction.

= OoAp,

a(d>'p)

= a>.p + I,

+ (_1)8,xp >'p /\ (dJ.tq), dod =

a is compatible with the

O.

No-grading, by its very

5.6 Z2-Graded Lie-Cartan Pairs

177

(5.6.14.6) Vz E Z2, k E £Z : ik E (DerK(A*(L, A)))z+1, L k E (DerK(A*(L, A)))i,

and d is some skew-derivation, with respect to the total grading 8 of A*(L, A).

and OOIA.(L,A) is some skew-derivation, with the total grading 8 of A*(L, A). Therefore K -lin span({d,Lk,ik;k E L}) is some subalgebra of the Lie superalgebra DerK(A*(L,A)) :=

E9 (DerK(A*(L,A)W, iE Z 2

with respect to the total grading 8.

(5.6.14.1) Obviously AA(L, A) is some unital subalgebra of A*(L, A) over K. For all k E L, Lk,ik, and d are reduced by AA(L,A). Therefore AA(L,A) is some bigraded differential algebra over K, which is graded-commutative. (5.6.14.8) These A-valued differential forms are the images of elements of the non-unital universal differential envelope of A, due to its universal property, as is shown in the following diagram.

A

ca_n_o_nl_·ca_'_e_m_b_e_dd_in-=g

I.

a

canonica' embedding

>

A "(L,A)

• D (A)

J

a ..

Here 0'* is some homomorphism in the sense of bigraded differential algebras over K, which is uniquely determined by 0'. Therefore one finds, that Vn E N, Vao, ... ,an E A, '1'0,,1 E K:

where the universal derivation of Q(A) is also denoted by d, for convenience.

(5.6.15) Consider again a graded A-bimodule E over K, and the vector spaces A*(L, E) and A*(L, A) over K 2 Q, char K = O.

178

5. Z2-Graded Lie-Cartan Pairs

(5.6.15.1) Vp E N, Ap E AP(L, A), Z E Z2, a E AZ, Vk ll ... , kp E L :

(aAp)(k 1 , . •. ,kp) := (-1) PZ aA p(k ll • •. , kp). Thereby using the wedge product, one obtains some graded A-bimodule A*(L,A) over K, with respect to the total grading a. One then establishes the K-linear mapping 1/, due to the diagram below.

p

{k 1 , ••• ,kp }

I

E

XL

v

EA' (L,E)

def

(- 1) pz X

).p

(I<, ,... ,kp)

E

E

Here "Ix E E, a E A: I/(x ®A a) = xa E E. (5.6.15.2) The next diagram allows for an establishment of the A*(L, A)right module E ®A A*(L, A) over K, such that "Ix E E, VA, J.L E A*(L, A) : (x ®A A)J.L = X ®A (A/\ J.L). EXA"(L,A)3{X,).}

__

L ~V X

MP)

xl8'A).eEl8'AA"(L,A)

E

E

~A

A·(L.A)

J

Hence, since A·(L, A) is graded-commutative with respect to the total grading a, one constructs the graded A·(L, A)-bimodule E ®A A*(L, A) over K, with the total grading of A*(L, A). (5.6.15.3) Let E be projective-finite, with the coordinatization = 1, ... ,n}. Then the A·(L, A)-right module E ®A A*(L, A) over K is also projective-finite. {ek,e k ; k

5.6 Z2-Graded Lie-Cartan Pairs

179

(5.6.15.3.1) The universal property of the tensor product allows for an isomorphism a in the sense of graded A-bimodules over K, Le., an even A-linear bijection

a E Hom~ (A I8lA A·(L, A), A·(L, A)) : A I8lA A·(L, A) 3 a I8lA ,\ +-----+ a

t\,\ E

A·(L, A).

'" (5.6.15.3.2) One then easily finds an appropriate coordinatization of E I8lA A·(L, A).

€k:= T~(ek,idA·(L,A))00'-1, gk:= O'oT~(ek,idA·(L,A)), n

L gk k=1

0

€k = id E I8lA A·(L, A).

(5.6.15.3.3) Since every element by

f

E

EI8lA A·(L, A) is faithfully represented

n

i

= L€k

0 fk, and i k=1 one concludes, that

= (i(eAW, n

"If E E I8lA A·(L, A) : f = L edeA) I8lA fk(eA).

k=1 (5.6.15.3.4) Obviously an appropriate coordinatization of A I8lA A·(L, A) is

{a-I, a}. (5.6.5.4) Then A·(L, E) is established as some graded A·(L, A)-right module over K. "Ix, ii, ZI,"" zp+q E Z2, Vk 1 E LZl, ... , k p+q E LZp+q, "Ix E E~,b E Ai/,'\p E (AP(L,E))X,J.l.q E (Aq(L,A))li:

(XJ.l.q)(kl,"" kq) := (-1)qX x J.l.q(k 1, ... , kq), (Apb)(k 1 , ... , kp) :=

(_1)y(Zl+oo+Zp) Ap(kl,

... , kp)b, (-1)qx (ApJ.l.q)(k 1, ... , kp+q) := - , - , p.q. L (-1) Tp+ L( l:5kJ!l Z;k Zi, +1I(Zit +oo·+z;p) l..,p+q] . . =:PEP p + q [ )1" ')p+q

Ap(kj1 ,·· .kjp)J.l.q(kjp+l"" ,kjp+q )' with the intrinsic degrees x of '\p, ii of J.l.q, and xb defined from E.

180

5. Z2-Graded Lie-Cartan Pairs

(5.6.15.4.1) An appropriate coordinatization of A*(L, E) is provided by the dual n-basis {€k' €k j k = 1, ... ,n}. For simplicity assume that the dual n-basis of E is homogeneous in the sense, that 'lIik: 3Xk E Z2 : Ck E Hom~l(A,E), c k E Hom~k(E,A), ckock E En~(E).

V}k, 'rip EN: AP(L, E) :;)

)..P

~(_I)PXkck o)..p E AP(L, A), E

AP(L,A) :;) IJop!!;L(-I)P Xkck olJop E AP(L,E), Ek

AO(L, E) :;) x ~ck(x) E AO(L, A) :;) a ~ck(a) E AO(L, E), ik

ik

€k E HomA'(L,A)(A*(L,E),A*(L,A)), €k E HomA'(L,A)(A*(L, A), A*(L, E».

(5.6.15.4.2) 'rifE E ®A A* (L, A) : n

° fk,

j = L€k

fk := gk oj E EndA'(L,A)(A*(L, A».

k=l n

V).. E A*(L,E) :,\

= L€k 0 )..k,

)..k:= €k

0'\ E EndA'(L,A)(A*(L,A».

k=1

Here A*(L, A) is considered as some A*(L, A)-right module over K. (5.6.15.4.3) These coordinatizations are used in order to establish an appropriate isomorphism 1/ : E ®A A*(L, A) +---+ A*(L, E), which explicitly is decomposed according to: n

A*(L, E) :;) ).. +---+ ,\ = L

n

€k o)..k +---+ L

k=l

de!

gk o)..k =: j

k=l

+---+ fEE ®A A*(L, A).

Obviously 1/-1

E HomA'(L,A) (A*(L, E), A*(L, A».

With the degrees

Xk

of Ck and c k , one then finds that 'rip E N: n

AP(L, E) :;) )..P ~ L (-1)P Xkck (eA) ®A (c k k=l

and especially: n

E:;) x ~ LCk(eA) ®A ck(x) E E ®A A, k=l

suppressing the involved embeddings.

° )..p),

5.6 Z2-Graded Lie-Cartan Pairs

181

(5.6.15.5) For the restriction vIE®AAA(L,A) one immediately finds the following diagram.

r~------,-V--4--<_ _1 (5.6.15.6) The usual K-endomorphisms of A*(L, E) can be calculated from those of A*(L, A), due to the diagram below.

A P+ 1(L,E)

~> E@AA P+ 1(L,A)

4<-_ _--«

v

ilk)j

[VO'Oilk)OV

P A (L,E) ....-----------v'

E@AAP(L,A)

1

Vx,z E Z2, Vx E E~,k E Lz,A E A*(L,A):

x ®A A

--+ (_l)%(l+z)X ®A v-1oi(k)ov

i(k)(A).

Vx E E, k E L, a E A: x ®A a --+ xa v

--+

i(k)

o.

(5.6.15.6.1) VX,z E Z2,VX E EX,k E LZ,A E A*(L,A),a E A:

x ®A >.

X ®A

--+ (-l)%Zx ®A V-1090(k)ov

A

--+ p(k)(x) ®A v-1op(k)ov

Oo(k)(A), X ®A a

--+ 0, 90(k)ov

A + (-l)%Zx ®A Po(k)(>'),

X®A a--+xa--+(p(k)(x))a+ (-l)%Zx(ka), v

(p(k)(x))a

p(k)

+ (-l)%Zx(ka) --+ p(k)(x) ®A a + (-l)%Zx ®A (ka), V-I p(k)(x) ®A A + (-l)%Zx ®A Opo(k)(>'),

x ®A A

--+ v- 109 p (k)ov

x ®A a

--+ p(k)(x) ®A v- 109 p (k)ov

a + (-l)%Zx ®A (ka),

and moreover, Vp EN, Ap E AP(L, A) :

182

5. Z2-Graded Lie-Cartan Pairs

X~Aa---+O, x~Aa---+O,

pov

6p ov

with the notation Px == p(x), which explicitly means: A1(L,E) 3 Px: L 3 k---+(-l)XZ p(k)(x) E E. de!

(5.6.15.7) Consider the graded A*(L,A)-right module A*(L,E) over K. For k E L, ik, Lk := Opo(k), Po(k), Oo(k), 150, and d are super-derivations of A*(L, A). Accordingly, the corresponding K-endomorphisrns of A*(L, E) are module-derivations of A*(L, E), in the sense of its total grading. "Ix, Z E Z2, 'Vp, q E N, 'V>'p E (AP(L,E))~,J-Lq E Aq(L,A),x E Ef,b E A,k E LZ:

+ (_l)(l+z)(x+ p)>'pi(k)(J-Lq) , Oo(k)(ApJ-Lq) = Oo(k)(>'p)J-Lq + (-l)z(x+ p)>'pOo(k)(J-Lq), p(k)(>'pJ-Lq) = p(k)(>'p)J-Lq + (_l)z(x+ p)>'pPo(k)(J-Lq) , op(k) (ApJ-Lq) = 0p( k)( >'p)J-Lq + (-1 )z(x+p)ApOPo (k) (J-Lq) , t5o(>'pJ-Lq) = t5o(>'p)J-Lq + (_l)x+ p>'pt5o(J-Lq), p(>'pJ-Lq) = p(Ap)J-Lq + (-l)x+P>'ppo(J-Lq), i(k)(ApJ-Lq) = i(k)(>'p)J-Lq

t5p(ApJ-Lq) = t5p(>'p)J-Lq + (-l)X+ PApt5 po (J-Lq), i(k)(>'pb) = i(k)(>'p)b, Oo(k)(>'pb) = Oo(k)(>'p)b, p(k)(>'pb) = p(k)(>'p)b + (_l)z(x+ p)>'p(kb) , Op(k)(>'pb)

= Op(k)(>'p)b + (_l)z(x+p) >'p(kb) ,

t5o(>'pb) = 150 (>'p)b, p(>'pb) =p(>'p)b+(-l)x+P>'p(db), t5p(>'pb) = t5 p(>'p)b + (-l)x+ P>'p(db), with the total grading of A*(L, E), a>.p = x

+ p,

i(k)(xb) = 0, Oo(k)(xb) = 0, t5o(xb) = 0, Op(k)(xb) = p(k)(xb) = p(k)(x)b + (-l) ZX x(kb), Pxb = Pxb+ (-l)Xx(db), i(k)(xl-L q) = (-l)(l+z)xxi(k)(J-Lq), Oo(k)(xJ-Lq) = (-l)ZX x Oo(k)(J-Lq), p(k)(xJ-Lq) = p(k)(x)J-Lq + (-1 )ZXxpo(k)(J-Lq), Op(k)(xJ-Lq) = p(k )(x)J-Lq + (-1 )ZXxOpo (k )(J-Lq) , t5o(XJ-Lq) = (-l)X xt5o(J-Lq), p(XJ-Lq) t5p(XJ-Lq) = PxJ-Lq + (-l)Xx(dJ-Lq).

= PxJ-Lq + (-1)x xPo (J-Lq),

5.7 Real Z2-Graded Differential Forms

183

These module-derivations are easily derived from the corresponding K-endomorphisms of E ®A A*(L, A), using the fact that 'Vx E E, 'V1l, 11' E A*(L, A) :

= (XIl)Il' ----+(x ®A 11)11' = x ®A (11 /\ 11')· V-I

x(1l /\ 11')

5.1 Real Z2-Graded Differential Forms (5.7.1) Consider the alternating algebra

E9 AP(E)

A(E):=

pENo

of the vector space E over a field K, char K f: 2. For PEN, any two multiindices {k 1 , ... , kp}, {ll' ... , lp}, each of them consisting of pairwise different indices E I, card I := dim E, are called equivalent, if and only if each of them is some permutation of the other one. Selecting one representative from every equivalence class, one obtains the set i p . For instance, in case of

i p := {{k 1 , ... ,kp};I::; k 1 < ... < k p ::; n},

dim E = n E N,

If {Ok;k E I} is a basis of E, then {Ok l basis of AP(E), for pEN.

/\ •••

/\Ok p ; {k 1 , ... ,kp }

E

l::;p::; n.

ip}

is some

(5.7.1.1) 'Vz E A(E) : z = "--'\J r~e +

~ L...J

Ck 1· .. kpOk k Ok 1··· kp := Oki /\ ... /\ Ok l ' " p' p '

{kl ..... kp}Eip,PEN

with the unique coefficients co, Ckl ... k p E K, and only finitely many coefficients f: O. Here coe may be called the body, and z - coe the soul of z E A(E). An explicit notation of the unit e of A(E) is conveniently suppressed. (5. 7.1.2) Obviously, dim E = n

=}

/\n+l(z - CO) =

o.

Moreover, in any case, 'VZ E A(E) : :3r EN: /\r(z -

eo)

=

o.

(5.7.1.3) Therefore one finds, that 'Vz = CO

+ soul

E A(E) : CO

f: 0 o¢::=} :3 unique

y E A(E) : z /\ y = eK;

here eK denotes the unit of K. In this case COy = eK

+L kEN

k

/\ (z ~ CO) , and then obviously z /\ y = eK. eo

184

5. Z2-Graded Lie-Cartan Pairs

(5.7.1.4)

° °

dim E = n EN=? Vz E A(E) : rVlk : OkZ = dim E ¢ N =? Vz E A(E) : rVk E I : OkZ =

=? z =

=?

Z=

Cl...nOl... n

1·

01-

(5.7.1.5) The alternating algebra A(E) is graded-commutative, with the Z2grading according to the direct sum

A(E) =

E9 AZ(E),

E9

AZ(E):=

zEZ2

AP(E),

pENonz

hence being compatible with its No-grading.

(5.7.1.6) Let K be equipped with an involution, Le., an idempotent is0morphism of fields: K 3 C +-----+ c· E K, and accordingly A(E) with the Kantilinear star operation: A(E) 3 Z +-----+ z· E A(E). Vz, y E A(E), Vc, d E K: (CZ

+ dy)*

= c· z·

+ d·y·, (z /\ y).

Assume that Vk E I : Ok

V{k 1, ... , kp }

E

ip

:

= y. /\ z·, (z·)* = z.

= Ok; then

0kl ... k p

= Ok

p /\ ••• /\

Okl

= (-l)~Okl ... kp.

(5.7.1.7) This sign sometimes being rather inconvenient, one might prefer some graded star operation, such that Ve,il E Z2,VZ E A(E),y E AiI(E),Vc,d E K:

(cz + dy)· = c· z·

+ d·y·, (z /\ y)*

= (_l)e'ly. /\ z·, (z·)* = z,

and z· E A(E). Furthermore, again assume Vk E I : Ok = Vp E N, {k ll ... ,kp } E i p : 0'k. ... kp =

Ok;

then

Okl ...k p '

and therefore Vz E A(E):

z = z· <==> here

eo,Ckl ... k p

reo

=~, Vp EN, {k ll ... , kp } E i p

: Ckl ... k p

= ckl ... kp 1;

denote the unique components of z.

(5.7.1.8) Appropriate odd derivations of A(E) are defined. Vk E I, {k 1 , ... , k p } E

{) 0 {)Ok k1 ...k p {) {)Ok

e := 0,

._ ·-

ip :

{O, k ¢ {k

1,

(-1)1+10kl /\

,

kp } /\ Okl_ l /\ Okl+ l /\ ... /\ Ok p ' k = kl '

5.7 Real Z2-Graded Differential Forms

*

and the unique K-linear extensions, V~ E Z2, Vz E Ae(E), y E A(E), Vk E I:

8 8()k (z

185

E Derk(A(E» C EndK(A(E)).

_ 8z e 8y 8()k t\ Y + (-1) z t\ 8th'

t\ y) -

8 Vk, lEI: 8()k

8 0

8

8

+ 8()l 0 8()k

8()l

= 0,

within the Lie superalgebra DerK(A(E». (5.7.2) Consider a real m-dimensional Coo-manifold M, and take K := R. The map I : M --+ A(E) is called infinitely differentiable, if and only if its components are smooth scalar fields: M ~q

f

L

lo(q)e +

!kl ... k"(q)()kl ...k,,

E

A(E),

{k1 •...• k"}Ef,,.pEN

with 10 E Coo(M), Vp EN, {k ll ... , k p } E I p : Ikl ...k" E Coo(M). The unit e of A(E) shall be suppressed for convenience. (5.7.2.1) The real vector space Am.E(M) of smooth mappings

I :M

--+

A(E), dim E

=: €,

is an associative superalgebra over R, the unit of which shows the image {I}. Moreover Am •E (M) is graded-commutative. I

(j

Am.E(M) = Am.E(M) EB Am.E(M), Vz E Z2: A~.E(M):= {f E Am.E(M); 1m I ~ AZ(E)}.

"Ii,Y E Z2,VI E A~.E(M),g E A~.E(M),Vq E M: (f t\ g)(q) := I(q) t\ g(q) = (_l)ZII(g t\ I)(q)· (5.7.2.2) VI E Am.E(M), VX E T(M),Vk E I,Vq EM:

Lxl(q)

:=

Lx/o(q)

+

L

Lx/{k}(q)(){k}'

{k}Ef"

8

{j(} f(q):= k

L.

8

f{k}(q){j(}(){k}' k

{k}EI"

Here obviously Lx and

-ik are super-derivations of Am.E(M), Le., .

0

8

I

"IX E T(M), Vk E I . Lx E DerK(Am.E(M», 8()k E DerK(Am.E(M». The corresponding generalization of differential calculus is an easy application of the concept of sheaves over a topological space.

5. Z2-Graded Lie-Cartan Pairs

186

(5.7.3) Let T be a topological space. T is called ringed space, if and only if there are mappings: topology

T

of T 3 V

{V, WET; V ~ W}

--+ --+

ring Rv, ring - homomorphism ptr : Rw

--+

Rv ,

such that the following properties hold: p~ = id Rv.

(i)

VV E

(ii)

VU, V, WET: U ~ V ~ W ~ p~

T :

0

ptr

= p~.

Vo: E A : Vo E T, UOE~ Vo =: V } I,g E Rv, Vo: E A: PvJf) = p~Jg) ~

(iii)

I _

- g.

Vo: E A: Vo E T, UoEA Vo =: V, 10 E Vo } v. v. VO:,(3 E A: Vo nV/3:f. 0 ~ Pv:nvfj(fo) = pJ:,nvfj(f/3)

(iv)

~

31 E Rv : Vo: E A: 10 =

P~Jf)·

Here A denotes a set. The family {Rv; VET} is then called sheaf over the topological space T. (5.7.3.1) Some examples of sheaves over an m-dimensional real Coo-manifold M are presented. T := M. T

3

V

--+

Coo(V), ptr : Coo(W) 3 I

T

3 V

--+

Am,€(V), with the restrictions of components.

T

3 V

--+

E(V), with the restrictions of smooth differential forms.

--+

Ilv E coo(V).

(5.7.4) Consider again the above defined manifold M, with an atlas {
(5.7.4.1) Obviously A mn is some free bimodule over the commutative ring coo(V), with the basis {Ok1 ...kp ; 1 ~ k 1 < ... < k p ~ n} U {eA n }' (5.7.4.2) On the other hand, A mn is some graded An-bimodule over R, and moreover A mn is some graded-commutative associative superalgebra over R. (5.7.4.3) An

¢9

Coo(V) 3 z

¢9

10

+-----+ real-linear

zlo

E A mn .

5.7 Real Z2-Graded Differential Forms

187

(5.1.4.4) Obviously the following sequence of real vector spaces is exact:

{O}

-+

A~~.

-+

InclusIon

Amn

-+ proJection

Coo(V)

-+

{O}.

Here A~~ denotes the ideal of nilpotent elements of A mn = Coo (V) EI1 A~~. Moreover A~~ is just the intersection of all the ideals J of A mn such that J

2 A~n'

(5.1.4.5)

a

axl

~i,k,~j,l:

a

(j

E

DerR(Amn ), ath

I

E

DerR(Amn ),

a a a a ax ax ax ax ' a a a a a a a a -0-+-0-=0, -0 ---0-=0. ath aO i aO i aOk ax l aOk aOk ax l

-0 -0 -= l0 l -j j

Here 7fxr, l = 1, ... , m, denote the usual Lie derivations with respect to the natural basis on the chart
(5.1.4.6) As left modules:

al;l = 1, ... {a x

,m}

{ aaO k ; k = 1, ... , n}

-+ free over

ex (V)

-+ free over

An

DerR(Coo(V)),

DerR(A n ),

with the corresponding module-multiplications defined quite naturally.

(5.1.4.1)

DerR(A mn ) = DerAn (A mn ) EI1 Dereoc(V) (A mn ), DerAn(A mn ):= {8 E DerR(A mn );8IAn = O}, Dereoc(V)(Amn) := {8

E

Der R (A mn );8Ie x (V) = O},

with an obvious suppression of the notation of embeddings.

(5.1.4.8) With the module-multiplication defined, such that

l//,g E A mn ,l/8 E DerR(Amn ): (/8)(g):= 18(g), one constructs the free left module DerR(Amn ) over A mn , with the basis:

;k=I, ... ,n;l=I, ... { aal'a: x k

,m}

-+ free over

A mn

DerR(A mn ).

188

5. Z2-Graded Lie-Cartan Pairs

(5.7.4.9) The Z2-graded Lie-Cartan pair {DerR(A mn ), A mn } is established with the real-bilinear mappings, such that Vf,g E A mn , "10 E DerR(A mn ) : of := o(f), (fo)(g) := fo(g).

(5.7.4.10) The elements of the graded-commutative associative superalgebra A*(D mn , A mn ), D mn := DerR(A mn ), are called Z2-graded differential forms. With respect to the generalized exterior derivation d defined previously for Z2-graded Lie-Cartan pairs, and with the total grading, both A* (D mn , A mn ) and A (D mn , A mn ) are bigraded differential algebras over R. Moreover these algebras are graded Amn-Ieft modules over R, with the module-multiplication defined such that "If E A mn , Vp EN, >. E AP(D mn , A mn ), VOt. ... ,op E D mn :

A",n

(5.7.4.11) Every element>. E A~",n (D mn , A mn ) is uniquely determined by its values on the basis {ik,~; k = 1, ... , nj l = 1, ... , dx

dx

l

(a~i )

l

(a~i) = dOk (a~i)

=

Oli e A",n' dO k

(a~i) =

m}. 'V'{k, i, 'V'{'l,j :

-Oki e A",n '

= 0,

l

dx E (A~",JDmn,Amn))I, dOk E (A~",JDmn,Amn))(j,

with respect to the total grading, and therefore

l

dx /\ dx i = -dxi /\ dx l , dOk /\ dO i = dOi /\ dOk, dx l /\ dOk = dOk /\ dx l .

(5.7.4.12) As some free Amn-Ieft module:

l

{dx , dOk; k

= 1, ... ,n; l = 1, ... ,m}

--4

A~",n (D mn , A mn ).

(5.7.4.13) With the generalized exterior derivation used above, such that Vx,y E Z2, "If E A~n'o E D~n : df(o) := (-l)X ll o(f),

one immediately finds, that

5.8 Cohomologies of Z2-Graded Lie-Cartan Pairs

189

(5.7.4.14) As free Amn-Ieft module, AA",n (D mn , A mn ) is generated by the basis 1\ .. 1\ dOk { eA"In' dxll 1\ ... 1\ dx l • 'dOk 1 r' l ll dx 1\ ... 1\ dx • 1\ dOk I 1\ ... 1\ dOk r'. 0

1 ~ k1 ~

...

~ kr ~

n, 1 ~ II < .0. < ls

~

m}.

(5.7.4.15) The so-called Berezin integral may be defined as the monomial of odd derivations

!3°0

v

0···0

n

!3°0 : Amn

V

1

3

f

---+ !Loon E COO(V)

---+. embeddmg

A mn .

5.8 Cohomologies of Z2-Graded Lie-Carlan Pairs (5.8.1) Consider the Z2-graded Lie-Cartan pair {L, A} over a field K, char K 1:. 2. The odd derivation d of A*(L,A), with respect to the total grading, explicitly reads, Vp ~ 1,Vx,fj,zO,ZI,""Zp E Z2,Vko E Lzo,o .. ,kp E LZp, V>'p E (AP(L, A))ii, Va E Af : da(k o ) := (-1) XZO koa,

(d>'p)(k o,k1, ... ,kp)

L

(-1 )i+j+(z'+Zj)(zo+oo+z._tl+Zj (z'+I +oo+Zj-tl

O=:;i<j=:;p

>'p([ki , kj ], ko,···, ~i'" 0' ~j"'" kp) p

\ (k 0,·· L..J ) oI\p + "(_1)j+Zj(lI+zo+··+Zj-tlk·

IJ,

0 , ", j' .. 0'

kp) ,

j=O

(d>'I)(k o, k1) = ->'1 ([k o, kID

+ (-1)ZOllko>'l(kd -

(-1)ZI(II+ZO)k 1>'I(ko).

The super-derivations ik o and Lko of A*(L, A) explicitly read, with the same notation of degrees as above, (iko(>'p+d)(k1, ... , kp) = (_1)(p+l+II)zo >'p+l(ko, k1, ... , kp), ik o(.X 1 )

= (-1)(l+II)zo>'l(ko),iko(a) = 0,

(Lko(>'p))(kl,"" kp) P

z zl+oo+Zj-tl\ (k -_(_1)1+p O"(_1)zo(lI+ L..J oI\p 1,· .. , k·)-It [k 0, k·] ) ' k·)+It· .. , k) P j=1

+ (-1 )PZO ko>'p(k 1, ... , kp),

190

5. Z2-Graded Lie-Cartan Pairs

(Lko(>\d)(kd = (-1)l+zo+ZOIlAl([ko,k 1]) + (-l)ZOkoAl(kl), Lko(a) = koa. (5.8.2) As indicated previously, K - lin span({d,Lk,ikjk E L}) is some subalgebra of the Lie superalgebra DerK(A$(L, A», with respect to the total grading of A$(L, A). (5.8.3) The elements of A$(L, A) are called A-cochains of L, those of ker d are called A-cocycles of L, and those of 1m d are called A-coboundaries of L. Obviously L k turns A-cocycles into A-coboundaries of L, because Lk = [d, ik] for k E L. (5.8.4) The cohomologies of the Z2-graded Lie-Cartan pair {L,A} are defined with respect to d, according to the complex of vector spaces over K:

... {O} d=i A 7 A1(L, A) -----. ... -----. AP(L, A) ~ AP+1(L, A) -----. .... Vp E Z: HP(L, A) := ker dPj 1m dP- l .

(5.8.4.1) The Z2-graded Lie-Cartan pair {L, A} is degenerate, if and only if HO(L, A) = Aj{O}. (5.8.5) More generally, one defines the cohomologies H~(L,E):= ker 0~j1m O~-l, p E Z,

with respect to the complex of vector spaces AP(L, E) over K, defining

AO(L, E) := E, Vq ~ -1 : Aq(L, E) := {O}, and correspondingly the natural restrictions o~ : AP(L, E) -----. AP+l (L, E) of Here the E-connection p is assumed to be flat, such that op 0 op = o.

op.

(5.8.5.1) Obviously Op(k) turns E-cocycles into E-coboundaries, generalizing straightforward the notation from the case of trivial Z2-grading. (5.8.5.2) In the case of p := 0, H8(L, E)

HJ(L,E)

+--->

+--->

E,

{Al E Al(L, E)jVk}, k2 E L: Al([k l ,k2 ]) = O}.

(5.8.6) Especially choose A := K, and moreover, Va E A, k E L : ka := O. With the module-multiplication of L inserted for the K-bilinear map: A x L -----. L, the defining conditions on Z2-graded Lie-Cartan pairs are reduced to the zero-map: L x A -----. {O} -----. A. Then every E-connection p is local. In case of p := 0, one finds the E-valued Chevalley cohomologies H&(L, E) with respect to 0o. In the special case of E := A := K, 0o = d.

6. Real Lie-Hopf Superalgebras

Several concepts have been developed in order to construct fermionic analogues of finite-dimensional real differentiable manifolds and real Lie groups, aiming at the description of physical systems with both bosonic and fermionic degrees of freedom, by means of additional nilpotent odd coordinates. The approach by B. DeWitt (1984) and A. Rogers (1986), the crucial point of which concerns the notion of superdifferentiable functions of countably many Grassmann variables, does not fit into a suitable axiomatic scheme of a theory of supermanifolds, as it was proposed on a sheaf-theoretic level by M. J. Rothstein (1986). The first attempts to construct some consistent framework of supergeometry are due to F. A. Berezin and D. A. Leites (1975), D. A. Leites (1980), and B. Kostant (1977), replacing the sheaf of commutative unital associative algebras of real-valued smooth scalar fields by the sheaf of graded-commutative unital associative algebras of smooth functions on the domains of charts of an atlas, with their values in the exterior algebra over some finite-dimensional real vector space. This sheaf-theoretic approach is performed in the detailed presentations of a theory of supermanifolds by Yu. I. Manin (1988), C. Bartocci, U. Bruzzo, D. Hermindez-Ruiperez (1991), and F. Constantinescu, H. F. de Groote (1994). There is up to now no satisfactory answer to the crucial question: What is a Lie supergroup? The concept of real Lie-Hopf superalgebras, which was developed by B. Kostant (1977), may serve for a precise version of the heuristic attempts by V. Rittenberg and M. Scheunert (1978). Lie superalgebras and Lie supergroups of supermatrices are treated extensively by J. F. Cornwell ((1989). The so-called classical Lie supergroups are classified by H. Boseck (1990) by means of his theory of affine Lie supergroups (1987). Another proposal to define a Lie supergroup and its action on a supermanifold can be found in the above-mentioned monograph by C. Bartocci et al. (1991). For the classical theory of Lie groups and their Lie algebras the reader is referred to the corresponding volumes by N. Bourbaki (1989, 1968), J. Dieudonne (1974), and for instance to the volume edited by A. L. Onishchik (1993). The underlying theory of differentiable manifolds is presented systematically by N. Bourbaki (1983), J. Dieudonne (1972, 1974), and also in several

192

6. Real Lie-Hopf Superalgebras

monographs, for instance that by S. Lang (1985), and in three volumes by W. Greub, St. Halperin, R. Vanstone (1972, 1973, 1976). Infinite-dimensional real differentiable manifolds are treated for example in the monograph by R. Abraham, J. E. Marsden, T. Ratiu (1983). Z2-graded distributions with finite support on a real Lie group are presented here as an instructive example of real Lie-Hopf superalgebras, and in particular of dualizing topological Hopf superalgebras. The latter aspect is one of the categorial sources of the theory of quantum groups.

6.1 Linear Forms on Graded-Commutative Associative Superalgebras (6.1.1) Let A be a real graded-commutative associative superalgebra with the unit eA, and consider the dual vector spaces

A*:= Homa(A,R), (A®A)*:= Homa(A®A,R), with the natural Z2-grading of A *. Due to the universal property of tensor products, "If, g E A* : 3¢> E (A ® A)* : Va, bE A :

L

¢>(a ® b) =

(-IYll f(aX)gY(b).

~,yEZ2

One then establishes an injection A*xA*

1/,

according to the subsequent diagram.

- - - - - - - - - - . . . A *@A*

w {f,g} - - - - - ) , tP

def

E

(A@A)*

J.

(6.1.2) The definition of a real-linear map: A* 3 f indicated in the next diagram.

A xA

~

J1(f) E (A ® A)* is

- - - - - - - - - - - . . . A@A

w

{a,b} _ _ _....,........,__), f(ab) def

E

R

~A(f)

Linear Forms on Graded-Commutative Associative Superalgebras

193

(6.1.3) Let J be a graded ideal of A. Then (A ~ J) + (J ~ A) is some graded ideal of A~A, and the following isomorphism of real unital associative superalgebras is established: A~A ['''] ['] ["] A A (A~J)+(J~A)3a~a +-----+a ~a EJ~]' inserting the tensor product of real associative algebras, or else the skewsymmetric tensor product of real associative superalgebras. (6.1.4) One easiliy finds the real-linear bijection:

due to the next diagram.

A

f

-------'-----_>

R

L AlJ-_f (6.1.5) Because of the implications, that

"If E A$ : fl J

= 0

<===> .1(f)I(A®J)+(J~M)

= 0,

one is led to the real-linear injective map:

with the aid of the subsequent diagram, which is commutative "If E Aj.

A <8lA

_ _ _ _ _ _ _L1---.:(~f)'____

L

A <8lA (A <8lJ) + ( J <8lA)

• R

AlJ

~M/J~

(6.1.6) Let especially AI J be finite-dimensional, such that one can make use of the real-linear bijection: A)$ ( -J

~

(A)$ J

3~'IjJ+-----+{E de!

(A A)$ -~J J

194

6. Real Lie-Hopf Superalgebras

A

A

J ® J 3 [a] ® [b]

e

¢([a])tP([b]) E R.

In this case one defines an appropriate comultiplication .1J by means of the following real-linear mappings:

Aj

3

.1 J

:

f .-- j

(A)· J

3

E

(~

r'

k j d;j ~ ¢I ® tPI

E

(A)· (A)· J ' J

®

k

Va, bE A : f(ab) =

L ¢I ([a])tPl ([b]). 1=1

(6.1. 7) The comultiplication .1J defined above is indeed coassociative. For an easy proof one needs the bijective structure map of R, and the linear bijection:

((~r ® (~r) ® (~r 3 (¢®tP) ®~ .--

T(T(¢,tP),~) E HomR ((~ ®~) ®~, (R®R) ®R),

due to the finite dimension of AI J. (6.1.8) With the counit fJ:

(~r 3 ¢~¢([eA]) E R

one establishes the graded-cocommutative coalgebra (4)· over R, because of .1 J = T 0 .1 J , with respect to the natural Z2-gradings, and the Z2-graded flip

for homogeneous elements ¢, tP of degrees

x, ii,

respectively.

(6.1.9) Consider a homomorphism of real unital associative superalgebras ----+ AI, and the corresponding real-linear map a· : Ai ---+ A 2, such that

a : A2

6.2 Real Lie-Hopf Superalgebras

195

Let J I be a graded ideal of AI, the codimension dim Ad JI of which is finite. Then J 2 := O-I(JI ) is some graded ideal of A 2 with finite codimension, due to an injective homomorphism of real unital associative superalgebras:

A2 J

2

3

[a2] -- [o(a2)]

Al E

JI .

Ai : !lIJl = 0 ~ o*(!l)IJ2 = o. Hence one obtains some homomorphism 0co of Z2-graded coalgebras over R:

V!l

E

(~: r

Va2

E

(~:

3 ¢I -;:: ¢2 E

A 2 : ¢2([a2])

:=

r'

¢I([al]), al:= o(a2)·

Here one again uses the real-linear bijection:

("*)

(6.1.9.1) Especially take ¢l, 'ljJ1 E *, let ¢l be group-like, and 'ljJi primitive with respect to ¢I. Then oco(¢t} is also group-like, and Oco('ljJI) is primitive with respect to Oco(¢I).

6.2 Real Lie-Hopf Superalgebras (6.2.1) Let G be a real Lie group, and L = LO EB L I a real Lie superalgebra. Their smash product is constructed, Le., the real Hopf superalgebra {F"., j.t"., 7]"., ~"., f".}, F". := R(G) ®". E(L), with the representation 1r of G on E(L) indicated in the next diagram. -~>

n(g): E(L)

r

4 - -_ _.....

E(L)

1

Here an automorphism in the sense of real Lie superalgebras 1rL(g) is lifted to an automorphism in the sense of real unital associative superalgebras 1r(g), ascending from L to the universal enveloping superalgebra E(L). (6.2.2) The real Lie algebra of left-invariant vector fields on G is inserted for the even subspace of L, such that LO := L(G), dim L O =: mEN.

196

6. Real Lie-Hopf Superalgebras

(6.2.3) Let L 1 be finite-dimensional, dim L 1 =: n E N, dim L = m + n. The representation 1rL is assumed to be a homomorphism of real Lie groups, in the sense of the chart with respect to a homogeneous vector basis for L: A := AutR(L)

+---+

GL(m + n, R)

---+ R(m+n)2. open ~'Ub~et

+---+

Here AutR(L) denotes the real vector space of real-linear bijections of L onto itself. (6.2.4) The real Lie algebra L(A) of left-invariant vector fields on A is identified with the real endomorphisms of L itself, by means of the real-linear bijection (3 defined in the diagram below. L(A)

•

p

def

• EndR(L)

[

]

T;d L(A)

Mat(m+n, R)

bundle chart

Here 1id(L) (A) denotes the tangent space of A at the unit id(L) of A. The real-linear bijection (3 obviously does not depend on the choice of vector basis for L. (6.2.5) The real-linear map Q' defined in the subsequent diagram is used in order to write down the assumption (ii) below.

G

3

exp(X)

---~. lr L 0 exp(X) E Aut R (L)

exp

exp L(G)

1 3X

1

a ----:--::~~ a (X) def

E

L(A)

•

p. •

EndR (L)

] Te (G) G

3[U O J

(6.2.6) One then assumes that, on the connected component of the unit ea of G, 1rL is an exponentiation of the adjoint representation of L 0 , in the sense of (i) and (ii) below. (i)

'rIq E G, X E L O : 1rdq)(X) = (Ad q)(X) := p(q).(X).

6.2 Real Lie-Hopf Superalgebras

(ii)

"IX E LO, Y E L : ({3 a a(X»)(Y)

197

= [X, Yj = Lx(YO) + [X, y I ],

inserting the product of L, or equivalently, the super-commutator of E(L). Here Ad: G --+ AutR(L(G» denotes the adjoint representation of G, which is induced by the inner automorphism of real Lie groups p(q), the latter acting as:

G

3

q --+ p(q) : G

3

p--+qpq-l de!

E

G.

Moreover Lx denotes the Lie derivation with respect to X E LO. (6.2.7) A triple {G, L, 11'} with the properties assumed above is called real Lie-Hopf superalgebra. (6.2.8) Let {G k , Lk' lI'kl, k = 1,2, be objects of this type, with the corresponding real Hopf superalgebras, F1rk := R(Gk) ®1rk E(Lk), k = 1,2. A homomorphism of real Hopf superalgebras : F1r1 --+ F1r2 is called homomorphism of real Lie-Hopf superalgebras, if and only if there is a homomorphism "( : G 1 --+ G2 in the sense of Lie groups, such that Vql E G1,X 1 E L?: ql ®XI--+"((ql) ®X2, ql ®eEI --+"((ql) ®eE2' ~

~

according to the diagram below. Here one denotes L2 := L(Gk), k

= 1,2.

L(G ) 3~ 1

I

1

One easily verifies that this assumption is compatible with the previous ones. (6.2.8.1) One then finds, that "In E N, Vql E G I , VX u , ... , X In E L~:

ql ® (XU'" X In ) --+ "((qt} ® (X2I ... X 2n ). ~

(6.2.8.2) Especially is called an isomorphism in the sense of real Lie-Hopf superalgebras, if and only if and hence "( are bijective.

198

6. Real Lie-Hopf Superalgebras

6.3 Z2-Graded Distributions with Finite Support (6.3.1) Consider the real vector space Ajin == Ajin(G) of distributions with finite support on an m-dimensional real Lie group G. The commutative unital associative algebra of smooth real scalar fields is denoted by A:= COO(G), A*:= HomR(A,R). Ajin:=

{¢

E A*j:3p E N,:3ql,'" ,qp E G,

:3 coefficients

{CL . km E Rjkl,···km E No;r = 1, ... ,P}: -# 0,

only finitely many coefficients

L

VfEA:¢(f)=Y;

r-l kl, ... ,k m ENo

CL.. kmL~10"'OL~:(f)(qr)},

inserting a basis {Xl,'''' X m } of the real Lie algebra L(G) of left-invariant vector fields on G. The set supp ¢:= {ql, .. ' ,qp} is called support of ¢.

(6.3.2) Due to the Poincare-Birkhoff-Witt theorem, the set {X~l ... x~m;kl, ... ,km E No}

is some R-basis ofthe universal enveloping algebra E( L(G)). Correspondingly A:n(G) := R( {L\\

0 .. · 0

L~:; k l , ... , km E No})

is the universal enveloping algebra of the real Lie algebra of Lie derivations R( {LXk; k = 1, ... , m}), as subalgebra of EndR(COO(G).

(6.3.3) Denoting Vq E G: A; := R -lin span({¢ E Ajin; Supp ¢ = {q} }), one finds an isomorphism of unital associative real algebras: A;

(6.3.4) More generally one constructs an R-linear bijection 'Y: R(G) ® E(L(G») 3 q ® X~l ... X~m

¢: COO(G) 3 f

--+

de!

L~

1

0· .. 0

+--+

¢

L~m (f)(q) E R. m

E

Ajin,

+--+

E(L(G)).

6.3 Z2-Graded Distributions with Finite Support

199

(6.3.5) The universal property of tensor products implies existence of an R-linear map L1 : A*

-----+

(A ~ A)* := HomR(A ~ A, R),

such that

V/,g E A, V E A* : L1((/g). (6.3.6) V E Ajin : 3p E N, 3{i., 1: E Ajin; k = 1, ... ,p} : p

V/,g E A: (/g) =

L 1. (/);:(g). k=1

Hence one establishes an R-linear map

Here one just uses the fact that, that Lx is some derivation on Coo (G), for an arbitrary smooth vector field X. (6.3.7) Moreover defining €Jin : Ajin 3 -----+ (eA) E R, with the unit eA of Coo(G), deJ

one establishes the cocommutative real coalgebra {Ajin' L1 Jin , €Jin}. (6.3.8) The linear bijection 'Y is an isomorphism of real coalgebras. (6.3.9) The adjoint representation Ad : G -----+ AutR(L(G)) is inserted into the real coalgebra {F".,L1".,€".}. Vq E G,X E L(G): 1l'dq)(X) := p(q)*(X) E L(G). F". := R(G) ~". E(L(G)).

(6.3.10) The structure mappings /L".,T/". are switched from F". to Ajin' (6.3.10.1) The real-linear form

'Y(eG ~ eE(L(G») = 1 : Coo(G) 3/ -----+ /(eG) E R deJ

is used as unit of the real associative algebra Ajin'

200

6. Real Lie-Hopf Superalgebras

(6.3.10.2) Let

¢(f)

:=

L'5l1

1/;(f)

:=

f

E COO(G), and q,p E G, k b

0 ..• 0

·.·,

lm E No; define

L~: (f)(q) E R, 1/;(f):= L1 1

0··· 0

L~", (f)(p) E R.

Then ¢

0

L kXl 1

0 .. · 0

L kX"'~

0

L IXl ,1

0· .. 0

L~ (f)(qp) E R, ~

with the transformed invariant vector fields X~ :=

(Ad q)(Xk )

E

L(G), k

= 1, ...

,m.

(6.3.11) One obtains the real Hopf algebra {Ajin' Ilfin, 1/fin, Ll/ in , €/in}, with the antipode afin, such that Vq E G,X E L(G) :

{f(q)j f

E COO(G)}

-----..{f(q-l); f

E COO(G)},

U/in

{Lx f(ec)j f E COO(G)} -----..{ -Lx f(ec)j f E COO(G)}, t7/in

(6.3.12) Therefore Ajin is some real Lie-Hopf superalgebra, in the sense of an isomorphism of real Hopf algebras "y : R(G) @11' E(L(G» ~ Ajin' with the trivial Z2-grading. (6.3.13) More generally, consider the real graded-commutative associative superalgebra

Amn(G) :=

{f :G -----..

smooth

An}, dim G =: mEN,

of Coo-mappings from the Lie group G into the alternating algebra

The above considerations are applied to this case, inserting the universal enveloping superalgebras:

LO:=L(G), L 1 L

:= LO EEl

L1

.

:=R({a~kjk=I, ... -----..

unlversal envelope

,n}), E(Ll)~An,

E(LO)®E(L 1) ~ E(LO)®A n .

6.3 Z2-Graded Distributions with Finite Support

201

(6.3.14) Consider the restriction of A:'nn(G) := HomR(Amn(G),R), which is defined by the next diagram. A~n(G)

real-linear • (COO(G))·

•

I

real-linear

A ~~n (G)

• de'

@

An

I A,;nrG)

•

An

@

Hence one finds an according isomorphism of real Z2-graded Hopf algebras:

with the graded-commutative Z2-graded Hopf algebra An, and inserting the adjoint representation of G for 71"0, such that "Iq E G: 7I"O(q)IL(G) = Ad q.

The elements of A;j~n(G) may be called Z2-graded distributions with finite support. (6.3.15) With respect to an appropriate homomorphism in the next diagram, with X E L(G) and 0 E R ( {

71"

of groups defined

-dk, ... ,-10:: }), one

then establishes an according real Lie-Hopf superalgebra A;j~n(G), and an isomorphism of real Hopf superalgebras: A;j~n(G) ~ R(G) ®1r E(L).

G 3q

• n(q): E(L)

L

nL(qj:

E(L)

Lx+a

1

- . (A d q)(X) + a € L

Here an isomorphism of real Lie superalgebras 7I"L(q) is lifted to an isomorphism of real unital associative superalgebras 7I"(q), for q E G. (6.3.16) One therefore obtains the real-linear bijection: A;j~n(G) ~

R({L';l

0 .. · 0 1

1~

L~'" Iq '"

s

1

~ n, ~ II

0

(c + d~ 001

0··· 0

1

< ... < l5

~ nj

c No}),

~) q E G' dE R , 00 . ' " .

1

k 1 , ... , km E

202

6. Real Lie-Hopf Superalgebras

the images of which are operating on Amn(G) as An-valued real-linear forms with finite support. (6.3.16.1) Let>. E An, f>.. E COO(G), and f(~) := f>..(q)>. for q E G. Then the real-linear form ¢, which corresponds to LJl1 0 · · · 0 L~: Iq 0 a with a E E(L 1), k , ... , k E No, is acting onto f according to m

l

¢U) = L~l

0··· 0

L~:f>..(q)(a>.).

Here the above defined isomorphism of real Hopf superalgebras explicitly means: A;,{~n(G) 3 ¢

+-+

q ® L~l

0··' 0

L~:

0

a +-+

E

R(G) ®,. E(L).

6.4 Real Lie Supergroups as Real Lie-Hopf Superalgebras (6.4.1) Let Gk, k = 1,2, be groups, and consider a homomorphism of groups ¢ : G 2 --+ Aut(G1 ), into the group of automorphisms of G 1 . On the Cartesian product of sets G I X G 2 define the map: (G 1 x G 2)

X

(G 1 x G 2) 3{{XI,X2}, {YI,Y2}} --+ {Xl(¢(X2)(Yl)),X2Y2} E G l x G 2.

With this product the set G l x G 2 becomes some group, the so-called semidirect product, denoted by G 1 x4> G 2 . In the special case of 1m ¢ := {id Gil, one obtains the direct product of these two groups. The unit of G l x", G2 is {el,e2}' Furthermore,

'v'XI

E

GltX2

E

G2: {Xl,X2}-l = {¢(x2'l)(x 1l ),x2'I}.

Obviously {{xl,e2}jXl E Gil is an invariant subgroup, and moreover {{elt X2}j X2 E G 2} is some subgroup of GI x", G 2.

'v'Xl E G1,X2 E G 2 : {xl,e2}{el,x2} = {Xl,X2}' (6.4.1.1) Let Gk, k = 1,2, be topological groups, and assume that the map: G l x G 2 3 {XltX2} --+ ¢(x2)(xd E G l is continuous with respect to the procuct topology of G l x G 2 . Then G l x4> G 2 is some topological group, with respect to the product topology of G 1 x G 2 .

Real Lie Supergroups as Real Lie-Hopf Superalgebras

203

(6.4.1.2) Especially let Gk, k = 1,2, be real Lie groups. If the map: G 1 x G 2 3 {Xl,X2}

----+

¢(x2)(xd

E

G1

is real-analytic on the product manifold G 1 X G2, then G 1 x'" G2 is some real Lie group. . (6.4.1.2.1) The corresponding left-invariant vector fields are obviously related by the real-linear bijections:

L(G 1

x",

G 2) 3 X .....- [u] .....- {[1I"1 0 u], [11"2 0

un

2

.....- {X1 ,X2} E EBL(Gk)' k=l

inserting the equivalence classes of curves

The projections 1I"k : G 1 x", G 2 ----+ Gk, k = 1,2, are continuous and open. (6.4.2) For an easy definition of real Lie supergroups one starts from a real Lie-Hopf superalgebra {Go, L, 11"}, F1f := R(G o) ~1f E(L), LO:= L(Go).

(6.4.2.1) En := An®E(L), as some unital associative superalgebra over R. Consider the real vector space

As subalgebra of the real Lie superalgebra EnL, M n is some real Lie algebra; it is called Grassmann envelope of L. Now use the direct sum M n = (R ~ LO) EB N n ,

with an appropriately chosen supplement Nn , such that N n is some solvable ideal of M n . With the choice N n := {

L

(Bit 1\ ... 1\ BI.)

~ XI1 ... I.; XI1 ...I. ELf}

1:S1 1< ...
one obtains some isomorphism of real Lie algebras: Mn/Nn .....- LO. (6.4.2.2) Let G 1 be a real Lie group, with the real Lie algebra L(Gd = N n .

204

6. Real Lie-Hopf Superalgebras

(6.4.2.3) A homomorphism of groups from 11":

Go

:3

go ----. 1I"n(gO) : En

:3

A® X

1I"n :

Go ----. Aut(En ) is constructed

A ® 1I"(go) (X)

f-----4

de!

E

En,

into the group of automorphisms of En, the latter in the sense of unital associative superalgebras over R. (6.4.2.4) For go E Go and z E Z2, A; ® LZ, M n and N n are invariant with respect to 1I"n(gO). A homomorphism of groups v : Go ----. Aut(Nn ) is constructed from 11" L:

Go

:3 90 ----.

v(go) : N n

:3

A® X

f-----4

de!

A ® 1I"L(go)(X) E N n .

Here Aut(Nn ) denotes the group of automorphisms of N n , in the sense of real Lie algebras. (6.4.2.5) Let 0: : Go ----. Aut(GI) be a homomorphism of groups, such that conditions (i) and (ii) are fulfilled. (i) is expressed in the diagram below.

-1----....·

~ 3go - _ . a (go): G1

(i)

ex p L . -_ __;>

r

v (go): Nn

I push

G,

1e xp

forward

!

--------;>

Nn

(ii) The map

Go x G 1

:3

{go,gd ----. o:(go)(gI) E G l

is real-analytic. Then one defines the real Lie group G := G l may be called real Lie supergroup.

Xo:

Go, which

(6.4.2.6) dim M n

= P + dim

p:= dim Go

N n = 2n- l (p + q),

= dim LO E N, q:= dim

L 1 EN.

(6.4.2.7) The Lie algebra of G is just the subalgebra M n of EnL. Hence one establishes an isomorphism of real Lie algebras:

L(G l

Xo:

Go)

f-----4

(A~ ® LO) $ (A~ ® L 1)

This theorem is proved subsequently.

----.

subalgebra

(A n 0E(L O $ L 1»L.

Real Lie Supergroups as Real Lie-Hopf Superalgebras

205

(6.4.2.7.1)

L(G)

..----..

real-linear

L(Go) $ L(G 1 )

..----..

real-linear

Mn ·

(6.4.2.7.2) The semidirect product with respect to VXo E LO,X1 E Nn,t E R:

Q'

implies, that

exp o(v 0 exp(Xo))(tX1 ) = exp(Xo) exp(tX1 ) exp( -Xo) = exp 0 Ad exp(Xo)(tXd. (6.4.2.7.3) VXo E LO,'x®Y E N n

Ad exp(Xo) (,X ® Y) = ,X ®

:

(1rL

0

exp(Xo))(Y).

(6.4.2.7.4) One then finds the diagram below.

Go

>

Aut (L) -

Aut R(L)

rexp K,

jex p

- - n --:-"'--d:-e-=-'--+> L(Aut R(L)) L

Here VXo E LO :

iJL 0

1l'~(Xo)

:L 3 Y

---+

[Xo, YJ,

inserting the product in the sense of L, due to the assumptions on the context of real Lie-Hopf superalgebras.

1r,

within

(6.4.2.7.5) Consider the adjoint representation

Ad; G

---+

AutR(L(G)) =: A,

and the exponential map in the subsequent diagram.

G

------~~

p

A

L(G) --p--:-,-d--;"'e----:'-· L( A) _.---"---::f3:--~' End R ( L( G))

VX,X' E L(G): (iJ 0 Ad'(X))(X') = [X,X'],with the product of L(G).

206

6. Real Lie-Hopf Superalgebras

(6.4.2.7.6) VXo E LO, A ~ Y E N n : (expoAd'(XO))(A ~ Y) = A ~ (expo1l'~(Xo))(Y), and therefore «(30 Ad'(XO))(A ~ Y)

= A~ «(3L

0

1l'~(Xo))(Y).

(6.4.2.7.7) Hence one finds, that the products of L(G), and of the subalgebra M n of EnL, coincide. (6.4.2.8) One thereby obtains the local diffeomorphism: L(G).

+--.

Inclunon

Vo 3 X.-- exp(X) E Ue . ~ G, .nclu~.on

of an open connected neighbourhood t'o of 0 E L(G), onto an open connected neighbourhood of the unit ec == e E G. Using basis vectors XOk of LO, and X 11 of L I, one then writes Vo 3 X

=

t(

+t( 1=1

L

e:J +

k-l

ct.l/Jll A 00. fdJlr )

l~ll <...
_dL.lr8hAoo'A8lr)~XlI'

L

l~ll<

~ X Ok

...
with the canonical coordinates of the first kind ~,ct.lr' dL.lr E real open intervals containing zero.

6.5 Linear Supergroups (6.5.1) Real or complex supermatrices are defined conveniently as special endomorphisms of real or complex supervectors. W:= An ~V, V:= VO EBV I , Vi'i:=K~, VI:=K t , s,tENo, s+t2:1,

An:=A(E), E:=K({8 1 ,oo.,8n }), nEN, K:=RorC. The elements of this real or complex vector space Ware written as columns, with the Z2-grading due to those of An and V, and called supervectors over the field K.

W = WO EB WI 3

W

= [

Z.l

Z~~t

] '

6.5 Linear Supergroups

V~+tI : Zl = Cl,o

I:

+ l~kl

207

Cl,k1 ...k/h 1 /\ ... /\ Bk p E An.

< ...
\.JZ. WE {:::::::}vI \.J8k 'V \.Js+t I . AE Az+I vzE 2·wE . 8 +1 .ZkE.l1n,ZIE.lln

EndK(W)

+---::--+ K-lmear

EndK(An)®EndK(V):) An®EndK(V) "!'

=: D,

using only An-multiples of the unit of EndK(A n ). Then

[A B]

I D i 3M= CD' D=D (j $D,

with the elements akl E A, . .. ,dij E D, such that V~ k, I, Vii, j :

akl

E

AE

.lIn'

bkj

E

d+z

.lIn

,Cil

E

AI+z

.lIn

,

dij

E

AE

.lIn'

Here and henceforth the above K-linear bijection is suppressed. With this Z2-grading, D becomes an associative superalgebra over K, with the unit eD == I := id W, the elements of which are called supermatrices over K.

(6.5.2) An appropriate notation of these tensor products is explicated for s=t=n=1. Vao, aI, boo, ... ,b l1 ,Vo, VI E K : (ao+ Ba l)0 [vo] = [(ao+Badvo] , VI (ao - Badvi

(ao + Bad 0 [boo bOI ] = [( ao + Badboo (ao - Bal)blO blO bl1 Vx, fi

E

Z2, VA

E

An, J.L

E

(ao + Badbol ] . (ao - Bal)bl1

A~, B E En4(V), V E V :

yielding an appropriate matrix multiplication of elements E D, W.

(6.5.3) Both Wand D can be established as graded An-modules over K. For instance, for s = t = 1, n = 2, Vvo, VI, boo, ... ,b ll E K,

A [vo] := A 0 [vo] = [(aD + alB I + a2 B2 + aI2 BI/\ ( 2)Vo] , VI VI (ao - alB I - a2 B2 + al2BI /\ (2)VI A [boo bOI ] := A 0 [boo bOl ]

blO bl1 blO bl1 (ao + al BI + a2 B2 + al2 Bl2)boo - [ (ao - alB I - a2 B2 + al2Bl2)b lO

(ao + al BI + a2 B2 + al2Bl2)bol ] (ao - alB I - a2 B2 + al2Bl2)b l1

'

208

6. Real Lie-Hopf Superalgebras

denoting (}12 == (}l 1\ (}2' More generally, with the isomorphism of unital associative K-algebras:

+ al (}l + a2(}2 + a12(}1 1\ (}2 f-----+ ao - al(}l - a2(}2 + a12(}1 1\ (}2 =: 5. E

A2 3 A =

ao

A2 ,

one defines VA E A2 ,w E W,M ED:

AW

:=

WA:=

[~ ~] w,

L

AM:=

[~ ~] M,

(_l)'ryAxWY, MA:=

L

(_I)X YAXMti = M

[~~] ,

x,yEZ2

x,yEZ~

with the coefficients A from the left and right, respectively.

(6.5.4) With an appropriate notation of matrix elements, such that akl

=

akl,O

L

+

V1 k, l

:

akl,k, ... kp(}k, 1\ ... 1\ (}k p ,

l~k,< ...
A o :=

[akl,Oj

k, l = 1, ... ,s],

etc., one finds the following lemma. VM ED: f3N ED: MN = N M = I {::::::} det (AoDo) '"

01.

In the case of M being invertible, N =: M- l is obviously unique.

(6.5.5) The supertrace str E HomK(D, An) is defined, such that Vi E Z2,M E D Z : str M:= tr A- (-I)Ztr D E A;.

+ M 1 ED: str M:= str MO + str M 1 E An. Vx,fi E Z2,VM E DX,N E DY: str(MN) = (-I)XY str(NM). VM = MO

(6.5.6) The superdeterminant is defined, such that VM E DO : det(AoDo) '" 0 =}

sdet M:=

det(A - BD-1C) det A det D = det(D _ CA-1B) E

A~.

VM,M' E DO: det(AoDoA~Db) '" 0 = } sdet(MM') = (sdet M)(sdet M'). _ 00 Mk _ VM E DO: exp(M):= kf E DO, exp(M)exp(-M) = I,

L

k=O

sdet 0 exp(M) = exp ostr (M).

6.5 Linear Supergroups

209

(6.5.7) With the norm, such that

Vz E An: Ilzll

:=

L

Izol +

IZ k 1 ••• k p l,

l$k 1 < ...
An becomes some Banach algebra, the topology of which is equivalent to the n natural one of K(2 ).

(6.5.8) Consider the group

G:= GLn(s,t,K):= {M E DOjdet(AoDo) =I- O} of even invertible real or complex supermatrices, where s, t, n E N are used as above. This so-called general linear supergroup G is established as the real Lie-Hopf superalgebra {Go, L, 1r}, such that G = G 1 Xo: Go, with the notation of the preceding chapter. Note here that also in the case of K := C the charts of the real-analytic manifold G are real-valued. (6.5.8.1) Go := { [

LO:=

~b~]

{[;~]

E

M at(s + t, K)j det(AoDo) =I- O} ,

jX E Mat(s,K), Y E Mat(t,K)}

= L(Go).

(6.5.8.2) G 1 := real Lie group of real or complex supermatrices M 1 :=

Is

L

+

L

lh1 ...lrAl1 ... lr

1$1 1 < ...
L

[

(}1 1 ••• l r

(}11 .. .lrBl1 ...l r ]

1$1 1 < ...
Cz 1 ••• l r

It

L

+

1$1 1 < ...
(}1 1...l r Dl 1... l r

1$1 1 < ...
with real or complex matrices Al 1...l r , ... , Dl 1...l r of appropriate dimensions, hereby suppressing an explicit notation of the skew-symmetric tensor product. Then Gl is an invariant subgroup of G. 1

L :=

{[i~ i~] E Mat(s + t,K)jX

L = LO (f) L 1

+-----+

K-linear

Mat(s

+ t, K).

1

=

X4 =

o}.

210

6. Real Lie-Hopf Superalgebras

(6.5.8.3) The group homomorphism a : Go ceding chapter explicitly reads, such that

--+

Aut(Gd used in the pre-

VMo E Go: a(Mo ): G I 3 MI""'-MoMIMol E G I . de!

One then finds the semidirect product G = G I that

X

Q

Go of real Lie groups, such

"1M = MIMo, M' = MfM~ E G: MM' = Mla(Mo)(MDMoM~ E G. (6.5.8.4)

».,

VMo EGo: a(Mo) 0 exp = exp o(a(Mo

».

with (a(Mo = I/(go) E Aut(Nn ) in the sense of real Lie algebras, using the notation of the preceding chapter. VMo E Go :

a(Mo): G I 3 exp(.A0X)""'- Mo exp(>'0X)Mo l E G I , (a(Mo». : N n Here N n

=

3 M~X.....A

de!

A

Mo(M9X)Mo-

I

=

>'®(MoXMo ) E Nn · I

A

L(G I ) is the solvable ideal of M n defined in the preceding chapter.

(6.5.8.5) The corresponding real Lie-Hopf superalgebra {Go, L, 11"} is described by the representation 1I"L : Go --+ Aut(L). VMo EGo: 1I"L(Mo) : L 3 X .....- MoXMo l E L. The derivation of 1I"L at the unit of Go obviously yields the product of the real Lie superalgebra L. (6.5.8.6) The real Lie algebra L(G) .....- (A~ ® LO) ED (A~ ® L 1 )

= (R ® LO) ED N n

is some subalgebra of the real Lie superalgebra (A n 0E(L»L.

(6.5.8.7) In case of K := R, the dimensions of these real Lie algebras obviouslyare dim {il

= S2 + t 2 ,

dim L(G)

= 2n -

l

dim L 1 = 2st, (dim LO

+ dim

L 1)

= dim LO + dim

Nn .

In case of K := C, the corresponding real dimensions are twice the above ones, of course.

6.5 Linear Supergroups

211

(6.5.9) Here one always uses, that the skew-symmetric tensor product An®Mat(s + t, K) just means an appropriately signed product of K-matrix elements with the elements of An, in the sense of an isomorphism of unital associative algebras over K :

(6.5.10) Consider the special case of s = t, n MEG = G 1 X Go is uniquely factorized.

= 2.

Every element

Q

:3 unique M o EGo, M 1 E G 1

'- { [Is + 012A~2 .01Q + 02C2 M

= M 1 M o,

Mo =

01B~ + 02B~] Is + 012D~2

. A' B' D' ( ' 12' 1"'" 12 E Mat s,

K)}'.

[~b~]'

In case of real supermatrices, obviously dim L( G)

= 8s 2,

dim N 2 = 6s 2.

(6.5.11) The special linear supergroup SLn(s,t,K):= {M E GLn(s,t,K);sdet M = I}, s, t E No, s + t 2: 1, n E N, is generated by the real Lie algebra sln(s,t,K):= {X E gln(s,t,K);str X = O}. Here gln(s, t, K) denotes the above described real Lie algebra of GLn(s, t, K). In the case of s, tEN, s i- t, and K := C, the complexification of the real Lie superalgebra L of the above special linear supergroup is just A(s - 1, t - 1). (6.5.12) The super-transpose of M:=

[~ ~]

E D is defined by

inserting the usual transposition of supermatrices over K.

212

6. Real Lie-Hopf Superalgebras

(6.5.12.1) Denoting

W:=[~ ~],vt:=[-~r

;],fors,rEN,t:=2r,

the orthosymplectic linear supergroup is defined for n E N by OSPn(s,t,K) := {M E GLn(s, t,K); M&tWM = W}. One then finds that for every element M of this supergroup, sdet M = ±l.

(6.5.12.2) The complexification of the corresponding real Lie superalgebra L is the orthosymplectic Lie superalgebra osp(s, t, C). (6.5.13) The special unitary supergroup is defined for n E N, s, t E No, s + t ~ 1, by SUn(s,t,C):= {M E SLn(s,t,C)jMMt = I}, inserting the adjoint complex supermatrix Mt := (M t )*, complex conjugation, and the C-antilinear star operation on An such that Ok = Ok'

(Okl /\ ... /\ Ok.)· = Ok. /\ ... /\ Okl' 1 ~ k 1 < ... < kr ~ n. (6.5.13.1) The complexification of the corresponding real Lie superalgebra L consists of all the matrices X E M at(s + t, C), which fulfill the conditions

str X = 0,

xt + X

=

o.

7. Universal Differential Envelope

Theories of fermions are concerned with graded-commutative algebras, containing both even and odd generators, the latter being nilpotent. Hence one is forced to construct supermanifolds and Lie supergroups, in order to develop some kind of differential and integral calculus with respect to both even and odd variables, and moreover to define supersymmetry transformations between bosonic and fermionic objects, for instance creation and annihilation operators, or currents consisting of them, in models of quantum field theory. From the mathematical viewpoint, such a theory of supersymmetry may be considered as a first step towards a theory of something like "noncommutative spaces". Already on the level of Z2-grading, non-commutative ingrediences should be incorporated rather in the spaces of functions on the involved differentiable manifolds, than in the definition of superspaces by means of even and odd coordinates. The structure theorem by I. M. Gel'fand on the duality of locally compact Hausdorff spaces and commutative C algebras, and the structure theorem for an arbitrary C· -algebra, indicate the possibility to characterize a topological space in terms of continuous mappings defined on the space. The theory of C·-algebras is presented in several monographs, cf. J. Dixmier (1982), O. Bratteli and D. W. Robinson (1979), M. Takesaki (1979), R. V. Kadison and J. R. Ringrose (1983, 1986). The view of C·-algebra theory as some kind of non-commutative topology is stressed for instance in the friendly approach to K-theory by N. E. Wegge-Olsen (1993). Whereas on the levels of measure theory and topology the theory of noncommutative spaces was developed in terms of C· -algebras and their Ktheory, an appropriate non-commutative differential geometry was originally constructed by A. Connes (1985, 1990, 1994). Derivations of associative superalgebras, which are odd and nilpotent, the images of which need not be elements of the algebra itself, are treated systematically, starting from an initial object: the universal differential envelope il(A) of an associative superalgebra A. Multilinear forms on Cartesian products of finitely many copies of A are written as linear forms on il(A), thereby not only reformulating the classical theory of Hochschild cohomologies of associative algebras, but moreover defining Hochschild and cyclic cohomologies of associative superalgebras. Cyclic cohomologies and their relation to the Hochschild complex

214

7. Universal Differential Envelope

were introduced by A. Connes in 1981, as an algebraic framework for the Chern character in K-homology. The global theory of elliptic operators by M. F. Atiyah, for which the reader is referred for instance to the monograph by B. Booss and D. D. Bleecker (1985), was generalized by A. Connes to the concept of Fredholm modules over an associative unital complex superalgebra A. As some easy guideline, consider an elliptic pseudo-differential operator

which maps sections of E 1 to sections of E 2, both E 1 and E 2 being complex vector bundles over a finite-dimensional real differentiable manifold M as base space. Assume that P can be extended to some invertible bounded operator

p: 1t I

---+

1t0 , p-l =: Q : 1t0 ---+ 1tI,

on complex Hilbert spaces 1t z , z E Z2' Then the bounded operator

F:=

[~ ~]: 1t0 tB 1t I =: 1t

+--->

1t

is idempotent and odd with respect to the indicated Z2-grading. Denote by K the Klein operator; then the super-commutators

6(A) := ifF, A]g = i(FA - KAKF), K:=

[~ ~I]'

A E B(1t),

with the identity operators of 1t z , z E Z2, both denoted by I, provide an odd nilpotent derivation 6, thereby establishing the C·-algebra of bounded linear operators on 1t as some Z2-graded complex differential algebra. Moreover let 1r : A ---+ B(1t) be a representation of A on 1t. Then the resulting A-right module 1t over C, such that fa := 1r(a)(f) for f E 1t, a E A, is called an involutive Fredholm right module over A, if and only if

6 0 1r(a) = i(F1r(a) - K1r(a)KF) E B oo (1t) for a E A, denoting by Boo (1t) the C·-algebra of compact operators on 1t, which is an ideal of B(1t). The idempotency of F may be replaced by the condition

1r(a)(F2

-

I)

E

B oo (1t) for a E A,

thereby violating the nilpotency of 6, such that the corresponding Fredholm module is no longer involutive. This more general algebraic scheme is achieved for instance from an elliptic pseudo-differential operator P with parametrix Q. The Schatten ideals of B(1t) are then used for a detailed investigation and various applications of Fredholm modules. For the study of norm ideals the reader is referred to the monographs by R. Schatten (1970) and B. Simon (1979).

7. Universal Differential Envelope

215

An appropriate generalization to a closable linear operator P, the closure of which is injective, is provided by the construction of so-called Connes modules. Excellent monographs on K-theory were written by M. F. Atiyah (1967) and M. Karoubi (1978). The theory of fibre bundles can be learned for instance from D. Husemoller (1966), or M. Nakahara (1990), the latter including applications to anomalies in gauge field theories, and to bosonic string theory. The relevance of differential topology for quantum field theory is also exhibited with respect to instantons and monopoles, bosonic strings, anomalies, conformal and topological quantum field theories by Ch. Nash (1991). An outstanding application of non-commutative differential geometry is provided by A. Connes' and J. Lott's version (1990) of the standard model of elementary particles, where the Higgs boson appears as some gauge field, which serves for the coupling between two copies of space-time. The latter is proposed as an orientable compact four-dimensional real differentiable manifold with Riemann metric, which is considered as the base space of some complex spinor bundle. More precisely, one constructs some representation 7r of the unital associative real algebra

A

:=

COO(M, R) ® (C EB H),

inserting the non-commutative field of Hamiltonian quaternions H, on the Hilbert space of leptons and quarks. The odd closed operator F, which is inserted into the super-commutators of an odd derivation 8, such that

807r(a) = ifF, 7r(a)]g for a E A, is constructed from some self-adjoint Dirac operator D, which is densely defined on the spinor states over M. Besides presentations of this model by A. Connes himself (1990, 1994), J. C. Va-rilly and J. M. Gracia-Bondia wrote on extensive review (1993). A related approach to the understanding of the Higgs boson as some gauge field was obtained by R. Coquereaux et al. (1991, 1992). The Lie algebra homology of matrices was studied by J.-L. Loday and D. Quillen (1984). An extensive presentation of cyclic homology of algebras, cyclic sets and cyclic spaces, the Chern character, homology of Lie algebras of matrices, and the relation to non-commutative differential geometry is due to J.-L. Loday (1992). The relationship between non-commutative geometry and quantum groups is discussed, among other topics, in lectures by Yu. I. Manin (1991, 1992). In particular the algebraic machinery of non-commutative differential geometry, and also its application to elementary particle physics, was presented quite explicitly by D. Kastler (1988, 1993). An introduction to non-commutative differential geometry, and an according survey over some of its applications to theoretical physics, is due to R. Coquereaux (1989, 1993).

216

7. Universal Differential Envelope

1.1 Non-Unital Universal Differential Envelope (7.1.1) Let A be an algebra over the commutative ring R. A is called graded with respect to the commutative monoid G, if and only if A = EBpeG AP, as the direct sum of R-bimodules, and \fp,q E G,\fa E AP,b E Aq: ab E Ap+q. With the unit Po of G, APO is some subalgebra of A. An element a E A is called homogeneous of degree pEG, if and only if a E AP. For an associative algebra A over R with the unit eA, obviously eA E GPo. Frequently the Abelian monoids No, Z, Z2 are used for grading; one denotes Z2 =: {a, I}. The gradings of A with respect to No and Z2 are called compatible, if and only if A=

EB AP, \fp E Z2 : AP = EB jiEZ2

AP.

pEjinNo

In the case of Z2, A is called superalgebra over R. The superalgebra A over R is called bigraded, if and only if A is also graded with respect to No in such a manner, that these two gradings are compatible. (7.1.1.1) The grading automorphism of a superalgebra A over R is defined as an R-linear bijection () : A 3 aO

+ ai

+---+ aO

def

- a i E A,

()2

= id A.

(7.1.2) Let A = AO$A i be an associative superalgebra over the commutative ring R, which may be unital or non-unital. In any case an appropriate unit is provided by the direct sum of R-bimodules R $ A =: A. With the R-bilinear mapping, such that \f).., J.L E R, \fa, bE A : {A, a}{J.L, b} := {AJ.L,)"b + J.La + ab} E

A,

one obtains an associative algebra over R with the unit {eR' OA} =: eA' With the isomorphism of unital associative algebras over R:

and an R-linear bijection:

AI

:= {{OR,

a}; a E AI}

+---+

AI,

A becomes an associative superalgebra over R. For convenience one usually writes \f).. E R,a E A:

7.1 Non-Unital Universal Differential Envelope

217

(7.1.3) Let D = DO EBD 1 be an associative superalgebra over the commutative ring R, and fJ : D three conditions.

--+

D an R-linear mapping, which fulfills the following

Vi E Z2 : 1m fJlDf ~ D Z+1 •

(i) (iii)

(ii) fJ 0 fJ

= O.

Vi E Z2,Va E DZ,b ED: fJ(ab) = fJ(a)b+ (-l)ZafJ(b).

Here fJ is an odd derivation of D. One establishes the closed complex of R-bimodules: DO

D1

--+

restriction of 6

DO.

--+

restriction of 6

Then D is called Z2-graded differential algebra over R.

(7.1.3.1) Especially let D be No-graded, i. e., D

=

EB D n, Vn,m E No, Va E Dn,b E D

m :

ab E D n+m .

nEN o

Then D is called bigraded differential algebra over R, if and only if these two gradings are compatible, and fJ is of No-degree 1.

(i)

"In E No : 1m fJlD n ~ D n +1 •

(ii) "In

E No : D n = (D n n DO) EB (D n

n D I ).

Obviously,

(ii) {::::::::} Vi

E Z2 : D

Z=

E9 (D

Z

n D n ).

nENo

(7.1.4) A homomorphism ¢ : D --+ D' of Z2-graded differential algebras over R is defined as an R-linear mapping, such that the following conditions are fulfilled.

(i) Va,b ED: ab--+¢(a)¢(b). 4>

(iii)

¢

0

fJ = fJ'

0

(ii) Vi E Z2: 1m ¢IDf ~D'z.

¢.

Here ¢ is a homomorphism of associative superalgebras over R, and also a zero-degree complex morphism, with respect to the above indicated complex of R-bimodules. If and only if both D and D' are unital, one also demands that ¢(eD) = eDt.

7. Universal Differential Envelope

218

(1.1.4.1) An isomorphism ¢ of Z2-graded differential algebras over R is defined as a bijective homomorphism in this sense; then ¢-1 is also an isomorphism in this sense. (1.1.4.2) Obviously the composition of such homomorphisms is again some homomorphism in this sense. (1.1.5) A homomorphism ¢ : D --+ D' of bigraded differential algebras over R is defined as a homomorphism of Z2-graded differential algebras over R, which also fulfills the condition, that

"In E No: 1m ¢IDn ~ D n '. Here ¢ is also a complex-morphism of degree 1, with respect to the complex of R-bimodules:

Do

--+ ... --+

Dn

--+ Dn + 1 --+ .... restriction of 6

(1.1.5.1) An isomorphism ¢ of bigraded differential algebras over R is defined as a bijective homomorphism in this sense; then ¢ -1 is an isomorphism in this sense too. (1.1.5.2) Obviously the composition of such homomorphisms is again some homomorphism in this sense. (1.1.6) The non-unital universal differential envelope of an as$ociative superalgebra A over R is defined as a homomorphism v : A --+ il(A) of associative superalgebras over R, which fulfills the following universal property: il(A) is some Z2-graded differential algebra over R, with the odd derivation d : il(A) --+ il(A). Let D be an arbitrary Z2-graded differential algebra over R, with the odd derivation h : D --+ D, and consider a homomorphism ¢ : A --+ D of associative superalgebras over R, which also means that ¢ is even. Then there is a unique homomorphism ¢. : il(A) --+ D of Z2-graded differential algebras over R, Le., ¢. 0 d = h 0 ¢., such that ¢. 0 v = ¢. Hence ¢. 0 d 0 v = h 0 ¢. (1.1.6.1) An explicit construction of il(A) in the next chapter shows that v is injective, hence may be viewed as an inclusion, as is indicated in the diagram below.

inclusion

A --------~>

n (A)

Explicit Construction of the Unital Universal Differential Envelope

219

(7.1.6.2) The usual arrow-theoretic argument shows, that any two such universal objects /I : A --+ f.?(A), /I' : A --+ f.?'(A), are isomorphic in the sense of an isomorphism: f.?(A) +--+ f.?'(A) of Z2-graded differential R-algebras. (7.1. 7) Let ¢ : A --+ B be a homomorphism of associative superalgebras over R. Then using abstract nonsense, there is a unique homomorphism ¢. : f.?(A) --+ f.?(B) in the sense of Z2-graded differential algebras over R, extending ¢, according to the diagram below.

A - - - - - - _ " " .Q(A)

B

_ _ _ _ _ _--»

.Q(B)

One easily verifies the covariant functorial properties:

(1/J 0 ¢).

=

1/J. 0 ¢., (id A).

=

id f.?(A).

If ¢ is bijective, then ¢. ist bijective too, and then (¢ -1).

= (¢.) -1 .

1.2 Explicit Construction of the U nita! Universal Differential Envelope (7.2.1) Let A = A 0EBA i: be an associative superalgebra over the commutative ring R, and REBA =: A the unital associative superalgebra over R, which was defined in the foregoing chapter. With the tensor product of R-bimodules, the unital universal differential envelope fi(A) is constructed. "In EN: fin(A) == fi n :=

fi(A) == fi:=

EB

A®

(0

A ) , fio(A) == fi o :=

A,

fin(A).

nEN o

(7.2.2.1) Due to the universal property of tensor products, "In E N, there is an R-bilinear mapping:

Ax

fin '" {a, Wn } --+ a 0 wn E fin,

such that Va, b E

A, Vb 1 , ... ,bn

E A :

a 0 (b ® b1 ® ... ® bn ) = (ab) ® b1 ® ... ® bn .

220

7. Universal Differential Envelope

(7.2.2.2) Again with the universal property of tensor products, Vn E N, there is an R-bilinear mapping:

fin

X

A :7 {w n , b}

---+

wn 0 bE fin,

such that 'IZl, ... , Zn E Z2, Val E AZl, .. . ,an E AZn, Vb E A, a E A :

(a ¢9al ¢9 ... ¢9an ) 0 b = a ¢9al ¢9 ... ¢9an-l ¢9 (anb)

+ (_l)n+ L~=I Zl

(aat) ¢9 a2

¢9 ... ¢9

an

¢9

b

(7.2.2.2.1) For n = 1,2, one obtains the following R-bilinear mappings. Va E A,al E AZl,a2 E AZi",b E A:

(a ¢9 al) 0 b = a ¢9 (alb) + (_l)1+zl (aat) ¢9 b, (a ¢9 al ¢9 a2) 0 b = a ¢9 al ¢9 (a2b) +(_1)2+Z1+Z~

(aat}¢9a2¢9b

+ (_l)1+Z~ a¢9 (ala2) ¢9 b. (7.2.2.3) An R-bilinear mapping:

fin

X

A :7 {w n , b}

---+

wn

0

bE

fin

is defined. Vn E N, VWn E fin, Vb:= {,B,b} =,B + bE A:

Wn 0

b := ,Bwn + Wn

0

b.

Vn E N,Vwn E fin, a E A2,b E A:

(Wn ¢9 a)

0

b = wn ¢9 (ab) - (-l)Z(w n 0 a)

¢9

b.

Here for convenience the following natural R-linear bijection is suppressed:

fin

¢9

A :7 (a

¢9

al

¢9 ... ¢9

an)

¢9

b +-----+ a ¢9 al

¢9 ... ¢9

an

¢9

b E fin+!.

(7.2.2.4) There is an R-bilinear mapping:

fin

X

fim :7 {en, 17m}

--+

en o17m E fi n+m ,

such that Vn, mEN, VW n E fin' b E A, Vb l ,

Wn 0 (b ¢9 bl

¢9 ...

I8i bm ) = (W n

0

b)

¢9

bl

,bm E A : ¢9

¢9

bm E fi n+m .

Explicit Construction of the Unital Universal Differential Envelope

221

(7.2.2.5) Most frequently, for R-bimodules E i , Fi, i E I, l E L, an R-linear bijection is used:

+-+{Xi®YliiEI,lEL}E

EB

(Ei®Fi)·

iEI,IEL

(7.2.2.6) There is an R-bilinear mapping: ii(A) x ii(A) 3 {~, il} ----. ~ 0 il == ~il E ii(A), such that for the elements

00

- +" '== bo L..J bn ® bn1 ® ... ® b~,

n=l

~ 0 il =

Ciobo + L (~n 0 bo + (aob n ) ® b; ® ... ® b~) + nEN

L

~n oiim·

n,mEN

Here for convenience the canonical embeddings are suppressed. Moreover all the multiple tensor products are taken without any brackets. (7.2.2.6.1) For instance, one obtains Va := 0: + a, b := {3 + b E A, Val E AZl, a2 E AZ2, Vb l , b2 E A :

(a ® al ® a2) 0 (b ® b1 ® b2) = {3o. ® al ® a2 ® bl ® b2 + a® al ® (a2b) ® b1 ® b2 + (_1)2+zl+z~ (o.al) ® a2 ® b ® b1 ® b2

+ (_l)1+z~ a® (ala2) ® b ® b1 ® b2 • _

_

_

k

(7.2.2. 7) V~n E {In, ilm E {lm, Vk E N, VWk E @A :

~n

0

(ilm ® Wk) = (~n 0 ilm) ® Wk

E iin+ m+ k .

222

7. Universal Differential Envelope

(7.2.2.8) This R-bilinear mapping provides an associative product, i.e., one finds, that

vt, il, ( E ii(A) : (t

= to

011) 0 (

(il 0

()

E ii(A).

An easy proof is obtained via an induction with respect to n, mEN, following the two steps indicated below. Va,b,eE A,Vall ... ,an E A,Vbl, ... ,bm E A,Vt E ii(A): (i)

«(a ® al

(ii)

(t 0 (b ® bl ® ... ® bm

® ... ® an)

0

b) 0 e = (a ® al ® ... ® an) 0 (be),

» e = to ((b ® b ® .. , ® bm ) 0

l

0

c).

(7.2.2.9) With the above defined bilinear mapping, the direct sum of Rbimodules ii(A) := A EB

~ ( A ® ( ®A) )

becomes an associative algebra over R, with the unit One conveniently writes e == eR.

e := {{ eR, OA}, 0, 0, ... }.

(7.2.3) The unital associative algebra ii(A) is Z2-graded, according to the definition, that Vz E Z2, Vn EN: ii~(A) :=

iiZ(A) := A Z EB

(EB

ii;(A»).

nEN

Obviously this Z2-grading is compatible with the associative product of ii(A) in the sense, that

(7.2.3.1) This Z2-grading f) explicitly reads, 'v'ZO,ZI,'" ,Zn E Z2,VaO E Azo,al E AZl, ... ,an E AZn :

(7.2.4) The natural No-grading v of ii(A), such that Viio E A, Val, ... , an E A : v(ao ® al ® ... ® an)

=n

E N, vao

= 0,

is also compatible with the above associative product. Moreover, by its very definition, the Z2-grading f) is compatible with v.

Explicit Construction of the Unital Universal Differential Envelope

223

(7.2.5) Due to the universal property of tensor products, there are R-linear mappings

do: A

--+

iit(A), dn : iin(A)

--+

iin+l(A), for n EN,

such that Va:= {a,a} == a +a E A,Vn E N,Val,'" ,an E A: do(a) == doa

= eA ® a,

dn(a ® at ® ... ® an)

= e A ® a ® at ® ... ® an'

(7.2.5.1) Obviously, "In E No: d n + t odn = O. Hence one obtains the following complex of R-bimodules: d:=

EB dn : ii(A)

--+

ii(A), dod = O.

nENo

(7.2.5.2) The Z2-grading of ii(A) fulfills the condition, that

"Ix, y E Z2, "It E ii:f(A), a

E AY : t ® a E ii x+y+t(A).

(7.2.5.3) This R-linear mapping d is some odd derivation of ii(A), Le., "It E iii (A) : ~ E iiz+l(A), Vij E ii(A): t

0

ij--+(~)

0

d

ij + (-l)zt 0 (dij).

An easy proof is provided by an induction with respect to n E N, t := a ® at ® ... ® an'

(7.2.5.4) An induction with respect to n E N leads to the convenient expression, that Va E A, Vat, ... ,an E A : a ® at ® ... ® an = a 0 (doat)

0 ••• 0

(doa n ) == a(dat) ... (dan),

where the embedding of A into A is usually suppressed. Especially one finds, that Vat, ... ,an E A: e A ® at ® ... ® an = (doad

0··· 0

(doa n ) == (dat)··· (dan).

(7.2.6) The R-linear submodule !1(A) := A EB

EB

nEN

!1n (A), !1n (A) := A ® ( ®A) ,

such that ii(A) = REB !1(A), is an ideal of ii(A). Obviously the canonical projection of ii(A) onto !1(A) is commuting both with the canonical projections of ii(A) onto iin(A), and with those onto iii(A), for n E No, z E Z2. Moreover 1m d

= 1m dl.a(A) ~ !1(A).

224

7. Universal Differential Envelope

(7.2.6.1) "In EN: diin(A) EB (A 0 diin(A» = fin+l(A), dA EB (A 0 dA) = Q(A) = dQ(A) EB (A

0

A0

A.

Q(A».

(7.2.7) Both the unital associative algebra ii(A), and its ideal Q(A), with the two gradings a and v defined above, are bigraded differential algebras over R with respect to d. fi(A) is called the unital universal differential envelope of A, and Q(A) is called the non-unital universal differential envelope of A. Note that A need not be unital. (7.2.8) The Z2-graded differential algebra Q(A) over R fulfills the universal property defined in the foregoing chapter. (7.2.8.1) Let D be an arbitrary Z2-graded differential algebra over R, with the odd derivation 0 : D ~ D, and consider a homomorphism ¢> : A - - D of assoc:iative superalgebras over R. Then the universal property of tensor products allows for the definition of an R-linear mapping ¢>. : Q(A) ~ D, such that "In E N, "lao, al,"" an E A : ¢>.(ao) = ¢>(ao), ao(dat}··· (dan)

~ ¢>(ao) (0 <1>.

(da l ) ... (dan)

~(8

0

¢>(at)) ... (0 0 ¢>(an »),

¢>(al» ... (0 0 ¢>(an».

0

<1>.

Here one uses, that "In EN:

= R -lin span{ao 0 al 0'··0 an, e A 0 al 0···0 an; ao,··., an E A}.

(7.2.8.2) One then easily finds, that ¢>. conserves the Z2-grading, and that ¢>. 0 d = 0 0 ¢>., using the fact, that "In E N, "lao, all'" ,an E A:

(7.2.8.3) In order to prove that ¢>. is a homomorphism of associative superalgebras over R, one proceeds in several steps. "lao, at, . .. ,an, bo, bl E A : ao(dat}··· (dan)(dbd (dad··· (dan)(dbd

~¢>.(ao(dad··· (dan»¢>.(dbt),

cP.

~¢>.((dal)··· (dan»¢>. (db l ),

cP.

ao(dal) ... (dan)b o - - ¢>.(ao(da l ) ... (dan) )¢>(bo), cP.

(da l )··· (dan)b o ~¢>.((dad··· (dan»¢>(b o), <1>.

via an induction with respect to n E N. These results are used for an easy finish of the proof.

7.3 R-Endomorphisms of f.?(A)

(7.2.8.4) Uniqueness of ¢. follows from ¢.

0

d=

80

225

¢•.

(7.2.8.5) Especially let D be bigraded, and assume 1m ¢ implies, that

~

Do, which then

Vz E Z2: 1m ¢IA< ~ D nDo· Z

Then the induced R-linear mapping ¢. is some homomorphism in the sense of bigraded differential algebras over R, Le., "In E No: 1m ¢.lnnA ~ Dn, ilo(A):= A.

7.3 R-Endomorphisms of !l(A) (7.3.1) Consider the non-unital universal differential envelope,

il(A)

:=

A EB

E9 iln(A), A

=:

ilo(A), "In EN:

nEN

iln(A) = R - lin span{(r + ao)(da l ) ... (dan); ao, ai, ... ,an E Aj r E R}, of an associative superalgebra A = All EB AI over the commutative ring R, with the Z2-grading o. (7.3.1.1) "In E N,Vao,al," .,an , b E

A,Vw,~,1]

E il(A),Vr E R:

n

o(ao(da l )··· (dan)) =

n + oao + Loak for homogeneous aO,al,'"

,an;

k=l

d(ao(dal)'" (dan)) = (dao)(dat}··· (dan); dod = 0; o(dw) = I + ow for homogeneous elements w; d(~1]) = (d{)1] + (-l)X~(d1]) for ~ E ilX(A),i E Z2' (7.3.1.2) For n

= 2,3, ... ,Val E AZl, ... ,an E AZn, "lao, bE A :

+ ao)(da l )··· (dan)b = (r + ao)(dad'" (dan-l)(d(anb)) + (-It+ z1 +"'+Zn (ral + aOal)(da2) ... (dan)(db)

(r

n-l

+ L( _l)n-k+ L~=Hl Zl(r + ao)(dat}··· (dak-d k=l

for instance,

(r + aO)(dal)b = (r

+ ao)(d(alb)) + (_l)l+Zl(ral + aoat} (db),

ao(dal)(da2)b = ao(da l )(d(a2b))

+ (-1)2+z1+Z~aoal(da2)(db) + (-1)l+Z~ao(d(ala2))(db).

226

7. Universal Differential Envelope

(7.3.1.3) 8(€17) = 8€ + 8"7, for homogeneous elements €,17

E

il(A).

(7.3.2) The R-linear mappings (3', (3, 'Y,).. : il(A) ----4 n(A), are defined with the tensor products: R x A ----4 R ® A, A x A ----4 A

~

A, iln(A) x A ----4 iln+l(A)j

"In E N,Vr E R,Vao E Azo,al E AZI,b E AZ,Vwn E il~n(A):

rdal ----4ra1, aodal----4(-l)Zoaoal, wn db----4(-l) Zn wnb, b----40, ~

~

~

~

rda1 ----40, aodal ----4(-1)ZO(aoa1 - (-1) ZOZl a1ao), 13 13 wndb ----4 (_l)Zn (wnb - (_lyn zlx.vn ), b ----4 0, 13 13 rdal----40, aoda l ----4aoda1 - (-1)(l+zdzo(da1)ao, '"f

'"f

wndb ----4 wndb - (-1) (l+z)Zn (db )wn , b ----4 0, '"f

'"f

rdal 7 0 ' aoda1 7 W

n

(_1)(l+zo)(l+zd a1 dao,

db ----4 (_l)(l+Zn)(l+Z)bdw oX

n,

b ----4 b. oX

(7.3.2.1) Hence one defines a:= (3' -(3, p:= e-'Y, e:= id n(A), and finds, with the notation from above, that: rda1 ----4ral. aoda 1 ----4 ( -1)Zo(l+zd a1aO , (> (> wndb ----4 ( _l)Zn(l+Z)bwn , b ----40, (>

(>

rdal----4rdal' ao da l----4(-1)Zo(1+zd( da 1)ao, p

p

wndb ----4 ( _l)Zn(l+Z) (db)wn , b ----4 b. p

p

(7.3.2.2) FUrthermore one defines the direct sums

$

A:=

An, 0':=

nEN o

$

O'n,

0'0 :=

0,

nEN o

n+1

. _ " k-n-1 n+l-kl 0 ._ O'n+l .- LJ P ap nn+l(A)' p .- e, k=l 0'1

= a!n1(A)'

0'2

= p-1ap + al n2 (A)'

n

An :=

L )..klnn(A)' k=O

)..0 :=

e, Ao := embedding of A into il(A).

7.3 R-Endomorphisms of fl(A)

227

(7.3.3) Here one uses, that p is an R-linear bijection of n(A) onto itself. Vi, y E Z2, "In E N, Vw E nZ(A), 1Jn E nR(A) :

wd1Jn ---+( _ly(1+ II ) (d1Jn)w. pn+l

(7.3.3.1)

dnn(A)) = nn+I(A)j dA EEl (A 0 dA) = nl(A), no(A):= A.

"In EN: dnn(A) EEl (A

0

EB dnn(A),

dn(A) = dA EEl

nEN

dnn(A) = sum ({(dad···(dan+djal, ... ,an+l E A}); n(A) = A EEl dn(A) EEl (A 0 dn(A)), A 0 dn(A) = (A 0 dA) EEl

EB (A

0

dnn(A)).

nEN

(7.3.3.2) Remember here the R-linear bijections, "In EN:

dA

+-----+

R 0 A, A 0 dA

+-----+

A 0 A.

(7.3.3.3) For an easy proof of p to be bijective, one needs the following R-linear bijections, which correspond to permutations of tensor products.

pldA

= embedding of dA into n(A);

"In EN: pldn (A) : dnn(A)

+-----+

dnn(A)

n

pldn(A) : dn(A)

+-----+

dn(A)

---+.

embeddmg

---+

embeddIng

n(A).

n(A).

(7.3.3.4)

dnn(A) 3 (da l )··· (dan+d ---+( -l)(l+Zn+J)(n+z J+...+Zn)(da n+d(dad··· (dan), p

for homogeneous elements ak E AZk, k = 1, ... , n + 1, for n E N. Hence one obtains, "In EN, Val, ... , an E A :

pn+lldnn(A)

= embedding into n(A).

228

7. Universal Differential Envelope

(7.3.3.5) One then easily finds the R-linear bijections, 'Vn E No : pin (A) : iln(A) ~ iln(A) n

-+.

embeddtng

il(A).

(7.3.4) In the special case of unital A, one defines an R-linear map il(A) -+ il(A). 'Vi E Z2, 'Vw E ilZ(A) :

T:

T(W)

:=

(-l)Zw(deA); (c - >')T =:

ro, r:= roA.

(7.3.5) One easily calculates, that 'Vn E N, 'Vao E AZo, ... ,an+! E AZn+l : ao(dat)··· (dan+!) -+(-lyoaoal(da 2)··· (dan+!) .

13

- (_1)(l+zn+d(zo+n+z1+"'+Zn)an+!ao(dad'" (dan) n

+ l) -l)k+zo+···+zkao(da l )··· (d(akak+l))'" (dan+l), k=l

(dad··· (da n+l ) -+al( da 2)'" (dan+!) 13

- (_l)(1+Zn+l)(n+zl+"'+zn)an+!(dal)'" (dan) n

+ L(-1)k+z1+...+zk(dal)'"

(d(akak+d)·" (dan+d,

k=l

>.ldn(A) = 0, ao(da l )··· (dan) -+( _l)(l+Zn)(n+zo+"+Zn-l)an(da o)'" (dan-l)' A

r

(7.3.6) The R-linear mappings d, (3', (3, 0:, a, T, To, are odd with respect to the Z2-grading G. On the other hand, "(, p, >., A are even with respect to G. (7.3.7) One uses the notation, that 'Vn E No, 'Vi E Z2 :

(7.3.8) These R-endomorphisms of il(A) fulfill the following relations.

= 0:(3 + (3'0: = 0:(3' + (30: = 0, (3' d + d(3' = e, o:d = >., o:d + do: = p, "( + do: = c - >., >.2 = >.p, p = do: + >., (3d + d(3 = "(, (3"( = "((3, (3p = p(3, d"( = "(d, dp = pd, (3p-l = p-l(3, dp-l = p-ld,

(32 = (3,2

+ 0: = 0, (3(c - >') = (c - >')(3', >'(3 + 0:"( = 0, (3'(>' + "() = (>' + "()(3.

(3)' - >'(3' (3'>. -

7.3 R-Endomorphisms of !1(A)

229

(1.3.9) A':=

E9 An+llnn(A)' Le. : nEN o

Q(A)

3

{Wni n

E

No}

A'

---+ de!

{An+1Wni n

E

No}

E

Q(A).

A'ldil(A) = 0, A'IA n(A) : A EB (A 0 dQ(A» = A 0 Q(A) o

(1.3.10) \:In E No, \:Iwn

E

---+

e~bedd.ng

Q(A).

Qn(A) :

pn wn = (e + p-l{3d)wn , p-n wn = (e - {3d)w n , pn+lWn = (e - d{3)w n , {3' chJJn = pn AWn. Here one uses the equations e - {3d = p + d{3, p + {3d = e - d{3, and that 1m

0:

= 1m A = A EB (A 0 dQ(A».

(1.3.11) 11" := Ap-l, e - 11" = dap-l, 1I"(e - 11") =

\:In E No : 1I"Wn = An+1Wn = ApnWn . e - 11" = A(e - A) = (e - A) A, 1m 11" 11"{311" = {311", A1I" = 1I"A = A.

o.

= 1m A = A

0

Q(A),

(1.3.12) One finds the following images of projections. 1m 11" = ker (e - 11") = A EB (A 0 dQ(A», ker d = ker ({3'd) = ker 11" = 1m (e - 11") = 1m (d{3') = 1m d = dQ(A), ker {3' = ker (d{3') = 1m ({3'd) = 1m {3' ~ 1m 11". Especially, 1m {3'

dl A n(A) : A o

0

= 1m 11" ===} A Q(A)

+---+

0

A

dQ(A)

= {O}, the trivial case. ---+.

e~beddmg

Q(A).

Q(A) = ker 11" EB 1m 11" = ker ({3' d) Ef7 1m ({3' d) = ker(d{3') EB 1m (d{3').

230

7. Universal Differential Envelope

(7.3.13) Vw E il(A), a E AX, bE A : w da db

-----+ (-IYwab, {3'p-l{3'p da db -----+ (-I)X ab. {3' p-l (3' P

"In E N, Vl

= 0,1, ... ,n -1, "lao E AZo, ... ,an+1 E A Zn+l :

aoda l ... da n+1 -----+( _1)(zo+,+zl+l+l+l)(zIH+"'+zn+1 +n-l) (da l+2)

pn-l

... (dan+d

ao(dal)'" (dal+l), aodal'" da n+1 -----+( _1)Zo(Zl+oo+Zn+l+n+I)(dal)'" (dan+dao. pn+l Hence one obtains, that "In ~ 3, VI = 2, ... , n - 1, Val E AZl :

aodal ... da n+1 -----+

pl-n{3' p-l {3' ppn-l

Val

ao

E AZl:

ao

dal da2 da3

(-I)Zlao(dal)'" (dal-dalal+1(dal+2)'" (dan+l)'

da l

{3'

da 2 da3

-----+

p- 1 {3' p3

-----+

p-l{3'p-l{3'p~

0, ao

da l

da 2

(-I)Zl ao al a2 da3' {3'

-----+

p-l {3' p~

0.

(7.3.13.1)

"In E N, Vl = 0, ... , n - 1, Val E AZl, ... , an+1 E A Zn+l : da l ·· . dan +1 ~(I)(zl+"+ZI+l+1+I)(ZIH+oo+Zn+l+n-l)da ~ 1+2'" da n+l da 1'" da1+1' pn-l

"In ~ 3, Vl = 2, ... ,n - 1, Val E AZl :

dal ... da n+1 -----+

pl-n {3' p-l {3' ppn-l

(-I)ZI(dat}··· (dal-l)alal+1(dal+2)'" (da n+1)'

7.3 R-Endomorphisms of n(A)

231

(7.3.14) "In E No, VW n E nn(A) : pn rrwn = /3'dpnwn , pn(c: - rr)wn = (c: - rr)wn . "In E No, VW n E nn(A) : /3' AWn - A/3wn = rr/3' pn wn .

Hence one concludes, that Arr

= rrA = A + rr - c:, /3' A - A/3 = rr/3'(c: - rr) = o:p-l,

o:p-lo: = 0, /3'p-l /3' p = /3p- l o:p + 0:/3. (7.3.14.1)

o:p -1/3 = o:p -1/3' = - /3' o:p -1 ,o:p-1/31 Ao!1(A) = 0. "In;::: 2, Val E AZl, ... ,an E AZ",an+l E A:

o:p-l/3(da l ... da n+1) = (_1)ZI +,,+z,,+n al (da 2) ... (da n )an+1' Val E AZl,a2 E A: o:p-l/3((dad (da 2))

= (-1)l+zl ala2 .

(7.3.15)

E:= /3(a - 0:)

+ (a + o:p-l)/3,

"In EN, VW n+1 E nn+l(A) :

n 'r' ( LJ -

o:p -1(3)Wn+l = "'" L..J Pl-n/3' P-1/3' ppn-I Wn+1' 1=1

Especially one finds, that

EIA$dA$(AodA)

= o:p-l/3!!1o(A)$!1I(A) = o.

(7.3.15.1) Therefore, "In;::: 3, Val E AZl, ... ,an E AZ", VaO,an+l E A: n-l = ~(-1)Zlao(dal)'" (dal-l)alal+l(dal+2)'" (dan+d

1=2

+ (-1)Z"ao(dad··· (dan-danan+l + (-1)Zl ao al a2 da 3 ··· da n+1' E(ao dal da 2 da 3) = (-1)Zlaoala2 da3 + (-1)Z2 ao da l a2 a3, E(ao da l da 2) = (-1Y 1 ao al a2·

232

7. Universal Differential Envelope

(7.3.15.2) E(dal da2) = 0, E(da 1 da2 da 3) = (-1) Z2 d(ala2 a 3)'

n-l E(dal'" dan+d

=

2) -l)ZI(dad'"

(da/-da/a/+l(da/+2)'" (dan+d

/=2

+ (-l)Zn(dal)'" (dan-l)anan+l + (-1)Zl a1 + (-lyl+"'+Zn+na1(da2)'" (dan)an+l'

a2 da3'" dan+l

(7.3.16) One then straightforward calculates, that Ed = dE. (7.3.17) In the special case of unital A one obtains the relations

T1r = 1rT = 1rT1r, r 01r = 1rrO= 1rr01r, r1r Td = -dT = rod, T/3' + /3'T = e.

= 1rr = 1rr1r,

Vw E n(A) : O:T(W) = eAW.

Note, that Va E A : eA da - da = {-eR' eA} i8l a. In the case of a field R one finds, that Va E A \ {OA} : eA da The tensor product of R-bimodules may collapse.

T/3'1r = -/3T1r = -/31rT, /3ro + Fo/3' = e - >., A1r/3(e - 1r) = 0, (/3r + r/3)1r = 0, r 2 = (e - >')T 2(e - 1r) = (e - 1r)T 2A(e - 1r), (1rr1r)2

:I da.

= 0.

(7.3.18) All these R-linear mappings are elements of the unital associative superalgebra EndR(n(A», with the unit id n(A) =: e. E is an even R-endomorphism of n(A), with respect to the Z2-grading a.

7.4 Hochschild and Cyclic Cohomology (7.4.1) Consider the dual bimodule of the non-unital differential envelope n(A) = n,

n*

= n*(A):=

HomR(n(A), R),

of an associative superalgebra A over R.

7.4 Hochschild and Cyclic Cohomology

(7.4.2) For any set T

{f

E

~

233

n(A) , one denotes

n*(A);'v'w E T: I(w) = O} =:

YO.

(7.4.3) The linear form I E n*(A) is called closed, if and only if I

0

d = 0;

I is called cyclic, if and only if lOA = f. C* == C*(A) := {f C~

E

n*(A); 1 0 d = O},

== C~(A) := {f E n*(A); lOA = f} ~ C*(A), since Ad = o:d 2 = O.

(7.4.3.1) FUrthermore, I E n*(A) is called graded trace, if and only if it vanishes on super-commutators, Le.,

(7.4.3.2) One easily calculates the following implications, 'v'1 E n*(A) :

I graded trace {:::::::} I 0 (3 = I 0 , = 0; I dosed, and I graded trace1{:::::::} I 0 d = I I cyclic {:::::::} I 0 d = I 0 , = 0; rIo (3 cydic, and I closedl {:::::::} I cydic.

r

0

(3 = 0;

Here one uses the convenient notation, that

'v'¢ E End R (n(A)), IE n*(A), wE n(A): 1 0 ¢(w) := I(¢(w)); hence 'v'1/J E EndR (n(A)) : (f 0 ¢)

0

'l/J =

1 0 (¢'l/J).

(7.4.3.3) 'v'n E No: n~ == n~(A) := {f E n*(A);'v'k E No : k

== C~(A) := C*(A) n n~(A), C~n == C~n(A) := C~(A) n n~(A)

i- n ===> Iln",(A)

= O},

C~

~ C~(A).

(7.4.3.4)

Z* == Z*(A) := {f E C*(A); 1 0 (31r = O}, B* == B*(A) := {f E n*(A); 3 g E C*(A) : go (31r = f} 'v'n E No :

Z~

B~

== Z~(A) := Z*(A) n n~(A), == B~(A) := B*(A) n n~(A) ~

~

Z*(A);

Z~(A);

here one uses, that (31r(31r = o. The elements of C*, Z* ,B* are called Hochschild cochains, cocycles, and coboundaries of A.

234

7. Universal Differential Envelope

(7.4.3.4.1) The Hochschild cohomologies of A are defined as factor modules over R.

H* == H*(A) := Z*(A)jB*(A); "In

E

No: H~ == H~(A);= Z~(A)jB~(A).

(7.4.3.4.2) The Hochschild coboundary operator b is defined as the R-linear surjective mapping:

C*(A)

3

I ~I def

0

(3rr

E

B*(A).

(7.4.3.5) Correspondingly the cyclic cochains, cocycles, coboundaries, and cohomologies of A are defined, and also the Hochschild coboundary operator is restricted to the cyclic one.

Z! == Z!(A)

= {f E C!(A); 1

0

{3

= O} = {f E C*(A); I

0

{3

= O}.

Z!(A) is the R - submodule of closed, graded traces on n(A). B! == B!(A) := {f E n*(A); 3g E C!(A) : go {3 = f} ~ Z!(A), due to the nilpotent endomorphism (3.

H! == H!(A) := Z!(A)jB!(A). Furthermore one defines "In E No :

Z!n == Z!n(A) := Z!(A) n n~(A), B!n == B!n(A) := B!(A) n n~(A) ~ Z!n(A), H!n == H!n(A) := Z!n(A)jB!n(A); C!(A)

3

I~ 1 0 (3 E B!(A). def

Obviously the R-linear mapping b).. is surjective.

VI E C!(A)

: b)..(f) = b(f).

(7.4.3.6)

VI E

n*(A) : I closed

<===}

loA cyclic.

(7.4.3.7) C!o(A) = CO'(A) = nO'(A), of course. In the cases of R := Q, R, or C, one finds that "In E No: C!n(A) = {g E n~(A); 31 E C~(A) : loA = g}, because Vg E C!n(A) : goA - (n

+ l)g =

0;

hence one obtains

C!(A)

= {g E

n*(A); 31 E C*(A) : loA

= g}.

7.4 Hochschild and Cyclic Cohomology

235

(7.4.3.8) Consider again the cases of R := Q, R, or C. Then one finds, that Vn E No, Vg E C~(A)

: goA

one uses, that Vn

fln,,(A) := n

~

= 0 {::::::} 3f E C~(A) : f

0

(e - A)

= gj

2:

~ 1go (ne + (n -

l)A + ... + 2A

n

-

2 + An -

1 )

Inn(A);

1

flndA) := 2glndA)" Hence one obtains, that Vg E

C$(A) : goA

= 0 {::::::} 3f E C$(A)

:f

0

(e - A)

= g.

(7.4.3.9)

Z$(A) = (Im dt n (1m Uhr)t; C~(A) = (1m d)O n (Im ')'t, Z~(A) = (Im dt n (1m ')')0 n (1m (3t = (Im d)O n (1m (3t; C~(A) = C$(A) 0 A, B$(A) = C$(A) 0 (31r, B~(A) = CHA) 0 (3j C$(A) n (1m At = C$(A) 0 (e - A).

C$(A)

= (dD(A)t,

(7.4.3.10) Obviously Vg E

C$(A) : go (3'1r E C$(A)j ((3'1r)2 = O.

If the associative superalgebra A is unital, one obtains that Vg E

C$(A) : go (3'1r

choose gOT

=:

= 0 => 3f E C$(A)

:f

0

(3'1r

= gj

f.

(7.4.4) Let A be unital. In order to relate Hochschild and cyclic cohomology, one defines

Vf E C$(A) : Bf := f 0 r1r. B (C~(A)) = {O}, B (C$(A)) = C~(A), B 2 = 0, Bob>. = -b>. 0 Bj B (Z$(A))

~ Z~(A),

B (B$(A))

~ B~(A).

236

7. Universal Differential Envelope

(1.4.5) Let R := Q, R, or C. The so-called periodicity operator S: C!(A)

---+

C!(A)

is defined, such that 'fIf E C!(A), 'fin E No, 'fIao, aI,

0

0"

an+2 E A:

Sf(ao) = 0, Sf(aodal) = 0, Sf 0 d = 0, 1

Sf(aoda l

da n +2) = - - f 0 E 0 A (aoda l · da n +2); n+3 for instance, 'fIzo, Zl, Z2 E Z2, 'fIao E AZo, . . ,a2 E AZi": 000

O'

0

1 Sf(aodalda2) = '~J 0 E

0

(c + >. + >.2)(aoda l da 2)

=~f (( -lYlaoala2 + (_1)(l+z2)(zo+zd+zoa2aoal +( _1)(1+z2)(Zo+Zd+(l+Zd(Z2+zo)+z2ala2ao)

0

(1.4.5.1) 'fIf E Z!(A) : Sf = foE E Z!(A) n B*(A); for instance, denoting the Z2-degrees of ak E A by

Zk,

k = 0, ... ,4,

Sf(aodalda2) = (_l)Zl f(aOala2), Sf(aodalda2da3) = (-lYl f(aOala2da3) + (-ly2 f(aO(dal)a2a3), Sf(ao da l da2 da 3 da4) = (-lYl f(aoala2da3da4) + (-lY2 f(aO(dal)a2a3da4 + (_1)Z3 f(ao(dad(da2)a3 a4).

(1.4.5.2) n+1 'fin E No, 'fig E C!n(A) : n + 3S(g 0 (3) = (Sg)

0

(3o

By means of S (B!(A» ~ B!(A) one defines an according R-linear mapping S : H!(A) ---+ H!(A) of equivalence classes. The De Rham cohomology of A is then defined as the R-bimodule

H* (A)'DR

0-

H!(A) Im(S _ id H!(A»

(1.4.6) For R:= Q, R, or C, the factorized R-linear mappings: H!(A)

~H*(A) ~H!(A) ~H!(A)

are defined by means of cyclic and Hochschild cocycles as representatives. Then the following sequence is exact: i

---+

0

iJ

---+

0

S

---+

0

i

---+

Graded Tensor Product of Bigraded Differential Algebras

237

7.5 Graded Tensor Product of Bigraded Differential Algebras (7.5.1) Let D, D' be Z2-graded differential algebras over R, with the Z2graded nilpotent derivations 8,8', respectively. The graded (skew-symmetric) tensor product D~D' is equipped with an R-linear mapping .1 , according to the commuting diagram below. Oi XO '3{a,S'}

•

8(SJ::LtJ

Z

s@s'eOi@O'

s@8'(S'JEO@O'

J.1

1

Oi @O'

With this endomorphism .1, D~D' is some Z2-graded differential algebra over R.

(i) (iii)

'Vi E Z2 : 1m .11(Di8!D')' ~ (D~D,)Z+l;

(ii)

.1 0.1 = 0;

Vz E Z2, "Is E (D®D')z, t E D~D' : .1(st) = .1(s)t

+ (_l)Z s.1(t).

(7.5.1.1) In the special case of bigraded differential algebras D and D', D~D' is also some bigraded differential algebra over R, with an appropriate N 0grading due to the direct sum

D~D' =

EB EB (D®D')~, nEN o

zEZ~

(7.5.2) Let f, f' be R-linear forms on D, D', respectively; their tensor product is defined as the R-linear form: D~D' 3

a (l) a'

--+

1M'

f(a)f'(a')

E

R,

due to the universal property of the tensor product of R-bimodules. Then one easily obtains the following implications.

f 0 8 = 0, f' 08' = 0 ===} (J (l) 1') 0 .1 = O. [Vx, fi E Z2, Va E D X , bE DY, a' E D,?t,b' E D'Y : f(ab) = (-1)x ll f(ba), f'(a'b') = (-l)X Il j'(b'a')l ===}[V x,fi E Z2,V s E (D~D,)?t,t E (D®D')Y: (J (l) f')(st) = (-It"(J (l) j')(ts)l

238

7. Universal Differential Envelope

(7.5.3) f E H omR (D, R) =: D* is called closed, if and only if f 0 {; = OJ f is called graded trace, if and only if it vanishes on super-commutators, Le.,

(7.5.4) Let the differential algebra Dover R be bigraded. The R-submodule 00

E9Dk =: D?n k=n

is an ideal of D. Hence one obtains the factor algebras D / D?n, and isomorphisms of associative R-algebras, such that n-l

'rfn EN: D/D?n 3 [d
with an obvious notation.

7.6 Universal Differential Envelope of the Graded Tensor Product of Associative Superalgebras (7.6.1) Consider the tensor product of direct sums of R-bimodules Ek = Fk fBFfc, k = 1,2:

E I i8l E 2

+--+ (FI R-hnear

i8l F2) fB (FI i8l F~) fB (F{ i8l F2 ) fB (F{ i8l F~).

Hence one obtains an isomorphism in the sense of R-bimodules: E I i8lE2

:)

sum{xi i8lx2jXI E F I ,X2 E F 2 } 3 Xl i8l X2 +--+ Xl i8l X2 E F I i8l F2·

(7.6.2) Consider the universal differential envelopes of associative superalgebras Ak over R, k = 1,2, and also that of the skew-symmetric tensor product Al ~A2, and use the universality indicated in the following diagram.

____e_m_b_e_d_d_in--"g'-s v

~

D(A , )@D(A

jv. L--

---:-------:--:-:-

embedding

•

Q

A

(A , @A

2

)

) 2

Universal Differential Envelope of the Graded Tensor Product

239

Here v* is some homomorphism in the sense of Z2-graded differential Ralgebras. (7.6.3) Since the tensor product of bimodules is compatible with the direct sum, Le.:

v* is some homomorphism in the sense of bigraded differential R-algebras.

(7.6.4) The tensor product of R-linear forms /k : Ek -----. R, on R-bimodules E k , k = 1, ... , n, is defined as the linear form Ii ® ... ® In, which is defined using the universal property of tensor products, such that n

n

n

Q!)/k : Q!)Ek :;) Q!)Xk -----. !l(xt} .. · In(x n ) E R. k=l k=l k=t (7.6.5) The so-called cup-product of linear forms /k E n*(A k ), k = 1,2, is defined as the composite mapping

Ii * 12

:=

(Ii ® h) 0

v*.

(7.6.6) With the nilpotent Z2-graded derivation L1 on n(Ad ® n(A2), this becomes some bigraded differential algebra over Rj here L1 is universally defined, in order to fulfill 'Vi E Z2, 'VWl E n Z (Ad,W2 E n(A 2) :

L1(Wl ® W2) = (d1wd ® w2

+ (-l)ZWl

® (d2W2).

With an obvious slight abuse of notation one then immediately finds, that = 1, ... , n, k = 1,2 :

Vr E R, Va~ E Ak, l

n(A 1®A 2) :;) (r + (a~ ® ag))d(at ® a~) ... d(al ® a2')-----. II.

n

(r + (a~ ® ag))

II((dIaD ® a~ + (-l)Zla~ ® (d2a~)) E n(Ad®n(A2), 1=1

for homogeneous elements (7.6.7) If Ik E n*(Ak),k product is also closed, Le.,

ai E AfT, l = 1, ... , n. =

1,2, are closed linear forms, then their cup-

Vii E C*(At)'h E C*(A 2) : Ii

* 12 E C*(A 1®A 2).

Here one easily calculates, that Vii E n*(Ad,h E n*(A2):

It 0 d1 = 0, 12 0 d2 = 0 ==:} (ft

* h) 0 d = (Ii ® h) 0 v* 0 d = (fl ® h) 0 L1 0 v* = o.

240

7. Universal Differential Envelope

(1.6.8) If Ik E .rr(Ak), k = 1,2, are graded traces, then their cup-product is also of this type, Le., '<1ft E n*(A t ), 12 E n*(A 2) :

11 0 {3t = ft

0

It

= 0, 12 0 {32 = 12 0 , 2 = 0

(11 * h) 0 {3 = (ft * h) 0 ,

~

= O.

One straightforward calculates the cup-product acting on super-commutators. (1.6.9) Since v* conserves both the Z2-grading and the No-grading, one finds that

'
E

No, '<111 E n~l (Ad, 12 E n~~(A2) : ft

* 12 E n~1+n~(At®A2)'

(1.6.1O) '
'<111

E

Z~nl (Ad, 12 E Z~n~(A2) : 11 * 12 E ZA,nl+n~(At®A2)'

(1.6.11) An easy calculation yields, that '
'<1ft

E

0

dA,

BHA t ), 12 E Z~(A2) : 11 * h(w) = (9t * h) 0 (3(w),

with the cyclic linear form 9t E C~(At) defined by the condition 9t o{3t = Here one denotes A t ®A 2 =: A.

!t-

(1.6.12) More generally one finds, that '
'
E Z~nl (Ad,

11 * 12

E

12

E B~n~ (A2)

:

B~,nl+n~ (A t®A2) , 12 * ft E Btnl+n~ (A2®At ) .

(1.6.12.1) The proof of this theorem by A. Connes is presented in his monograph "Noncommutative Geometry" (Chap. II!.!. Cyclic Cohomology, Theorem 12). (1.6.12.2) This theorem allows for the definition of an according cupproduct of cyclic cohomologies, such that '
'
E H~nl (At),

[121

[ft] * [121 = [ft * 12]

E

E H~n~(A2)

:

H~,nl+n~ (A t®A2) ,

with an obvious notation of equivalence classes. (1.6.12.2) Note that the cup-product of cyclic cochains need not be cyclic.

Universal Differential Envelope of a Commutative Ring

241

7.7 Universal Differential Envelope of a Commutative Ring (7.7.1) The commutative ring R is also considered as the free R-bimodule, which is generated by the unit eR. Moreover R is considered as unital associative superalgebra, with respect to the trivial grading according to R = RO EB Rl, Rl := {OR}. (7.7.2) With respect to this trivial Z2-grading of R, its universal differential envelope is constructed.

ii(R)

REB Q(R) = REB REB dQ(R) EB R

dQ(R) = R - lin span ie, e, (de)··· (de), e (de)··· (de); n EN}, 0

~ ~

n

n

time~

time~

with the unit e of ii(R). Here the unit eR of R is used in the following sense.

e := {eR,O, ... }, e := {O,eR,O, ... }, de = {O,O, {eR,O} ® eR,O, .. .}, e de = {O,O, {O,eR} ® eR,O, .. .}, (de)n := (de)··· (de) = { 0, ... ,0 ,{eR,O} ® eR ® ... ® eR,O, .. .}, ~ "-v-" ---..-n time~ (n+l) time~ n time~ e(de)n ={ 0, ... ,0 ,{O,eR}®eR®···®eR,O, ... }. "-v-" ---..-(n+l) time~ n time~ (7.7.3) One easily finds the following rules of calculation.

ee = e, ee = ee = e, e(de) = (de)e = de, e2 := ee = e, e(de)

+ (de)e = de.

(7.7.3.1) "In EN:

- e)(de)n, n odd (d e )ne -_ {(e e(de)n, n even' e(de)n e = (7.7.3.2)

ii(R) = R -lin span{(de)n, e(de)n j n with the notation

e =: (de)o.

E No},

{ae(de)n,

n odd n even

242

7. Universal Differential Envelope

(7.7.3.3) "In,m E No:

n odd n even'

- e)(de)n+m, (d e) ne(d e)m = {(e e(de)n+m,

e(de)ne(de)m = {O('d )n+m n odd . e e , n even

(7.7.4) In the sense of the inner direct sum

EB

n(R) = R ffi D(R), D(R) =

Dn(R),

nENo

one obtains the R-bases: {e} _

R, {e} ---+ R, {de} -

"In EN: {(det}

---+

dR, {e de}

---+

d~R, {e(de)n} -

R

dRj

0

R

0

dfi o.~. 0

n time a

n timea

n(R) = R ffi R ffi dR ffi (R 0 dR)ffi

... ffi

(d~R)

ffi (RodRo ... odR) ffi"',

n time a

{(de)n, e(de)n; n E No}

-

free over R

n(R).

The No-grading of n(R) explicitly reads as follows. n(R) =

EB nn(R),

no(R) :=

R := R ffi R

= R( {e, e}),

nENo

"In E No : nn+l (R) = dDn(R) ffi R D(R) =

0

dDn(R) = Dn+l (R).

EB Dn(R), Do(R) := R = R({e}). nENo

"In E No : dnn(R)

= dDn(R) = R( {(de)n+l}),

RodDn(R) = R({e(det+ 1 }).

(7.7.5) The Z2-grading of n(R) is non-trivial, Le., Vi E Z2: nZ(R)

= R({(de)n,e(de)n;n E No,n = z}).

(7.7.6) "In E No : d (e(de)n)

=

(de)n+l j de

= o.

d,R.

Universal Differential Envelope of a Commutative Ring

243

(7.7.7) Obviously the canonical embedding, of an associative algebra A over R into its unital extension A := REBA, is an ideal of A. The corresponding factor algebra is unital, due to the isomorphism of R-algebras:

A/A 3

[{r, a}]

+---+

r

E

R.

Moreover, since n(A) is the unital extension of n(A), this is an ideal of n(A). (7.7.8) "In EN:

j3'(e) = 0, j3'(de) = j3'(e de) = e, j3'(e(de)2n) = 0, j3'(e(de)2n+l) = e(de)2n, j3'( (de)2n) = (e - e)(de)2n-l, j3' «de)2n+l) = e(de)2n, a(e) = 0, a(de) = a(e de) = e, a(e(de)2n) = a«de)2n) = _e(de)2n-l, a(e(de)2 n+l) = a«de)2n+l) = e(de)2n, ..\(e) = e, ..\(e de) = -e de, ..\(e(de)n) = (-lte(de)n, ..\ 0 d = 0, j3(e) = j3(de) = j3(e de) = 0, j3(e(de)2n+l) = j3«de)2n+l) = 0, j3(e(de)2n) = e(de)2n-l, j3«de)2n) = (2e - e)(de)2n-l, p(e) = e, p(de) = de, p(e de) = de - e de, p(e(de)n) = (_l)n+l(e - e)(de)n, p«det)

= (-It+1(de)n,

'}'(e) = '}'(de) = 0, '}'(e de) = 2e de - de, '}'«de)2 n+l) = 0, '}'«de)2n) = 2(de)2n, '}'(e(de)2n)

= (de)2n,

'}'(e(de)2n+l)

= (2e -

e)(de)2 n+l,

T(e) = e de, T(de) = -(de)2, T(e de) = -e(de)2, T(e(de)n) = (_l)ne(de)n+l, T«de)n) = (_l)n(de)n+l, Fo(e) = 2e de, Fo(de) = -(de)2, Fo(e de) = 0, Fo(e(de)2n+l) = 0, Fo(e(de)2n) = 2e(de)2n+l, Fo«det) = (_l)n(de)n+l. (7.7.9) VI E n·(R), Vrll r2 E R, "In EN:

10 j3(rle(de)2n + r2(de)2n) = (rl + 2r2)/(e(de)2n-l) - r2f«de)2n-l). Hence for every linear form

1 0 j3 =

°

<===}

Ilnl(R)

I

E

n·(R) the following implications hold.

= 0.

(7.7.10) VI E n·(R), "In E No : 1 0 (e: - ..\)(e(de)n) =

{~f(e(de)n),

n even n odd'

244

7. Universal Differential Envelope

Therefore,

I is cyclic, if and only if it is some closed graded trace; Le.,

VI E n*(R) : I E C~(R) {:=:} 1 0 13 = 1 0 d = 0 {:=:} IE Z>'(R)

~

1 0 'Y = o.

(7.7.10.1) "In E No : C~n(R)

{O},

= Z~n(R) = { C~(R)

~ R,

R-hnear

n odd

n even·

(7.7.11) B;..(R) = {O}, and therefore "In E No : H!n(R)

~

R-linear

Z!n(R)

~

R-linear

{{O}, n odd R, n even

7.8 Universal Differential Envelopes of Finite-Dimensional Clifford Algebras (7.8.1) Consider a symmetric K-bilinear form 9 : Ex E --+ K, over a pdimensional vector space E,p EN, over the field K of char K f. 2. One can choose an orthogonal basis {e I, ... , ep } of E, such that "If. k, l:

The form 9 is non-degenerate, if and only if m = p; otherwise {em+l' ... ,ep } is some basis of its kernel ker g.

(7.8.2) The Clifford algebra Cover E with respect to 9 is constructed by the usual universal diagram, such that

"Ix, y E E : xy + yx = 2g(x, y)ec. Then C is some associative K-superalgebra with the unit ec == eo, which is polynomially spanned by Ej the Z2-grading of C is defined such that E ~ C i . The K-dimension of C equals 2P , due to an isomorphism 'Y of unital associative superalgebras defined below.

(7.8.2.1) For 1 ~ k ~ p, let Ck be the Clifford algebra over the onedimensional vector space E I := K ({ cd), with the unit eCk =: 4,k), with respect to the symmetric K-bilinear form gk : E I x E I --+ K, such that gk(cl,cd :=Dk· 2

CI

= DkC2(k) ,cIC2(k) = C2(k) CI = el,

( (k)) 2

C2

= C2(k) , k = 1, ... ,po

Universal Differential Envelopes of Finite-Dimensional Clifford Algebras

245

(7.8.2.3) With respect to the Z2-grading of Ck, k = 1, ... ,p, such that k ) == C1 is odd, one establishes an isomorphism, in the above sense:

ci

~

, :C

E 3 ek +---+ (1) (k-1) (k+1) (p) C· .C C2 ® ... ® C2 ® C1 ® c2 ® ... ® C2 E 1® ... ® P'

An appropriate K-linear basis {ekl ... ek r ; 1 $ k 1 < ... < k r $ p} u {eel of C is then established, due to the images ,( ec) =: 1) ® ... ® p ), and

4

4

,(ek ···ek )=:cl(1)® ... ®cl(p) vr:k:I k :={1, kE{k 1, ... ,kr } 1 r i p ' t 2 otherwise

.

(7.8.3) Within the universal differential envelope n(C), one easily calculates the following realtions, ek being odd for 1 $ k $ p, and eo == ec even. n

_

(d eo ) eo -

{(deo)n - eo(deo)n, (d)n eoeo,

n = 1,3,... uJ)k' 2 4 ... . Vi . n="

+ dekdeo),

(dek)ek - ekdek = Okdeo, (dek)2ek - ek(dek)2 = ok(deodek

V'fn E N: n

(dek)n ek

= Ok

L dek'"

dekdeodek'" dek

+ eddek)n E

nn(C),

1=1

with the factor deo at the position I. Furthermore Vik: (deO)ek

+ eOdek = dek,

(deo)2ek

= eodeOdek,

(deo)3ek

= (deo)2dek - eo(deo)2dek,

(de )n e o k

= { (deo)n-1dek -

eo( de o)n- 1dek,

eo(deo)n-1dek,

(dek)eo - ekdeO = dek, (dek)2eo = ekdekdeO

n = 3,5, n=2,4,

. '

+ (dek)2 + Ok(deO)2,

V'fn EN: (dek)neO = eddek)n-1deo

+ (dek)n

+ Ok ((dekt-2(deo)2 + (dek)n-3deodekdeo + ... + deo(dekt-2deo) . Vik =/;1 : (dek)el - ekdel

+ (del)ek

- eldek = O.

246

7. Universal Differential Envelope

(7.8.4) Consider as an example the first order cyclic and Hochschild cocycles and coboundaries of the Clifford algebra C for p = 1, with the unit eo and one odd generator ell e~ = 151eo· One easily calculates that .Ql(C) n 1m /3 = K ({eodeo, eldeO, eoded) , .Ql(C)n 1m(/31r) =K({eodeo,eldeo,15leoded). Denoting the dual basis vectors of {ekdel ; k, l = 0,1} by ekl, such that

one therefore finds that

Z~l (C) = K ({eu}),

Z;(C)

= {~~~:~~})~d),

~~; ~

One then obtains the cohomologies: H~l(C) ~ Z~l(C) for 15 1 = 0,

H;(C)

~

Z;(C) for 15 1

= 0,

dim

H~l(C) = 0 for 15 1

dim H;(C)

= 0 for 15 1

# OJ

# O.

(7.8.5) The second order cyclic cocycles, again for p = 1, for 15 1 = 0 are the elements of Z~2(C) =

K ({ eooo, elOO - eolO + eOOl, eud) ,

denoting by eijk the dual basis vectors of eidejdek, i,j, k = 0,1,2. Since the cyclic coboundaries vanish both on eldeldel and eodeodeo,

(7.8.6) In the case of p = 2, with two odd generators el and e2, such that e~

= e~ = el ez + e2el = 0,

the first order cyclic cocycles of this four-dimensional Grassmann algebra are the elements of Z~1 (C)

= K ({ eu, e12 + e2ll e22, el(12) + e(12)1' e2(12) + e(12)2}) ,

denoting by eH the dual basis vectors with respect to the K -linear basis:

{eii:dei; k, f = 0,1,2, 12}

--+

Co dC, e12 := ele2,

such that for instance

el(12) (e l d(ele2)) = 1, e(12)1(ele2det} = 1. Since /3 vanishes on the support of f E

Z~l (C),

one finds that

B~l (C)

= {O}.

Universal Differential Envelopes of Finite-Dimensional Clifford Algebras

247

(7.8.7) The zero order cyclic and Hochschild cocycles and coboundaries, in the cases of p = 1,2, are calculated easily. For p = 1, f o13(el de l) = -2f(e~) dim Z!o(C)

= dim

= -2od(eo)

= 0,

Z~(C) = {;: ~~ ~ ~

, B!o(C) = B~(C) = {O}.

For p = 2, one again finds that B!o(C) = Bo(C) Z!o(C)

= Z~(C) = Homc(C, K)

Z!o(C) = Z~(C)

= K ({c12})

for 01

= {O};

= 02 = 0;

for 0102 -; 0,

with the dual basis vector C12 of ele2. (7.8.8) Consider again the case of p = 1 and 01 -; 0, e~ = OleO. With the above notation of dual basis vectors, such that ck(el) = Okl, k, l = 0,1, one calculates that Z!o(C)

= K ({cd), B!o(C) = {O}, dim H!o(C) = 1;

Z!l(C) = K ({Cll}) = B!l(C), dim H!l(C) Z!2(C)

= 0;

= K ({clOO - cOlO + c001,clll}),

B!2(C) = K ({clOO - cOlO

SCl = C100 - COlO

+ COOl + 301Clld),

dim H!2(C) = 1;

+ cOOl - Olclll·

(7.8.9) More generally one calculates, that for an arbitrary non-degenerate Clifford algebra C, with 01 .. ·Op -; 0, Z!O(C)

=K

H!,n+2(C) =

({Cl...p}), B!o(C)

S(H!n(C)) for

= {O},

n E No,

inserting the correspondingly factorized periodicity operator S, and denoting by Cl...p the dual basis vector of el··· ep- For odd n E N, H!n(C) = {O}.

248

7. Universal Differential Envelope

7.9 Differential Envelopes of Real Scalar Fields The unital universal differential envelope ii(A) of an associative superalgebra A over the commutative ring R can also be constructed using multiple tensor products over the direct sum A := REBA. (7.9.1) Consider the R-linear mapping ~

: A® A 3

A ® A,

a ® b - + a ® b - jL(a ® b) ® e A E

with the graded structure mapping jL : A ®

A3

a ® b - + (-1)1'ab E

A for

Due to its properties ~ 0 ~ = ~, ker ~ = an R-linear bijection

bE

AY, Y E Z2-

A® R,

1m ~ = ker jL, one obtains

'l/J1 : ii1(A) 3 a ® b +---+ a ® b - jL(a ® b) ® e A E ker jL.

Then the R-linear mapping

60 : A 3 a - + e A ® a - (_1)Xa ® e A = e A ® a - ( _1)X a ® e A E ker jL, de!

for a =:

Q

+ a E AX,

Q

E

R, obviously equals 60

= 'l/J1

0

do-

(7.9.2) With the left and right module-multiplication, such that E Z2, Va E A, b =: /3 + b E AY, e =: 'Y + c E AZ :

Vy, i

'l/J1(a ® b)

0

e = 'l/J1 «a ® b)

0

c) = a ® (be) - (-1)II(iib) ® e,

a 0 'l/J1(b ® c) = 'l/J1(a 0 (b ® c)) = (ab) ® c - (-I)Z(abe) ® e A, one establishes an A-bimodule 1m 'l/J1 = ker jL over R.

(7.9.3) The n-fold tensor product over usual, such that for instance

A of such objects is

constructed as

ker jL®A ker jL 3 ('l/J1(a®b) og) ®A 'l/J1(c®d)

= 'l/J1 (a ® b) ® A (g 0 'l/J1 (c ® d)) -

(7.9.4) With the above module-multiplications, and the tensor product over A, being used as product, the R-direct sum

H(A):=

E9 Hn(A),

n

Hn(A)

:=

Q?) Aker jL for n ~ 1,

Ho(A)

nEN o

becomes an associtative unital superalgebra over R. For instance, 'l/J1(a®b) o 'l/J1(C® d) ='l/Jl(a®b)®A'l/J1(C®d)

E

H2 (A).

:=

A,

7.9 Differential Envelopes of Real Scalar Fields

249

(7.9.5) The Z2-grading of h(A), and also its No-grading, are switched to

H(A) via an R-linear bijection

a+ ... + ao @al @... @ an + ... +---+ a+

'l/J : h(A) 3

... + 'l/Jl(iio @aI) @A 'l/Jl(e A @a2) @A'" @A 'l/Jl(e A @an ) + ... E H(A). (7.9.6) Moreover'l/J is an isomorphism of Z2-graded differential algebras over R, with respect to the odd derivation 6 := 'l/J 0 d 0 'l/J-l. For instance, 'Vao E Azo,al E AZI: 6(iio)

= 60 (ao) = 60 (ao) = e A @ao -

(-l)ZOao @e A'

'l/Jl(ao @al) = ao @al - (-lY1aoal @e A = ao o6(al) ~'l/Jl(eA @ao)@A 'l/Jl(e A @aI) = 6(ao)@A 6(aI). 6

(7.9.7) Now let especially A be unital. Identifying eA with the unit eA of A, one obtains the following surjective homomorphism II : H(A) ----+ H(A) of Z2-graded differential algebras over R, with respect to the odd derivations 6 and 6, such that 6011 = II 0 6.

A3

a =:

ker ji

3

Hn(A)

0:

+a =

0:

e A + a ~ 0: eA

'l/J(a@b) ----+(0: eA v

3

(g/ A k=l

'l/J(~k) ~

de!

+ a E A,

+ a) @b -

(g/

II 0

(0:

eA

'l/J(~k)

+ a)b( -l)Y @eA E

for b E AY,

Hn(A).

A k=l

A3a~eA@a-(-lYa@eAfor aEA", 6(eA) =0; de!

60 II 0 'l/J(iio @al @... @ an) = 6(ao) @A 6(aI) @A ... ®A 6(a n }, for ao =: 0: + ao E A, alt ... ,an E A. Note that one uses here, that ker II is some graded ideal of H(A), and moreover ker II <; ker (II 0 6). Obviously 6 is compatible with the natural No-grading of H(A) in the sense, that

6 (Hn(A)) <; Hn+l(A), H(A):=

E9 Hn(A), nENo

Ho(A)

:=

A.

250

7. Universal Differential Envelope

(7.9.7.1) As in the case of the odd derivation d of ii(A) one finds, that ker(1I 0 'l/J)

~

ker(1I 0 'l/J 0 {3) n ker(1I 0 'l/J 0 (Jrr).

Therefore the R-endomorphisms j3, j31r of .Q(A) can be pushed onto H(A), and Hochschild cochains, cocycles, and coboundaries of A can be defined with respect to the differential envelope H(A), and conveniently denoted by the same letters as those with respect to .Q(A). C*(A) :=

{J

E H*(A)

== (H(A»*; I 0 0 = o}

Z*(A) := {f E C*(A)j I

0

j

j31r = O}, HH(A) := Z*(A)/B*(A),

B*(A) := {f E H*(A)j 3g E C*(A) : go j31r = J} ~ Z*(A);

VI E H*(A)

: bl := 1 0 j31rj 1m blco(A) = B*(A).

Since j31r is compatible with II, the above Hochschild coboundary operator b in the sense of H(A) is acting similarly as that of .Q(A). (7.9.7.2) Unfortunately in general oX is not compatible with II. Therefore one cannot define cyclic linear forms in the sense of H(A), as one does with respect to .Q(A). For instance, for a E A't, x E Z2, one finds that 110 'l/J(a ® eA) = 0, but

(7.9.8) As an example consider the commutative unital associative algebra of real scalar fields,

H 1 = sum ({a

0 o(b) =a®b-ab®eja,bEA}) ..-----.. sum ({{a(x)(b(y) - b(x»;x,y E Rm}ja,b ~ c oo (R 2m ).

E

coo(Rm )})

Generally, factorizing with respect to the tensor product over A, one finds that H n , n E N, is the real vector space of sums of real smooth mappings of the kind aoo(ad'" o(an) ..-----.. {ao(xo)

tl

(ak(xk) - ak(xk-d) j Xo,···, Xn E R m

with ao, ... ,an E Coo (Rm) . For instance, H2

--+ {

{t(x, y, z) - t(x, y, y) - t(x, x, z) t E COO (R3m ) }.

+ t(x, x, y); x, y, z E R m };

}

7.9 Differential Envelopes of Real Scalar Fields

251

(7.9.9) On this purely algebraic level the closed de Rham currents over Rm can be related to Hochschild cocycles in the following way. (7.9.9.1) The inclusion of Coo (Rm) into the Z2-graded real differential algebra E (Rm) of differential forms, with the exterior derivation denoted by .1, is extended by universality to some homomorphism: n(A) --+ E (Rm) of such objects. This homomorphism vanishes on the kernel of v 0 'l/J, thereby allowing for an according homomorphism of Z2-graded real differential algebras: H(A) --+ E (Rm). This homomorphism is then used in order to construct an injective real-linear map from closed de Rham currents to Hochschild cocycles:

{4> E Harna (E(R m »; 4> 0.1 = O}

=:

C :3 4>

--+

1 E Z*(A) c H*(A),

such that 'v'aO,al,'" ,an E A: I(ao) = 4>(ao),

(7.9.9.2) Conversely the usual A-linear basis:

{eA

= I, .1(xk1 ) 1\ ... 1\ .1(xkp )j I

~ k1

< ... < kp ~ m}

E(R m )

--+ free over A

is used to construct an R-linear map: Z*(A) E A: 4>(ao) = I(ao),

:3

1 --+ 4> E C, such that

'v'aO,al,···,a n 4> (ao.1(aI)

1\ ... 1\

.1(an » =

~!

L

(-Itp 1 (aoo(ajl) @A ... @A o(ajn» ,

PEP n

~] E P n , and denoting the sign In of P by T p . Since this real-linear map vanishes on B*(A), one obtains an according real-linear map: inserting the permutations P:=

HH(A)

:3

[I]

--+

4>

E

[~Jl

C.

(7.9.9.3) One easily verifies the composite mapping: C :3 4>

--+

[I]

--+

4>.

(7.9.9.4) Hence one obtains an injective real-linear mapping:

C :3 4>

--+

[I]

E

HH(A).

(7.9.9.5) Under certain topological conditions, A. Connes constructed an isomorphism between continuous Hochschild cohomologies and closed de Rham currents.

252

7. Universal Differential Envelope

7.10 Universal Differential Envelope as Factor Algebra In order to extend conveniently a graded star operation, which is defined on an associative superalgebra A over R, to its unital differential envelope n(A), the latter is reconstructed as an appropriate factor algebra.

(7.10.1) E := A EB (R ~ A) =

eO EB E 1,

E I := AI EB (R ~ A Z + 1 ),

with an obvious linear bijection.

T(E) =: T = TO EB T 1, with the usual Z2-grading of the tensor algebra. The

set S

:=

{a ~ b - ab, (eR ~ a) ~ b + (-lya ~ (eR ~ b) - eR ~ (ab); a E A I, b E A; Z E Z2} c T,

with the canonical embeddings being conveniently suppressed, generates an ideal D of T, which is graded with respect to the Z2-grading of T.

D:= sum{tst'; s E Sj t, t' E T} =

E9 (D n T

I

).

IEZ~

F := TID = FOEBF 1 , with the Z2-grading defined according to the following implications. 'lit = to

+tI

E T:

[t] E F Z ¢:=>tz+ 1 ED¢:=> 3t' E [t] : t' E T I . de!

(7.10.2) The universal property of tensor products is used to define R-linear mappings .1p : TP(E) ---+ TP(E), such that 'lip E N, 'vZ1, ... ,zp E Z2 :

.1p

(~(ak + Tk ~ bk») P

k-l

k-l

I-I

= L( _l)Zl+oo+Zk_l @(al + TI

~ bl ) ~ (eR ~ ak)

P

~

@

(al

+ TI ~ bl)

E TP(E).

l-k+1 With the linear mappings

.1 1 : E

3

a + T ~ b ---+ eR ~ a E R ~ A, .10 : R ---+{O} ,

an R-linear mapping

de!

de!

7.10 Universal Differential Envelope as Factor Algebra

253

is established, with the following additional properties. Vz E Z2 .. 1mJ11 T· C Tz+l ,.

.1 0 .1

= O',

(7.10.3) This Z2-graded differential algebra T(E), with the nilpotent Z2graded derivation .1, is factorized with respect to D, and .1 is compatible with this factorization.

1mJ1ID ~ Dj F 3 [t] £![J1(t)] E F. Again the latter is some nilpotent Z2-graded derivation; hence F is some Z2-graded differential algebra over R. (7.10.4) Now take an arbitrary unital Z2-graded differential algebra Dover R, and consider an even algebra-homomorphism : A --+ D, Le., Vz E Z2 : 1m!A. ~ DE. is extended to F according to the commuting diagram below.

even algebra-homomorphism

A

----~.

E

- - -.... >

T(E)

J.l

de'

- - -.... >

T(E)/D

al@b UJ

iP(a) + o( iP( b)) D

Here the definition of an even R-linear map t/J is possible, due to the universal property of the tensor product R ® A. The homomorphism of unital superalgebras t/J. is factorized by D. Suppressing the embeddings, Vp EN, VXl, ...

,xp E E, Vr E R : t/J. (Xl ® ... ® Xp ) = t/J(Xl) ... t/J(Xp ), t/J.(r) := reD;

1m t/J.lo = {O}j "It E T(E) : .([t]) := t/J.(t).

254

7. Universal Differential Envelope

One therefore obtains some homomorphism of unital superalgebras ¢., which especially fulfills: ¢.08=00¢.; [{eR,O, ... }] =:eF--+eD.
(7.10.5) For every homomorphism of unital superalgebras ¢ : A --+ D, into a unital Z2-graded differential algebra Dover R, there is a unique homomorphism in the sense of unital Z2-graded differential algebras over R, namely ¢., such that ¢. ° J.L = ¢. (7.10.6) Since J.L is a homomorphism of unital superalgebras, due to the above factorization, one finds an isomorphism in the sense of unital Z2-graded differential algebras over R, according to the diagram below.

1 A

---->

D(A) --------. Q (A)

=R (9 D(A)

T(E)/ D

Here one defines: 0 (Q(A))

3

{r,O,O, ... } --:---.reF = [{r,O,O, .. .}]

0

E F .

/1-.

Both il. and {3. are homomorphisms in the sense of unital Z2-graded differential algebras over R. Hence one obtains:

il. ° {3.

= id

F, {3. ° il. = id ii(A)j eii(Al

+-----+

eF.

(7.10.6.1) With all the canonical embeddings written down, Le., an explicit notation of direct sums, Va, b E A, Vr E R :

{r, {a, eR ~ b}, 0, ... } + D --:-+{{r, a}, {eR' OA} !3.

~

b, 0, ...},

with the unital universal differential envelope written as direct sum over R, ii(A):= (REBA) EB

EB ((REBA) ~0A).

nEN

Conveniently one writes Va, b E A, Vr E R :

T(E)jD

3 [r

+ a + eR ~ b] --:-+r + a + db E !3.

ii(A).

Extension of Graded Star Operations

255

(7.10.7) There is an isomorphism (3* of unital Z2-graded differential algebras over R: tensor algebra of (A EEl (R 181 A)) ideal of S

+---+

R EEl'o(A).

1.11 Extension of Graded Star Operations to the Universal Differential Envelope (7.11.1) Let A = AD EEl Al be an associative superalgebra over the commutative ring R, and take an involutive isomorphism of rings: R 3 T +---+ fER, for instance complex conjugation. The bijection: A 3 a +---+ a* E A is called graded star operation on A, if and only if the following conditions are fulfilled. Va,b E A, VT,S E R: (Ta + sb)* = fa* + sb*, a** := (a*)* = aj Vi,y E Z2: a E AX,b E Ail ===} a* E AX, (ab)* = (-l)XYb*a*.

(7.11.2) Let T(E) be the tensor algebra over the direct sum of R-bimodules E = ED EEl E I , and consider an even antilinear involution: E 3 x +---+ x* E E, Le., Vx, Y E E, VT, S E R : (TX

Vi E Z2 : X E E

Z

===}

+ sy)*

x* E E

Z

= fx*

+ sy*,

XU =

x,

•

(7.11.2.1) T(E) is Z2-graded, such that

Vp EN, Vi'k : Xk E EZk

===} Xl

181··· 181 X p E EZt+··+ zp ,

and the embedding of TJ(E) := R into T(E), HT, O,···}; T E R} ~ (T(E))D. This Z2-grading of the unital associative algebra T(E) is defined with the R-linear bijection: p

TP(E) := @E +---+

E9 ZE Z 2

E9 Zl, ... ,zpEZ2,

Zt

(EZI 181 .. · 181 E9· +"'+Zp-Z

(7.11.2.2) There is a unique graded star operation: T(E) 3 t such that
= {f,O, .. .},

Vx E E: {O,x,O, .. . }*

+---+

t* E T(E),

= {O,x*,O, .. .}.

(7.11.2.2.1) The tensor product of R-bimodules E I , ... , E p can be used to lift a p-antilinear map f : E I x ... x E p ---+ F to an R-antilinear map f* : E I 181 ... 181 E p ---+ F, into an R-bimodule F, over the commutative ring R with an involutive isomorphism of rings.

256

7. Universal Differential Envelope

(7.11.2.2.2) There is an R-antilinear map: TP(E) such that \fZ1, ... , zp E Z2 :

---+

TP(E), for pEN,

EZI ® ... ® EZp 3 Xl ® ... ® X p ---+

(_1)L19
(7.11.2.2.3) Of course, if the bimodules EfJ, EI are free over R, one can use their bases for an appropriate definition of this antilinear bijection of T(E) onto itself. (7.11.2.2.4) The remaining properties of a graded star operation on T(E) are easily shown. (7.11.3) Let A = AD EB AI be an associative superalgebra over R, with the graded star operation *. There is a unique graded star operation on the universal differential envelope il(A), such that '
which explicitly means that {OA,e A ®a,O, ... }* = {OA,e A ®a*,O, ... }, eA := {eR,OA}.

Here the graded star operation on il(A) is again denoted by *. This so-called canonical graded star operation explicitly reads, '
(dal ... da n )* = (_1)L19
(dai)ao, (dai),

(7.11.3.1) Due to the universal property of the tensor product, there is an antilinear map * : il(A) ---+ il(A) with the above properties; obviously it preserves the Z2-grading of il(A). Moreover uniqueness follows from the very properties of graded star operations. (7.11.3.2) The idempotency, Le.: property that

* * 0

=

c := id il(A), follows from the

\fZ1,z2 E Z2,'
For an elegant proof of the latter, in order to avoid straightforward, but tedious calculations, one uses the factor algebra representation of REB il( A).

Extension of Graded Star Operations

257

(1.11.3.2.1) The graded star operation on A is extended, with the universal property of R ~ A, onto:

E

3 {a,r~b} +-----+ {a*,f~b*} E

E:= AEB

(R~A).

This even antilinear involution on E is then extended to a unique graded star operation on T(E).

Is

(1.11.3.2.2) 1m * ~ S, and therefore 1m the graded star operation: T(E)/D 3

It] +-----+[t*]

* 10

~ D. Hence one obtains

E T(E)/D.

*

(1.11.3.2.3) Hence one defines an even antilinear map: ii(A) 3 {r,w} --{f,w*} E ii(A) = REBn(A) .

•

(1.11.3.2.4) This even antilinear map * on ii(A), and the graded star operation * on T(E)/D, are compatible with the isomorphism

/3. : T(E)/D +-----+ REB n(A), which was defined in the previous section, due to the following commuting diagram:

*0

/3. = /3.

0

*,

because Va, b E A, Vr E R :

{r, {a, eR ~ b}, O, ...} + D --:-+{{r, a}, {eR' OA} ~ b, O, ... }, {3.

with * acting on both sides, such that a, b, r are replaced by a· , b· , f. Extending this diagram to (r + ao)dal ... dan, for ao, al, ... , an E A, r E R, yields the desired result: * 0 /3. = /3. 0 *, with the notation * used on both algebras. (1.11.3.2.5) V[t l ] E (T(E)/D)Z1, [t2] E (T(E/D)Z2 :

[tl][t2] ~(-1 Y1Z2 [t;][ti]

~(-1)ZlZ2W2Wi = * 0 /3. ([tl][t2])

E (R

EB n(A»Zl+Z2 .

258

7. Universal Differential Envelope

(7.11.4) In the special case of trivial Z2-grading of A, Ai := {O}, such that Vn EN, Vao,all'" ,an E A:

one obtains immediately Vr E R :

((r

+ ao)da 1 ... da n )*

= (-1) n ( y l )

(da~)··· (dai)(r + aii),

(aOdal)* = (dai)aii, with an obvious slightly modified notation, for convenience. In this case, EO = A, E i = R ® A.

(7.11.5) 'VZ], Z2 E Z2, VWl E !?Zi'(A),W2 E !?%2'(A) :

[Wl,W2]g := WIW2 - (-1)ZlZ~W2Wl ~(-1)ZlZ~[W2,wi]g. Vw E !?(A) : (dw)* = d(w*) == dw*; one equivalently writes * 0 d = do *. Vw E !?(A), bE A : (wdb)*

= p(w*db*).

p-l{3o*=_*o{3, p-1'Y0*=-*0'Y, po*op=*, op-l 0 * = * 0 {3',

11'

° * = * ° {3'd.

(7.11.6) On !?*(A) := HomR(!?(A) , R), an even antilinear involution is defined, such that

VI E !?*(A),w E !?(A) : j*(w) := I(w*);

(j*)* = f.

Here one uses the Z2-grading according to the direct sum

!?*(A) =

E9 (!?*(A»i,

(!?*(A»Z := {I E !?*(A); 1m Iln;:;:T(A) = {O}}.

iEZ~

(7.11.6.1)

j*od=(fod)*, j*o'Y=_(fo'Yp-l)*, j*o{3=_(fo{3p-l)*. Hence one obtains the following implications.

I closed

~

j* closed, I cyclic

I graded trace

* : C*(A)

+-----+

~

~

j* cyclic;

j* graded trace.

C*(A), CHA)

+-----+ C~(A),

ZHA)

+-----+ Z~(A).

(7.11.6.2)

VI E C~(A) : j* ° {3 =

-(f ° {3)*; hence *:

B~(A) +-----+ B~(A).

7.12 Cycles over Associative Superalgebras

259

7.12 Cycles over Associative Superalgebras (7.12.1) Let D be a Z2-graded differential algebra with the derivation b, over a commutative ring R. The linear form lED· := Hom R (D, R) is called cycle and usually denoted by J, if and only if it is some closed graded trace, which explicitly means that

Va,b ED: lob(a)

=

!

b(a) = 0, ![a,b]g = 0,

with the super-commutator

[a,b]g :=ab-

L

(-l)XlIbiia~.

~,iiEZ2

(7.12.1.1) Especially let D be bigraded. Then the cycle cycle, if and only if "In

I- kENo: IID

k

I

is called an n-

= 0.

l'

(7.12.2) The tensor product J ® of cycles, on the skew-symmetric tensor product Drf9D' of Z2-graded differential algebras D and D' over R, is again some cycle. Remember that Vi, y E Z2, Va E D, a' E D'~, b E Dii, b' E D' :

(a®a')(b®b') = (-l)XII(ab) ® (a'b'), b®a' E (Drf9D')x+lI, .:1(b ® b') = b(b) ® b' + (-l) lI b ® b'(b'). If both D and D' are bigraded, then

Vi,y E Z2,Vk,l E No,Va E D:,a' E D'f: a®a' E (Drf9D')%~r. VI E D·,f' E D'·,a E D,a' E D': I®f'(a®a') = I(a)f'(a').

(7.12.3) Let

I

= J be a cycle on D, and A an associative superalgebra over

R. A homomorphism of associative superalgebras ¢ : A extended to some homomorphism ¢. : il(A) differential algebras over R, Le., Vz E Z2: 1m ¢.lnS(A) ~ D

Z

,

-----+

-----+ D is uniquely D in the sense of Z2-graded

¢. od = bo¢•.

One then finds, that "In E N, "lao, aI, ... ,an E A, Vr E R:

(r + ao)dal ... dan -----+(r + ¢(ao))b 0 ¢(at}··· 8 0 ¢(an ). ¢.

260

7. Universal Differential Envelope

(7.12.3.1) The cycle 1 is pulled back to the linear form l' := to the diagram below.

D(A) Vw E n(A) : 1'(w)

;=

character

1 0 ¢.(w) ==

def

f

lo¢., according

• R

J

¢.(w).

The family {D, 0, A, ¢, f} == {o, ¢, f} is called cycle over A. The pull back E n·(A) is called the character of this cycle.

l'

(7.12.3.2) Especially let D be bigraded. The family {c5,¢,f} is called Nocycle over A, if and only if 1m ¢ ~ Do; in this case

"In E No: 1m ¢.lnn(A) ~ Dn; here no(A)

:=

A.

(7.12.3.3) Obviously the character T is some closed graded trace, which explicitly means that T 0 d = T 0 (3 = O.

T

E Z.x(A),

(7.12.3.4) Especially let {c5,4>,f} be a No-cycle over A. If 1 is a n-cycle for some n E No, then the character l' is some cyclic n-cocycle of A, T

E

Z.xn(A)

:=

Z.x(A) n

n~(A).

{7.12.4} Conversely every closed graded trace 1 E Z,X(A) is the character of the cycle {d, 4>, f} over A, which is denoted by the diagram below. A

• n(A)

,

~ ~ ~Zi(A) n(A)

f

(7.12.5) Let

{c5: D

--+

{c5' : D'

D,¢: A

--+

--+

D', ¢' : A'

D,f: D --+

--+

R},

D',!, : D'

--+

R}

'

R

7.12 Cycles over Associative Superalgebras

261

be two cycles over A and A', respectively. The character of the tensor product of these two cycles is the cup-product of their characters, according to the commuting diagram below.

v..

(1.12.5.1) The tensor product of these two cycles over A and A', respectively, is defined as the cycle over A0A', Le., the family

A, ,~, A A, ,T(¢,'¢) ,f ® f '} , {D®D A®A with the usual derivation L1 on D0D'. Here the notation ¢ ® ¢', instead of T(¢, ¢'), would be misleading. (1.12.5.2) The R-linear map

T(¢, ¢') : A0A'

3

a ® a' - - ¢(a) ® ¢'(a')

D0D'

E

is some homomorphism of associative superalgebras. Hence it is extended uniquely to some homomorphism (T(¢, ¢')) .. in the sense of Z2-graded differential algebras over R. (1.12.5.3) On the other hand, since T(¢ .. , ¢~) in just this sense, one finds that

T(¢.. , ¢~)

0

0

v.. is some homomorphism

v.. = (T(¢, ¢'))...

(1.12.5.4) Hence one obtains, with the characters r := the desired result

(f ® /')

0

(T(¢, ¢')) ..

= (r ® r')

0

v..

= r * r'.

f 0 ¢ .. , r' := /' 0 ¢~,

262

7. Universal Differential Envelope

1.13 Fredholm Modules (7.13.1) Let A be an associative algebra over the field C of complex numbers.

(7.13.1.1) A is called *-algebra, if and only if an involution exists, which is defined as an antilinear bijection: A 3 A +-----+ A- E A, such that

VA,B

E

A,Va,/3

E

C:

In this case, VA E A: A = 0 <===} A- = o. Moreover in this case, a subset of A is called self-adjoint, if and only if VAt EAt: Ai EAt.

At

(7.13.1.2) A is called normed algebra, if and only if it is some normed space, and the following product inequality holds. VA, B E A, Va E C :

II All

~ 0,

IIA + BII

IIAII ~

= 0

<===}

A = 0,

IIAII + IIBII, IIABII

IlaA11 = lalllAII, ~

IIA II IIBII·

In this case, A is called Banach algebra, if and only if it is complete with respect to this norm. (7.13.1.3) Let A be a Banach algebra with involution. A is, called B-algebra, if and only if

A is called C- -algebra, if and only if

in this case,

(7.13.1.4) If the *-algebra A is unital with the unit I, then I

= I-

and

11111 = 1; here A is assumed to be non-zero, which equivalently means I:f:. O. (7.13.1.5) Let A be a B--algebra. Its unital extension A:= CEBA, with the unit i:= {1,OA}, is some B--algebra, with the involution and norm defined such that

Va E C,A E A: {a,A}-

:=

{a,A-},

II{a,A}11 := lal + IIAII.

7.13 Fredholm Modules

263

(7.13.1.6) If A is a C·-algebra, then its unital extension, with the unit and involution defined above, becomes some C·-algebra, with the norm such that

Va. E C, A

E A:

11{a., A}II

:= sup{IIa.B

+ ABII; BE A, IIBII

= I}.

(7.13.1. 7) Let I be a left or right ideal of the *-algebra A. If I is self-adjoint, then obviously I is some two-sided ideal of A. (7.13.1.8) The B·-algebra A is called simple, if and only if the only closed two-sided ideals of A are A itself and {O}. (7.13.2) Let 'H. be a Hilbert space with complex coefficients. The set B('H.) of bounded operators on 'H. is some C·-algebra, with the adjointness t of operators, and with respect to the operator norm. (7.13.2.1) Obviously every closed self-adjoint subalgebra of the C·-algebra

A is some C· -algebra. (7.13.2.2) The set Boo('H.) of compact operators on 'H. is an ideal of B('H.) , and Boo ('H.) is some C·-algebra. (7.13.3) Let 'H. = 'H. 0 EB 'H. I be a separable Hilbert space with complex coefficients, with the corresponding orthogonal projectors NO, N I , and the socalled Klein operator K := NO - N I = I - 2N I . (7.13.3.1) The C·-algebra B('H.) is decomposed into the direct sum of closed linear subspaces,

B('H.) =

E9 BZ('H.), ZE Z 2

BZ('H.) := { A E B('H.); "Ix E Z2 : 1m AI1i.f ~ 'H. x+ z } . "If

= fO + fI

E 'H.,A

= AO + AI

which equivalently means, that AO= ~ ANo + N I ANI, A I

E B('H.), Vi,x E Z2 :

p

A1(A(P»x+z,

= N I ANo+ NO ANI.

(7.13.3.2) This direct sum is equivalently written in matrix notation. VA = AO + AI E B('H.) :

° [A0

OO

A =:

K :=

0]

All' A I =:

[~ ~I ],

onto [

[0

~~]

AIO E

'H..

A0 0

1 ]

'

264

7. Universal Differential Envelope

(7.13.3.3) Obviously the closed subalgebra BO('H) is self-adjoint, hence some C* -algebra. (7.13.3.4) The unital associative algebra B('H) is Z2-graded, with the grading automorphism K:

B('H) 3 A

= AO + A l

+-----+ de!

AO - AI

= KAK E B('H),

K

2

= id B('H).

(7.13.4) Let A = AO EB Al be an associative superalgebra with complex coefficients. Consider some representation 1r of A on 'H, Le., a homomorphism of complex associative superalgebras:

(7.13.4.1) If A is unital with the unit e, one demands 1r(e) = I. In this case one obtains an A-right module 'H over C, with the definition that

"If E 'H,a E A: fa:= 1r(a)(f). (i) 'H is some right module over the ring A. (ii) "If E 'H,a E A,a E c: a(fa) = (aJ)a = f(aa). Here the product of B('H) is used in the sense, that VA, BE B('H), f E 'H : (A

0

B)(f) := B(A(f)).

(7.13.4.2) With the grading automorphism () : A 3 aO

+ al

+-----+ aO -

*1

aI E A

'

()2 =

id A

,

the Z2-grading of 1r is written such that Va E A:

1r 0

()(a) =

K

o1r(a) = K1r(a)K.

(7.13.5) Let F be an odd bounded operator on 'H, F =:

[~ ~],

KFK = -F.

The A-right module 'H over C is called Fredholm A-right module, if and only if 2

Va E A: 1r(a)(F

-

I) E Boo('H), F1r(a) - K1r(a)KF E Boo('H).

In this case, 'H is called involutive, if and only if one especially finds F2 = I.

7.13 Fredholm Modules

265

(7.13.6) Let {Tn;n E N} be the sequence of repeated singular values of

T

E

Boo('H).

)0, +oo[ n spectrum (TtT) =: {T~; n EN}, "In EN:

°~ TtT

°<

Tn+! ~ Tn;

00

T~ < gn I > gn,

= L n=l

with an arbitrary orthonormal basis of eigenvectors gn'

<

(7.13.6.1) For a compact operator T E BooCH), and 1~ p is defined by

(t, T~

o ~ IITII. ,~

)

1/.

~

00,

the p-norm

00.

The usual operator norm on B(1i) is denoted by IITlloo

==

IITII·

(7.13.6.2) 00

ITI := (TtT)1/2 = LTn

< gnl > gn ~ 0, ITIITI = TtT.

n=l

VI ~ p

°~

< 00

:

0 ~ IITll p = IITtll p = IllTlll p = IIIT t lli p ~

IITII = IITt II = II ITI II =

(7.13.7.1) V T 1 , T 2 E B(1i), V 1 IITlilp

<

00

==>

II ~

ITt I II <

p<

00.

00.

00 :

max {IIT1 T2I1p, IIT2Tlllp} ~ IIT1 1l p1lT2 11.

Here T 1 is assumed to be compact, such that both T 1T 2 andT2T 1 are compact.

(7.13.7.2) V T 1 ,T2 E BooCH),Vp,q,r E [1,+00[: 1

1

1

-p + -q = -r ==>

IIT1T211r ~ IIT1 ll pllT2 11q ~

00.

(7.13.8) The Schatten ideals B p (1i) , 1 ~ p < 00, are defined as the ideals

of B(1i). With respect to the p-norm, B p (1i) is some Banach space. VI ~ p


00 :

B p (1i) C B q (1i) C B oo (1i) C B(1i).

266

7. Universal Differential Envelope

(7.13.8.1) The trace class B l (1t) is equipped with the trace norm·IITlll, and the Schmidt class B 2 (1t) with the Schmidt norm IITI12' for T E B oo (1t). One then obtains, that

(7.13.9) The trace of T E B l (1t) is defined as 00

tr T:=

L

< gnlTgn

>E C,

n=l

which obviously is independent of the choice of an orthonormal basis {gn; n E N} for 1t. One then finds, that

Vp, q

E

1

1

]1, +oo[ ,V T l

E B p (1t), T 2 E B q (1t) :

-p + -q = 1 => tr(Tl T 2 ) = tr(T2 T l ). V T E B I (1t) : Itr TI :5 IITIII; VI :5 p <

00, "IT

E B p (1t) : IITll p

= (tr(ITIP»l/P.

(7.13.10) The Fredholm A-right module 1t is called p-summable, for 1 :5 p < 00, if and only if one finds that Va E A:

(7.13.11) The super-commutator algebra (B(1t»L, of the unital associative superalgebra B(1t), is defined with respect to the super-commutator.

Obviously 1f' provides an according representation of the super-commutator algebra AL on 1t, Le., a homomorphism 1f': A L ----> (B(1t»L in the sense of complex Lie superalgebras. (7.13.12) F

2

=

[PoQ

QOp] , F Ft

=

[P

~t Q~t],

F 2 = I {:::::::} rp: 1t 1 ----> 1t0, p-l

= Q1.

F

[Q~Q

Ft F

=

= Ft

{:::::::} P

F is unitary, if and only if both P and Q are unitary.

p?p ] .

= Qt.

7.13 Fredholm Modules

267

(7.13.13) For A E B(1t) and p E [1, +00]' assume that

FA - KAKF

E

Bp('H)j

then one easily finds, that

(F 2 - 1)A E Bp('H)

A(F2 - I) E Bp('H).

{:=:::>

(7.13.14) VA E B('H) : 2 [AOO(PQ - I) A(F - I) = A 10(PQ - 1)

(7.13.15) Remember the well-known criterion: 'liTE B(1t) : T compact {:=:::>

fVg E 1t:<

gl/n >

----4 n~oo

0 ==>

liT/nil n-+oo 01· ----4

Hence one finds, that 1t is some Fredholm A-right module, if and only if the above eight bounded operators, AOO(PQ - I), . .. ,QA01 + A 10 P, are compact for all A E 1m 11'. (7.13.16) The supertrace of T E B 1 (1t) is defined as an element of Bi(1t):= Homc(B 1(1t),C) 3 str: B 1('H) 3 T

----4

tr(KT)

E

C,

IIKTlll = IITI11. One easily finds, that 'liTE B 1(1t) n Bl(1t) : str T = O. With the convenient notation, that V1~p

< 00, 'liz E Z2 : B;(1t)

:= Bp('H) n BZ('H),

one finds, that 'lip, q E ]1, +oo[ ,"Ix, fj E Z2, V T 1 E B:(1t), T 2 E Bg('H) :

!p +!q = 1 ==> str(T1T2) = (-I)Xllstr(T2TI), which just means, that the supertrace vanishes on super-commutators. (7.13.17)

V T l E B 1(1t), T E B(1t) : tr([T1,T]) = 0, str([T1,T]g) = O.

268

7. Universal Differential Envelope

(7.13.18) The Z2-graded derivation 0 : B(1i) such that 'VA E B(1i):

---t

B(1i), which is defined

o(A) := i[F, A]g = i(FA - K AKF) = iK[KF, A] E B(1i), fulfills the subsequent conditions.

'Vi E Z2 : 1m OIBR('H) ~ B + (1i). Z

1

'Vi E Z2, 'VA E B Z (1i), BE B(1i) : o(AB) = o(A)B + (_l)Z Ao(B). 'VA E B(1i) : 00 o(A) = [A, F 2]. Hence one obtains some Z2-graded differential algebra. F 2 = I===}0 0 0 = 0, o(F) = 2iI. (7.13.19) 1 'V T E B 1 (1i) : str 0 o(T) = 0, str T = 2i str(Fo(T».

(1.13.20) One uses the following terminology. Let D = DO EB D 1 be a Z2graded differential algebra over the commutative ring R, with the Z2-graded derivation O. The linear form fED· := HomR(D, R) is called closed, if and only if f 0 0 = 0; f is called graded trace, if and only if it vanishes on super-commutators, such that

(1.13.21) As an example, assume F2 = I. In this case B(1i) is some Z2graded differential algebra over C. With the restriction 01 : Bl (1i) ---t Bl (1i) of 0, B 1 (1i) again is some Z2-graded differential algebra over C. The supertrace str E Bi (1i) is some closed graded trace, because of

str 0

0!B1('H)

= o.

(1.13.22) Let the Fredholm A-right module 1i be involutive, F 2 = I. The representation 7r induces some homomorphism 7r. : n(A) ---t B(1i) in the sense of Z2-graded differential algebras over C, according to the diagram below. Jf

inclusion

embedding" n(Jf)

~/ S(.9f) ' .

.

mcluslon

-----_" C sfr

7.13 Fredholm Modules

269

(7.13.22.1) Vn E N, VaO,al,'" ,an E A:

(1

+ ao)da l ... dan ----+{1 + 1r(ao))i n [F, 1r{al)]g'" [F,1r{a n )]g. 1r.

(7.13.22.2)

r.?tr{A):= {w

E

r.?{A);1r.{w)

E

B I{1t)}, t:= stro1r.ln,r(.A)'

One easily finds that r.?tr{A) is an ideal of r.?{A). Moreover r.?tr(A) is some Z2-graded differential algebra over C, because of 1m dln'r(A) ~ r.?tr(A), Le., Vw E r.?tr(A) : 1r.(dw) = b 0 1r.(w) = ifF, 1r.(w)]g E B I (1t). The linear form t E r.?;r(A) := Homc{r.?tr(A), C) is an even closed graded trace. Vw E r.?{A) : 1r.{w) E BI(1t) ==> t{w) = 0; Vw E r.?tr{A) : to d{w) = str 0 b 0 1r.(w) = 0; VWI,W2 E

r.?tr{A) : t{[WI,W2]g)

=

str([1r.(wd,1r.(W2)]g)

= O.

Here the super-commutator on r.?{A) is defined according to its Z2-grading. (7.13.22.3) Vn E N, VaO,al,'" ,an E A:

aodal'" dan E r.?tr(A) ==> Tn{ao,al, ... ,an) := t(aoda l .·. dan) =

i ntr(K1r(ao)[F,1r(ad]g'" [F,1r(a n )]g);

especially,

aao + aal These C -

+ ... + aan + ii = I ==> Tn{aO,al,'" ,an) = O. (n + 1) -linear forms Tn are called cyclic n-cocycles of A.

(7.13.22.4) The linear form t := stro1r.ln,r(A) is an example of (generalized) cycle. (7.13.23) Let the involutive Fredholm A-right module 1t be p-summable in the sense, that

:3 1

~

P < 00: VA

E 1m 1r:

[F,A]g = FA - KAKF

E

Bp{1t).

Then one easily obtains, that Vn EN:

n

~

p ==> r.?n{A) = sum{aoda l ··· dan, dal'" dan; aO,al, ... ,an E A} ~ r.?tr{A).

270

7. Universal Differential Envelope

(7.13.23.1) Here one uses, that

Vn E N, VAl, ... , An E B p (1t) : n 2: p 2: 1 = } AI'" An E B I (1t)· (7.13.24) Let the involutive Fredholm A-right module 1t be p-summable. Then Vn 2: P 2: 1 :

.f.?n(A) ~ .f.?tr(A), to ,lnn+1(A) = to .Blnn+dA) = 0, to plnn+dA) = t 0 >.1 nn+l (A) =

tl nn +dA),

which explicitly means, with the J notation of closed graded traces on Z2graded differential algebras, that Vao E AZo, ... , an+l E AZn+l :

J

aoda l ... da n+1 = (_l)(1+zn+1)(n+1+ zo+ .. + zn)

J

an+1 da Oda l'" dan.

(7.13.24.1) The construction of closed graded traces is extended to .f.?p-I(A), using the definition: ffi

W

.f.?n(A)

3

t'

i 2

W ----+ --str(F(6 0 1r.(W)))

n~p-I~O

de!

E

C.

One then finds, that Vn 2: P - 1 2: 0, Vao, ... , an E A :

t'ldom t'n(dn(A)unl(A»

= 0;

t' - tltn 'l7n+l~p~l

Aodnn(A)

= O.

(7.13.24.2) For P 2: 2, VWI, W2 E dam t' :

t'([Wll W2]g) = str([1r.(wd, 1r.(W2)]g)

= 0,

because in this case WIW2 E .f.?tr(A). Here one uses, that VZi',z2 E Z2,VWI E .f.?:zt(A),W2 E .f.?Z2'(A):

F(6 0 1r.([Wl,W2]g)) = i([1r.(W.(WI),1r.(W2)]g - (-1)Zl+z2F[1r.(WI),1r.(W2)]gF).

7.14 Connes Modules

271

7.14 Connes Modules (7.14.1) Let 1t = 1t0 EB 1t I be a separable Hilbert space with complex coefficients, with the Klein operator K. Let D be densely defined and closable on the Z2-graded linear subspace V = (V n 1t0 ) EB (V n 1t I ), and assume D to be odd, which explicitly means that Vi E Z2, "If E V n 1t z : D f E 1tZ+1. These two assumptions equivalently mean, that KV=V:=domD, V=1t, KD=-DK.

(7.14.1.2) For a closable linear operator A : 1t 2 dam A --+ 1t, one finds easily the following lemma. If K dam A = dam A and K AK = (-1) Z A, then K dam A = dam A and K AK = (_1)Z A. Therefore especially Kdom lJ

= dam

lJ, KlJ

= -lJK.

(7.14.1.3) Let fJ be injective; then D- 1 is closable, D-l = lJ-l. Moreover let fJ- 1 be bounded; then ran lJ = ran D. Assume ran D = 1t; in this case lJ- 1 E B(1t); due to the above lemma, lJ- 1 E B I (1t). Finally assume, for some 1 ~ p < 00, that

lJ- 1

E B~(1t) := B p (1t)

n B I (1t).

(7.14.1.4) Let A = AO EB AI be an associative superalgebra with complex coefficients, and consider a representation 1r : A --+ B(1t). If A is unital with the unit e, one demands 1r(e) = I := id 1t. (7.14.1.5) Furthermore assume, that VA Elm 1r : A dam fJ ~ dam lJ, [A, lJ]g bounded; then [A, D]g E B(1t). The family {1t, D, A, 1r} is called p-summable Connes module.

(7.14.1.6) Let A be equipped with some graded star operation *, with respect to complex conjugation, Le., an even antilinear involution * : A +-----+ A, such that

"Ix, y E Z2, Va E AX, bE Aii : (ab)*

=

(-1) XY b*a*.

Let the representation 1r be compatible with * and the adjoint in the sense, that Va E

A : 1r(a*)

= (1r(a))t j

in this case 1r is called graded *-representation of A on 1t. The above Connes module is called self-adjoint, if and only if D is essentially self-adjoint; in this case both lJ and lJ- 1 are self-adjoint.

272

7. Universal Differential Envelope

(7.14.2.1) Consider the p-summable Connes module {'H, D, A, 11"}, for some 1 ~ p < 00. The separable Hilbert space 1i EB 'H is equipped with an appropriate Klein operator L. 'H EB 'H =: .c = .cO EB.c 1, VZ E Z2 : .c f := 1i z EB 'H f

L :=

1],

[~

;

2

L = id.c.

An according representation of A on L is defined:

A

3

P

a d,;j

[1I"(a) 0

0]

1I"(a)

E

B(.c).

If 11" is some graded *-representation, then p is also some graded star representation: A 3 a$ ---+(p(a»f E B(.c). P

(7.14.2.2) The following odd operator F is densely defined and closable.

F:=

[D~l ~],

dom F:= ran DEBdom D, domF='HEBdomD, 1'=

[V~l ~].

One easily finds, that

L dom F

=

dom F, LFL

=

-F, L dom I'

=

dom 1', LFL

=

-F.

(7.14.2.3) Va E A:

[F,p(a)]g = LD-?,A]g E B(.c),

with A := 1I"(a), because of the assumption that VA Elm

11":

[V,A]g:= VA - KAKD, [D,A]g E B('H).

Hence one obtains, denoting A:= 1I"(a), B:= 1I"(b), that Va,b E A:

7.14 Connes Modules

273

(7.14.2.3.1) One easily proves, that VA,B E B(1t):

[~ ~]

E

Boo(.e) <==> A and B

E

Boo (1t).

(7.14.2.3.2) Considering carefully the non-repeated singular values with their multiplicities, one finds that V 0 < p < 00,

VA, BE B oo (1t) : A and B E Bp(1t) <==>

[~ ~]

Here an ideal B p(1t) of B(1t) is defined for 0 < p <

B p(1t)

{T

:=

E

Boo (1t);

E Bp(.c).

00,

~ T~ < oo} ,

inserting the repeated singular values Tn of T.

(7.14.2.4) Let al, ... ,am E A, and denote \/"tl : 1l'(aL) =: AI. One immediately finds that, for even m,

[F,p(at}]g'" [F,p(am)]g -

--1

=

--1

-

--1

-

--1

!~:' D]g~ _.~~ [~' Am-1]g~ _1[A m , D]g~

[D' A1]gD [

-

o

D

m ~ 2n ~ p ~ 2

[A 1,D]gD ==}

[D,A 2]g·ooD

[P, p(a1)]g ... [F, p(am)]g

0

] . __ 1 [A m- 1,D]gD [D,Am]g E B 1(.c).

(7.14.2.5) Obviously,

p2 =

[~ II

] , dam p2

0

= Pdam P = dam P, F2 = F2 = id.c.

dam D

(7.14.2.6)

VA,BEB 1 (1t):

[~ ~]

EB 1 (.C),

tr[~ ~]

=trA+trB.

(7.14.2.7) "In E N, "lao E AZo, ... ,a2n E A~:

2n ~ p ~ 2, ZO + ==}

+ Z2; = 0 [F, p(a1)]g'" [P, p(a2n)]g E B 1(.c),

T2n (ao , a1, =

00

•

00

,a2n)

•

:=

(-l)ntr (Lp(a o)"""'[F""",-p(:-a-:1):-]g ... [F, p(a 2n )]g)

(-1)L::=1 za-1tr (Kb- 1[D, Ao]g

00.

b-1[b,A2n_1]gb-1[D, A 2n ]g) ,

with the notation, that nl : Al := 1l'(al)' Here one uses, in order to commute under the trace, that V 0 < p < q < 00 : B p(1t) C B q(1t) C B oo (1t).

'i5

274

7. Universal Differential Envelope

(7.14.3) For simplicity at the moment, let D = D E B 1 (1i); then F

=F E

B 1(£), F 2

= id £.

(7.14.3.1) B(£) 3 T ~ i[F, Tl g = i(FT - LTLF) E B(£). de!

With this nilpotent Z2-graded derivation 8, B(£) is some Z2-graded differential algebra over C. Vz E Z2, VS E B Z (£), T E B(£) : 8(ST) Vz E Z2 : 1m 8!B«.c) ~ B Z + 1 (£);

= 8(S)T + (_l)Z S8(T);

808 = O.

(7.14.3.2) VT E B 1 (.£) : str T:= tr(LT)

1

= 2istr(F8(T)),

str

0

8(T) = 0,

because the supertrace vanishes on appropriate super-commutators. Obviously str

IB (.c)nBI (.c) = 1

O.

(7.14.3.3) The universal property of the non-unital differential envelope is used for the subsequent diagram. inclusion

n t r (J'l)

/.~ B(£)

•

.

I

With the intertwining property: p. Vm EN, VaO,al,'" ,am E

.

mc us/on 0

B, (£) - - - - -... C sfr

d = 8 0 p., one then obtains that

A:

One immediately verifies, that 1m dln'r(A) ~ ntr(A) := p;1(B 1 (£)). The Z2-graded differential algebra ntr(A) is an ideal of n(A).

7.14 Connes Modules

275

(7.14.3.4)

t

:=

str

0

p.lntr(A)

E

,Wr(A)

:=

Homc(iltr(A), C).

One immediately finds, that

Moreover this Z2-graded trace t is even and closed.

tlntr(A)nnl(A)

= 0,

= 0.

to dlntr(A)

(7.14.3.5) "In E N, "lao E AZo, ... , a2n E

AZ~n

:

+ ... + Z2n = 0 ~ aOdal ... da2n E iltr(A), t(aoda 1 ... da2n) = (-l)ntr (Lp(ao)[F, p(at}]g'" [F, p(a2n)]g) 2n ~ p ~ 2, Zo

= T2n(ao,al,'" ,a2n); 2n ~ p ~ 2, Zo

+ ... + Z2n = I

(7.14.3.5.1) "In E N, "lao, 2n ~ p ~ 2 ~ aoda 1

~

t(aoda 1 •.• da 2n ) = 0.

,a2n+l E A : da2n+l E iltr(A), t(aOdal'" da2n+t} = 0.

(7.14.3.6)

EB

ilm(A) ~ iltr(A). "1m E No : 1m 'l'ln",(A) ~ ilm(A),

m~2n~p~2

because the restrictions of p and oX onto Ao dilm (A) are acting as linear bijections onto this linear subspace, the restriction of ponto dilm(A) is bijective onto the latter, the restrictions of p and oX onto A are bijective onto A, and oX 0 d = 0. (7.14.3.7) Therefore one obtains "In EN:

n - 1 ~ p/2 ~ 1 ~ to '1'1",

W~~2n

t

0

oXl$",~~nn",(A)

=

n

(A)

~~

= to 111m

w~~2n

n

(A)

~~

= 0,

t 0 pl$",~~nn",(A) = tl$",~~nn",(A)'

Explicitly one finds, that for n - 1 ~ p/2 ~ 2n, "lao E AZo, ... ,am E AZ'" :

~

1,

"1m

J

aOdal ... dam = (_l)(1+z",)(m+ Zo +'''+zm-ll

J J

= (_l)(l+z",)(l+m+ Zo +,.·+z",-ll

amdaO'" dam-l damaoda 1 ... dam-l'

7. Universal Differential Envelope

276

(7.14.3.8) Here in the sense of cycles over associative superalgebras, t is written as fermionic integral. Vi E Z2, Vw E ntr(A) n n!(A) :

J

w == t(w) = (-l)%t(w) = str

0

P.(w).

(7.14.4) In order to omit the above assumption on D to be bounded, consider the linear mapping: 1m P 3 6'

[~ ~]

.----.

-;;;; l[F, T]g =

=:

[

T 0

b-1[A, b]gb- 1

l

-] [b,A]g 0

E B(£).

Under the assumption, that 1m 6' ~ 1m P, define 6: 1m P that 6' = /I 0 6, with the inclusion /I of 1m pinto B(£).

--+

1m p, such

(7.14.4.1) The associative superalgebra D(£) := 1m p, with the Z2-grading defined by the direct sum D(£) =

EB D

Z

(£), D Z (£) := D(£) n B Z (£),

zEZ~

is some Z2-graded differential algebra over C with the super-derivation 6, because VA E 1m 1f, "IS, T Elm p, Vz E Z2 : KAK = (-l)%A

==}

K[A,b]gK = (-l)%+1[A,b]g,

due to an according lemma on closable linear operators cited above.

= (-l)%S ==} 6(S)T + (-1)%S6(T)ldom P = i[F,ST]g; 60 6(T)ldom P = o. LSL

Here one uses, that "IT Elm p: T dom F ~ dam F. If A is unital, then D(£) 3 id £ is unital too.

(7.14.4.2) Consider the following diagram. A

>

n(A)

I\}

S( L)

inclusion

c

nt, (A)

~~.....

t

0 .§·/0

c;

'0inC "us/on D( L) '.inC I us/on . D1 ( L) - - - - - , C sfr

7.14 Connes Modules

277

(7.14.4.3) 'im E N, 'iao,al,'" ,am E A:

(1

+ ao)da l ... dam -----+ i m(l + p(ao»[F, p(al)]g'" [F, p(am)]g, p~

with the super-derivation:

D(£)

T~i[F,T]g

3

E

D(£), fJ op~

=

p~ od.

(7.14.4.4) Obviously n:r(A) is an ideal of n(A) . Furthermore one finds, that 'iA E 1m 1r n B I (1i) :

[D,A]g

E

B I (1i) ~ 1m

fJID1(.C)

~ D I (£) ~ 1m d\:a;r(A) ~ n~r(A).

In this case n:r(A) is some Z2-graded differential algebra over C. Here, as before for BI (£), one must carefully show that D(£), D I (£) are Z2-graded linear subspaces of B(£), Le.,

D I (£)

=

E9 Di(£),

Di(£)

:=

D I (£) n B Z (£),

!E Z 2

which in turn implies, that

E9 (n~r(A) n n!(A»).

n~r(A) =

!E Z 2

(7.14.4.5) 'in E N, 'iao E AZo, . .. ,a2n E A Z2n

aoda l

...

da 2n

:

If 2n 2: P 2: 2, then

E n~r(A),

and the following implications hold.

zo + zo +

+ Z2n = 0 ~ t(aoda l + Z2n = I ~ t(aoda l

da2n)

= T2n(ao, ... , a2n);

da2n)

= O.

(7.14.4.5.1) 'in E N, 'iao, al, .. " a2n+l E A :

2n 2: P 2: 2 ~ aodal ... da2n+1

E n~r(A),

t(aOdal'" da2n+l)

= O.

(7.14.4.6) Let d be reduced by ntr(A)j then the even graded trace t is closed.

'iwi

E n~r(A),W2 E

tln;r(A)nnf(A) =

n(A) : t([WI,W2]g)

= 0;

OJ to dln;r(A) = O.

(7.14.4.7)

E9

nm(A) ~ n~r(A).

m~2n~p~2

Therefore the restrictions of t 0 d, to" t E9m~2n nm(A), vanish for n - 1 2: p/2 2: 1.

0

{3, t

0

A - t, top - t, onto

8. Quantum Groups

Following the terminology of V. G. Drinfel'd (1987), an enormous variety of algebraic, topological, and also differential structures is subsummed under the notion of "quantum groups" , due to the at least four-fold origin of according research. One main starting point was the study of solutions of the parameterdependent classical and quantum Yang-Baxter equation (CYBE, QYBE) , which led M. Jimbo to the definition of an appropriate q-deformation of the simple complex Lie algebra associated with a symmetrizable generalized Cartan matrix (1985). The two sources of this new construction were the monograph by V. G. Kac on Kac-Moody algebras (second edition, 1985), the first edition of which had already been printed by Birkhauser, Boston, 1983, and in particular the pioneering papers by P. P. Kulish, N. Yu. Reshetikhin, and E. K. Sklyanin on the parameter-dependent CYBE and QYBE (1981, 1982, 1983, 1984), were the universal enveloping algebra of 8l(2, C) was originally q-deformed. The solutions of the parameter-dependent CYBE, with their values in a simple finite-dimensional complex Lie algebra, were classified by A. A. Belavin and V. G. Drinfel'd (1983). The geometric significance of the CYBE with respect to Poisson-Lie groups was pointed out by V. G. Drinfel'd (1983). The study of quadratic algebras, the relations of which can be written in terms of an R-matrix, which solves the parameter-independent QYBE, was motivated from the quantum inverse scattering method (QISM), which had been developed by L. A. Takhtajan and L. D. Faddeev (1979), P. P. Kulish and E. K. Sklyanin (1982), and collaborators. In their fundamental paper of 1989 (1990), N. Yu. Reshetikhin, L. A. Takhtajan, and L. D. Faddeevestablished quadratic algebras corresponding to the four series of non-exceptional simple finite-dimensional complex Lie algebras, and also the duality between these Hopf algebras and the q-deformations of the universal enveloping algebras of the involved Lie algebras. These quadratic algebras were also used in order to transform quantum spaces. The q-deformed universal enveloping algebras were finally reconstructed as quantum doubles. The view of quantum groups as super-bialgebras of transformations of quantum superspaces was pointed out by Yu. 1. Manin (1989), who in the

280

8. Quantum Groups

sequel also investigated related aspects and examples of non-commutative differential geometry (1991, 1992). From the aspect of model construction, J. Wess and B. Zumino proposed a new differential calculus on the so-called quantum plane (1990), which was generalized to quantum spaces by A. Schirrmacher (1991). In his fundamental contribution to the Proceedings of the International Congress of Mathematicians at Berkeley (1987), V. G. Drinfel'd used the quantum double construction in order to write down explicitly the universal R-matrix of the quasitriangular Hopf algebra Uq (sl(2, C)), and to indicate its generalization to Kac-Moody algebras. Starting from the commutative unital associative algebra of continuous functions on a compact topological group, S. L. Woronowicz (1987, 1989) developed his theory of so-called compact matrix pseudogroups, including in particular an appropriate bicovariant differential calculus, as some concept of quantum groups on the level of C·-algebras. A purely algebraic approach to compact quantum groups is provided by M. S. Dijkhuizen and T. H. Koornwinder (1994). Quasitriangular Hopf algebras were introduced and exhaustively investigated by V. G. Drinfel'd (1990). Important contributions are also due to D. E. Radford (1992, 1993). The coassociativity of comultiplication, and accordingly the conditions on the antipode, were weakened by V. G. Drinfel'd (1990) to the definition of quasi-Hopf algebras, in order to provide an algebraic framework for the construction of gauge transformations in models of conformal field theory. Quasi-Hopf algebras, which are related to the topological gauge field theory of R. Dijkgraaf and E. Witten (1990), were constructed and studied by R. Dijkgraaf, V. Pasquier, and P. Roche (1990). This concept of quasi-Hopf algebras was further weakened by G. Mack and V. Schomerus (1992), omitting the condition L1(e) = e ~ e for the unit e of the Hopf algebra in question, and thereby truncating the tensor product of representations. This truncation then allows for the construction of quantum symmetries of covariant field operators, which obey local braid relations. Starting from an algebraic scheme of desired relations, for instance those of Uq (sl(2, C)), which were invented and used in order to describe the symmetries of models as for instance the open XXZ quantum spin-~ chain, one is forced to introduce an appropriate topology for at least three reasons. (i) Some relations of q-deformed enveloping algebras are non-polynomial. Since these relations are of exponential type, and also the momentum-like relations are not homogeneous, one cannot use the topology of formal power series with relations over C, but one succeeds with the choice of the commutative ring of coefficients C[[h]], h = lnq. (ii) This choice in turn brings about complications with respect to the representation theory of quantum enveloping algebras, which should be performed preferably on real or complex vector spaces. Therefore representations

8. Quantum Groups

281

of quantum groups are usually studied on the "image level" , forgetting about the homomorphisms, which lead down from the level of abstract generators to that of real- or complex- linear operators. Whereas for generic values of the deformation parameter q the representation theory of semisimple finitedimensional complex Lie algebras can be mimicried, quite new types of representations appear for q being a root of unity. In this case the centres of quantum enveloping algebras become much larger, which indicates an according reduction of such algebras. In particular there are finite-dimensional representations, which are not of highest weight type; these new representations, which are called cyclic (or periodic), are useful for the construction of models, for instance of the chiral Potts model. On the other hand, aiming at representations in the strict sense of homomorphisms, and avoiding all the troubles of bimodules over commutative rings, one might try to equip quantum groups with locally convex topologies, which are nuclear and either of Frechet type or the strong dual of this type. (iii) In terms of the involved exponentials as new generators, one obtains polynomial relations of quantum enveloping algebras, but the definition of the universal R-matrix enforces an appropriate completion, anyhow. Although a consistent Z2-grading of quantum groups is straightforward on an abstract level, a detailed performance, Le. application to examples, turns out to be extremely difficult, due to the problems of (i) establishing the full set of relations of basic classical Lie superalgebras, (ii) constructing PBW-like linear bases, and (iii) calculating the universal R-matrix. Besides of topological problems, the duality of quantum enveloping superalgebras and matrix quantum super-semigroups, in the sense of superbialgebras, is defined with respect to bilinear forms, which may be degenerate. Despite of this up to now unsolved problem of degeneracy, the duality of super-bialgebras seems to serve as some very useful guideline for the construction and analysis of such objects, in particular with respect to the so-called classical limit of q -----t 1.

q-deformed universal _q.L.-_-

1~. Lie superalgebra

enveloping superalgebra dual as Lie super-bialgebras

dual as super-bialgebras

mat rix quantu m super-semigroup

-

-

-

~

Lie superalgebra with semiclassical MeR

Here the semi-classical main commutation relations (MeR) are written in terms of some classical R-matrix, which is the coefficient of In q in the power

282

8. Quantum Groups

series expansion of some finite-dimensional representation of the involved universal R-matrix. Some parameter-dependent version of semi-classical MeR appears within the context of the classical spectral transform method for the non-linear SchrOdinger equation, as was pointed out by P. P. Kulish and E. K. Sklyanin (1982). The problem of q-deformation of the universal enveloping algebras of infinite-dimensional complex Lie algebras, in particular of affine Kac-Moody algebras, in order to obtain a family of quasitriangular Hopf algebras with similar triangular decompositions and PBW-like linear bases as in the undeformed case, is by far unsolved on the level of mathematical rigor. Linear bases for a certain class of unital associative algebras over a commutative ring may be constructed by means of the so-called diamond lemma, following G. M. Bergman (1978). This method can be used for instance for an easy construction of a C-linear basis of the Manin algebra Mat q (2,C). More generally R. Berger (1992) established a class of unital associative Ralgebras with Poincare-Birkhoff-Witt-like linear bases, including many examples of quantum groups. Here R denotes the commutative ring of coefficients, which with respect to q-deformed universal enveloping algebras is specialized to complex formal power series in h = In q. The involved C[[h]]-algebras are assumed to be generated by finitely many generators with inhomogeneous quadratic relations, thereby restricting oneself to