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 quantum superspaces and their transformations, and to the upper (or lower) part of the triangular decomposition of Uq (sl(2, C)). The first very complicated proof of an expected PBW-like C[[h]]-linear basis of the upper (or lower) part of the triangular decomposition of Uq(sl(n+ 1, C)), n E N, is due to M. Rosso (1989), who constructed from these bases via the quantum double the universal R-matrix of this quantum group. An appropriate q-deformation of the Weyl group by means of the so-called Lusztig automorphisms was used by A. N. Kirillov and N. Yu. Reshetikhin (1990) to factorize the universal R-matrix of the q-deformation Uq (£) of an arbitrary simple finite-dimensional complex Lie algebra £. The quantum Weyl group and factorization of the universal R-matrix were also investigated by S. Levendorskii, Y. Soibelman, and L. L. Vaksman (1991). The duality of the q-deformed universal enveloping algebra of sl(n, K), for n = 2,3, ... , which is written in terms of polynomial relations, with the corresponding matrix quantum group, in the sense of Hopf algebras over a field K, char K # 2, was investigated by M. Takeuchi (1992), who solved the problem of degeneracy of the involved K-bilinear form in this case, for generic q. S. M. Koroshkin and V. N. Tolstoy (1991, 1992) calculated the universal R-matrix of Uq (£) for basic classical Lie superalgebras £, although the full set of relations for £ was established by H. Yamane (1991, 1994). The latter author proposed a PBW-like linear basis and constructed the universal Rmatrix for Uq (£), inserting an arbitrary basic classical Lie superalgebra £,

8. Quantum Groups

283

over the commutative ring C[[h]] of coefficients, using thereby carefully the h-a
284

8. Quantum Groups

Oscillator and spinor respresentations of the quantum enveloping algebras of the series Am, B m, em, Dm, mEN, were constructed by T. Hayashi (1990). Representations of Uq(sl(m, n, C)), m, n E No, m+n ~ 1, in terms of creation and annihilation operators of finitely many bosonic and ferrnionic qdeformed oscillators were used by M. Chaichian, P. P. Kulish, and J. Lukierski in order to study multimodes of systems of such q-oscillators, and for the definition of systems of finitely many q-deformed superoscillators in terms of q-deformed CCR and CAR, and of an according Hamiltonian (1990, 1991). Algebraic aspects of the QYBE were investigated extensively by L. A. Lambe and D. E. Radford (1993). V. V. Bazhanov (1987) used certain classes of trigonometric solutions of the parameter-dependent QYBE, which are associated with the non-exceptional simple finite-dimensional complex Lie algebras, for the construction of two types of integrable quantum systems: quantum Toda lattices, and integrable models of magnetics. A guideline from integrable models to quantum groups is drawn for instance in lectures by L. D. Faddeev (1990, 1993). An overview of integrable models and conformal field theory was presented by H. Grosse in several lectures (1990). Many original papers on the CYBE, QYBE, and integrable systems are reprinted in a volume edited by M. Jimbo (1990). Quantum groups as quantum deformations of Poisson algebraic groups are introduced and studied for instance in lectures by L. A. Takhtajan (1990), and those by C. De Concini and C. Procesi (1993). Systematic presentations of quantum groups are provided by the monographs by J. Fuchs (1992), J. Frohlich and T. Kerler (1993), G. Lu~ztig (1993), St. Shnider and S. Sternberg (1993), V. Chari and A. N. Pressley (1994), A. Joseph (1995), and Ch. Kassel (1995).

8.1 Duality of Z2-Graded Hopf Algebras (8.1.1) Let {'Hk, JLk, 17k, L1 k , cd, k = 1,2, be Z2-graded bialgebras over a commutative ring 'R. These two Z2-graded bialgebras are called dual, if and only if there is an R-bilinear form fJ : 'HI x 'H 2 3 {aI, a2} -+ (al/a2) E 'R., which fulfills the following conditions. Val, bl E 'HI, Va2, b2 E 'H2: (alla2 b2) = (.1 1 (al )la2 ® b2), (a l bl la2) = (al ® blIL12(a2)), (ella2) = c2(a2), (al!e2) = cl(al), with the units ek of 'Hk, 'vZ1 f Z2 E Z2 ; 1m fJl....,*l x ....,&2 = o. n.~ Tl. t

Here the universal property of tensor products is used for defining an Rbilinear form: ('HI ® 'HI) X ('H2 ® 'H2) -+ 'R., again denoted ( I ), such that 'vZ1, Z2 E Z2, Val E 'HI, bl E 'Hft, a2 E 'H~\ b2 E 'H2 :

(al ® bl la2 ® b2) = (-1Y1Z~(alla2)(bllb2).

8.2 Quasitriangular Z2-Graded Hopf Algebras

285

(8.1.1.1) Let especially Hk, k = 1,2, be Z2-graded Hopf algebras over n, with the unique antipodes O'k. These two Z2-graded Hopf algebras are called dual, if and only if they are dual as Z2-graded bialgebras over nj then

(8.1.2) Consider a pair Hk, k = 1,2, of dual Z2-graded bialgebras over n. Then the left kernel K l , and the right kernel K2 , of the 'R.-bilinear form (3 are both Z2-graded ideals and coideals of Hl and H 2 , respectively. K l := {Ul E Hlj'VU2 E H2: (allu2) = O}, il l := Hl/K l ,

K2 := {U2 E H 2j'VUl E H l : (ullu2) = O}, il 2 := H2/K2. One obtains the Z2-graded bialgebras {il k ,ilk,i7k,.1k,€k},k = 1,2, over

n.

(8.1.2.1) The n-bilinear form i3 : ill x il 2 -+ n defined below is nondegenerate, Le., its left kernel and right kernel both are the null space.

i3: ill x il 2 3 {ii l

=Ul + K l , ii2 =u2 + K2}

----+

(8.1.2.2) These two Z2-graded bialgebras ilk over respect to i3.

(ullu2) =: (ii l lii2)

n, k =

En.

1,2, are dual with

(8.1.2.3) Let especially Hk, k = 1,2, be Z2-graded Hopf algebras over n, with the unique antipodes O'k. Then the corresponding Z2-graded Hopf algebras ilk over n, with the unique antipodes (1k, are dual.

8.2 Quasitriangular Z2-Graded Hopf Algebras (8.2.1) Let {H, Il, TI,..1, e} be a Z2-graded Hopf algebra over the commutative ring n, with the antipode 0'. It is called quasitriangular, if and only if there is an even element R E H®H, which fulfills the conditions (i), (ii), and (iii) below.

with the unit ert. of H, the inverse obviously being unique. (ii) Denoting R =: L:~=l R~ ~ R~/, define Rkl, 1 :::; k < l :::; 3, as elements of .

H®H®H by i

i

R 12 := LR~~R~/~ert., R23:= Lert.~R~~R~/, ~l

~l

286

8. Quantum Groups i

R 13 :=

2: R~ Ci9 e1t Ci9 R~'. i=i

Then demand 1'£@1'£@1'£

3

R13R23

+-+

T(L1, id l'£)(R)

E

(1'£@1'£)@1'£,

1'£@1'£@1'£

3

R13R12

+-+

T(id 1'£, .1)(R)

E

1'£@(1'£@1'£),

with the natural 'R.-linear bijections.

(iii) Va

E

1'£ : 1'£@1'£ 3 .1(a) +-+ R.1(a)R- i E 1'£@1'£, T

with the Z2-graded flip r, such that "Ix, y E Z2 : l'£ z Ci9l'£ Y 3 a Ci9 b +-+ (-1 )XII b Ci9 a E l'£ Y Ci9l'£ z .

(8.2.1.1) If {l'£,J.L,TJ,.1,c} is quasitriangular with respect to R, then it is also quasitriangular with respect to r(R-i), and {l'£,J.L,TJ,r 0 .1,c} is then quasitriangular with respect to r(R). (8.2.1.2) In the special case of r(R) Hopf algebra over R.

= R-i, 1'£ is called triangular Z2-graded

(8.2.1.3) More generally, one conveniently uses an appropriate embedding, for 1 $ k < l $ n.

i

:=

2: e1t Ci9 ... Ci9 e Ci9 R~ Ci9 e1t Ci9 ... Ci9 e1t Ci9 R~' Ci9 e1t Ci9 ... Ci9 e1t, i=i

inserting

R~

at the position k,

R~'

at the position l.

(8.2.2) Consider the above quasitriangular Z2-graded Hopf algebra 1'£ over R, with the unique antipode a. One calculates easily, with the unit e'R, of R, that

(i) T(c, id l'£)(R) = e'R, Ci9 e1t,

T(id 1'£, c)(R) = e1t Ci9 e'R,.

(ii) Furthermore RT(a, id l'£)(R)

= R-iT(id 1'£, a)(R- i ) = e1t Ci9 e1t,

because of RT(a, id l'£)(R) = T(J.L 0 T(id 1'£, a), id 1'£) 0 T(.1, id l'£)(R) = T(TJ 0 c, id l'£)(R),

8.2 Quasitriangular Z2-Graded Hopf Algebras

287

using R 13 R 23, the antipode property, and the above equations (i). Hence obviously T«(1, (1)(R) = R,

T«(1, (1)(R- 1) = R- 1.

(iii) Moreover one finds the Z2-graded quantum Yang-Baxter equation (QYBE)

For an easy proof one calculates i

T(id 7-£, r

0

~)(R) = L

R~

@

(R~(R~')R-l),

i=l

i

L

R~ @ (R~(R~'»

= (e'H @ R)T(id

7-£, ~)(R) ~ R23R13R12'

i=l

(8.2.3) The square of the antipode (1 of a quasitriangular Z2-graded Hopf algebra {7-£, J.L, TI,~, e} is generated by an even invertible element u, which can be constructed from the so-called universal R-matrix R, following V.G. Drinfel'd. i

R=: LR~@R~', i=l i

i

u:= L(-1y:z:'(1(R~')R~, (1(u)

,

i=l

= (1-1(U) = LR~(1(R~'),

,

i=l

u- 1 = L(-1)z:z:'R~'(12(RD, (1(u- 1) = (1-1(u- 1) = L(12(RDR~', i=l

i=l

= (r(R)R)-l(u ® u) = (u@ u)(r(R)R)-l, ~ 0 (1(u) = (r(R)R)-l«(1(u) @(1(u» = «(1(u) @(1(u»(r(R)R)-l; ~(U)

Va E 7-£ : (12(a) = uau- 1 •

Here z~, zr denote the Z2-degrees of R~, R~', i = 1, ... , I, respectively. Obviously (12, hence (1 itself, is bijective. Moreover, UC1(U) = (1(u)u is central. U- 1(1(U) =: g, (1(g) = g-l, ~(g) = 9 ® g, e(g) = e'R,.

288

8. Quantum Groups

8.3 Matrices with Non-Commuting Components (8.3.1) Let A be an associative algebra with the unit eA, over a commutative ring R. For n E N, An is defined as an A-left module over R, with the basis:

o o 1

o

=: E~; i = 1, ... , n

----+ free over A

o and the notation indicated below. Al

V

Xl] [:

EAn,rER:r

= A.

[Xl] [ : :=

Xn

TXI ]

Xn

:

EA n .

TX n

(8.3.1.1) Obviously, with the direct sum of bimodules over R: n

An 3 LXiE~ i=l

+-----+ R-linear

(8.3.2) For n E N, Mat(n,A) the basis:

{E:?; i,j = 1, ... , n}

{Xl, ... ~

n

,Xn }

E

ffiRA W

+-----+ R-linear

A

@

Rn .

defined as an A-left module over R, with

----+ free over A

Mat(n, A),

Mat(I,A) = A.

Here E:! E M at( n, R) denotes the matrix with all the components zero, except the component eR in the i-th column and the j-th row, for i,j = 1, ... , n. One conveniently uses the notation cited below. n

Mat(n,A) 3 [aij;i,j = 1, ... ,n]:=

L

aijE:!.

i,j=l

The module multiplication with elements E R is defined naturally. V[aij;i,j = 1, ... ,n] E Mat(n,A),T E R: T[aij;i,j = 1, ... ,n]:= [raij;i,j = 1, ... ,n] E Mat(n, A).

8.3 Matrices with Non-Commuting Components

289

(8.3.2.1) With the R-bilinear mapping: 2

II Mat(n, A) 3 {[aij; i,j = 1, ... , n], [bij ;i,j = 1, ... , n]} n -+

[Lailblj;i,j = 1, ... ,n]

E

Mat(n, A),

1=1

Mat(n,A) becomes an associative algebra over R, with the unit

(8.3.2.2) One then establishes an isomorphism of unital associative algebras over R: n

Mat(n, A)

3

[aij; i,j =

1, ... , n]

L

f-----+

aij

@

E:!

E

A @ Mat(n, R).

i,j=1

(8.3.2.3) The following homomorphism of unital associative algebras over R is injective.

Mat(n, A) ¢: An 3

3

[aij; i,j = 1, ... , n]

[~1] dJ (a Xn

1

-+

¢

E

EndR(A n ),

al~Xn]

;Xl

+

+

an}x}

+

+ annXn

E

An;

if especially A is commutative, then ¢ E EndA(An). (8.3.3) The elements E:! @E~l E Mat(n, R)@Mat(n,R),i,j, k, l = 1, ... , n, with the tensor product of vector spaces over R, are used as basis of an A-left module Mat(n,A) @A Mat(n,A) over R: {E:!@E~I;i,j,k,l=I, ... ,n}

-+

free over A

Mat(n,A)@AMat(n,A).

Again the module-multiplication with coefficients E R is defined naturally. (8.3.3.1) One conveniently uses the notation indicated below. 2

0AMat(n,A) 3[aij;i,j

=

1, ... ,n]@A [bkl;k,l

=

1, ... ,n]

290

8. Quantum Groups

(8.3.3.2) With the unique R-bilinear mapping: 2

Il(Mat(n,A)@AMat(n,A)) - - Mat(n,A) @AMat(n,A),

such that

\fli 1, ... ,1 2 , Val,a2 E A:

{a 1E ni t31

10. '¢I

k1h a2Ei232 En ' n

10. '¢I

E nk212} __

. 61 1 k2 E ni 1h ala2 63112

10. '¢I

E nk112,

Mat(n,A) @A Mat(n,A) becomes an associative algebra over R, with the unit

(8.3.3.2.1) Obviously there is an isomorphism of unital associative algebras over R: Mat(n, A) @A Mat(n, A) .....- A

@

(Mat(n, R)

@

Mat(n, R)).

(8.3.3.2.2) This unital associative algebra over R must not be confused with the usual tensor product of algebras over R. For instance one finds the following consequence of assuming A to be actually non-commutative; in this case there are elements E Mat(n,A) @A Mat(n, A), such that

[aij;i,j = 1, ... ,n] @A [bkl; k,l = 1, ... ,n]

(8.3.4) The bijection:

{Eni t31

10. '¢I

E ni2h ;Zl,···,)2= . . 1, ... ,n.....} {Eij.. n 2 ;Z,)= 1, ... ,n2} ,

such that for instance:

[~ ~] [~ ~] @

.....-

[H ! ~],

is usually

000 0

lifted to an isomorphism of unital associative algebras over R: Mat(n,R)@Mat(n,R).....- Mat(n 2 ,R).

Moreover, this bijection of bases is also lifted to an isomorphism of A-left modules over R, Le., an A-linear and R-linear bijection:

8.3 Matrices with Non-Commuting Components

291

Mat(n, A) 0A Mat(n,A) ~ Mat(n 2,A), such that for instance:

Mat(2,A)0AMat(2,A) 3

[~ ~]0A[~ ~]

~

[H ~ ~] o

0

0

Mat(4,A).

E

0

This A-linear and R-linear bijection is also an isomorphism of unital associative algebras over R.

(8.3.5) Consider the unital associative superalgebra Mat(m, n, R) := Mat(m

+ n, R)

over R, with the Z2-grading such that

Eij

E

m+n

{(Mat(m,n,R»)O for 1 ~ i,j ~ m, or m (Mat(m,n,R))I otherwise

+ 1 ~ i,j

~

m +n

(8.3.5.1) There is an isomorphism of unital associative algebras over R: Mat(m, n, R)®Mat(m, n, R) ~ Mat«m such that for instance in the case of m

Mat(l,l,R)®Mat(l,l,R) 3

~

0:'10:'2 0:'1'Y2 [ ,10:'2 -,1'Y2

O:'Ifh 0:'1 82 '1/32 -'1 8 2

+ n)2, R),

= n = 1:

[~~ ~:] 0 [~: ~:]

/310:'2 -/31'Y2 810:'2 81'Y2

/31/32] -/31 82 E Mat(4, R),

81132

8182

which may be used in order to establish some Z2-grading of the latter too.

(8.3.5.2) Let A be a unital associative superalgebra over R. The unital associative R-algebra Mat(m

+ n, A) ~ A 0

Mat(m, n, R)

is Z2-graded by demanding this isomorphism of unital associative R-algebras to be compatible with the Z2-grading, and then denoted by Mat(m, n, A). Here the ungraded tensor product of unital associative superalgebras over R is inserted on the right hand side. With this convention, quantum transformations of finite-dimensional quantum superspace will be constructed. With respect to Z2-graded main commutation relations, one should rather use the graded tensor product of A with matrices over R.

292

8. Quantum Groups

(8.3.5.3) FUrthermore the R-bimodule Mat(m+n,A) ~A Mat(m+n,A) is established as some unital associative superalgebra over R, and denoted by Mat(m,n,A)~AMat(m,n,A), such that the R-linear bijection:

Mat(m, n, A)~AMat(m,n,A) f---+ A ~ (Mat(m, n, R)~Mat(m, n, R)) is an isomorphism in the sense of unital associative R-superalgebras. For instance in the case of m = n = 1, Val, ... , d4 E A:

(8.3.5.4) Correspondingly one establishes an isomorphism of unital associative superalgebras over R: Mat(m,n,A)~AMat(m,n,A)

f---+

Mat((m+n)2,A),

(8.3.6) Let B be a unital associative superalgebra over R, which is generated by the homogeneous elements bl, ... , bn , and the set S of homogeneous relations. (8.3.6.1) The tensor algebra over the R-bimodule R( {bl, ... , bn }) with the basis {b l , ... , bn }, Fn := T(R( {b l , ... , bn })), is used as the free algebra over this basis, with the following universal property. V map "Y : {b l , ... , bn } ----4 any unital associative algebra Cover R : :I unique homomorphism "Y. : F n ----4 C of unital associative algebras over R, such that the following diagram is commutative.

{ b l'

... ,

free over R

bn }

embedding

. )

F"

I_>c_ . _I r

r,.

8.4 Transformations of the Quantum Plane

293

(8.3.6.2) B := F n / J, J := ideal(S),

with the ideal J generated by the set S ~ Fn , J = sum({tst'j

S

E S; t, t' E Fn }).

(8.3.6.3) The tensor algebra Fn is naturally Z2-graded by the proposed degrees of homogeneous basis vectors. Since all the elements E S are assumed to be homogeneous, the ideal J is graded. Therefore the factor algebra B is Z2-graded too. (8.3.6.4) Let A also be Z2-graded. An A-point of B is defined as subset

{al, ... ,an} of A, the elements of which are homogeneous and fulfill the relations of the set S in the following sense. Assume "If k : degree ak = degree bk, and consider the unique homomorphism a of unital associative superalgebras over R: Fn --+ A, such that

"Ifk : a(bk) = ak· Then the above indicated property means, that 1m

al s =

{O}.

(8.3.6.5) Such an A-point {al, ... , an} of B is called generic, if and only if

a is injective. (8.3.6.6) These definitions are applied to unital associative algebras A, B over R by their trivial Z2-gradings, taking their odd parts = {O}.

8.4 Transformations of the Quantum Plane (8.4.1) Take the vector space E 2 := K({x,y}) with the basis {x,y} over the field K. The tensor algebra over E2 is constructed, F2 := T(E 2), as the free algebra over the set {x, y}, with the coefficients E K. The corresponding universal property means, that 'V map a : {x, y} --+ any unital associative algebra A over K : :3 unique homomorphism a. : F2 --+ A of unital associative algebras over K, such that the following diagram is commutative. {x,y} free over

K..

- - - - -....) A

a

~

4<

embedding.

if? --'

294

8. Quantum Groups

(8.4.1.1) Let J;'O == J q be the ideal of F 2, which is generated by the relation

S:,o == Sq

C F 2 , J q :=

ideal(Sq) = sum({tst'jS

E

Sq;t,t'

E F 2 }),

Sq := {xy - q-lyx}, q E K \ {O}. The quantum plane is defined as the corresponding factor algebra

K:'O == K q := F 2 /Jq, [xy - q-lyX] = J q = O. K q = K({x~y~;k,l E No}), denoting "It E F2 : tq == [t] := t

+ J q,

t~:= eq := eF2

+ Jq =

eKq •

(8.4.1.2) Due to the universal property of the tensor algebra, and the above relation, there are homomorphisms

.1q : K q ---+ K q ® K q,

Cq :

K q ---+ K

of unital associative algebras over K, such that:

xq ---+ lK, yq Eq

---+ Eq

0, eq ---+ lK· Eq

Hence one obtains the bialgebra {Kq,J.Lq,1]q,.1 q,Cq} over K, with the structure maps J.Lq, 1]q of the factor algebra F 2 / Jq, without an antipode. (8.4.1.3) This relation can also be written with the K-linear bijection Pq: E 2 ®E2

~

E 2 ®E2 ,

which is defined on the basis of E 2 ® E 2 by the map:

x®x---+x®x, x®y--+qx®y, y®x--+q-ly®x, y®y--+y®y. Then obviously

and therefore

ideal( {v ® W

-

pq(w ® v)j v, WE E 2 }) = ideal(Sq) = Jq,

with the ideal generated as indicated above.

8.4 Transformations of the Quantum Plane

(8.4.1.4) With the homomorphisms eq from above, and ciative algebras over K, such that:

..j~

295

of unital asso-

one obtains the cocommutative bialgebra {Kq,j.Lq,l]q,..j~,eq} over K, which does not admit an antipode. (8.4.2) The so-called matrix quantum semigroup of format {O,O} over the field K is defined for q E K \ {a}.

Mat q(2,O,K) == Mat q(2,K):= T(E4 )/I;'0, E4:= K({a,b,c,d}), 1;,0 == 1q := ideal(T;'O),

Ti'o == Tq := {ab - q-1ba, ac - q-1ca, bd - q-1db, cd - q-1dc, ad - da + qbc - q-1cb, ad - da + qcb - q-1bc}. The corresponding equivalence classes are denoted byaq == [a] := a + 1q, .... (8.4.2.1) For q2 :f. -lK, bc-cb E K -lin span(Tq); in this case the element [ad - q-1bc] = fda - qcb] is central, Le., it commutes with all the elements of Mat q(2,K). (8.4.2.2) For q2

:f.

-lK,

Mat q(2, K) = K( {akb1cmlfl

+ 1q;k, ... , n

E No}).

This statement follows from an application of the so-called diamond lemma to the case of unital associative K-algebras with quadratic relations. (8.4.2.3) The mappings:

abcd-

aq ® aq + bq ® cq, a-lK, aq ® bq + bq ® dq, b-O, cq ® aq + dq ® cq, c-O, cq ® bq + dq ® dq, d - l K ,

induce an appropriate comultiplication ..jq and counit eq, according to the diagram below.

296

8. Quantum Groups

p

Mit q(2,O,K) ®Mat q(2,O,K)

:1.1

algebrahomomorphism

Ima p

q

!

{a,b,c,d} - - - ' " T(K( (a,b,c,d}))

'" Mat q(2,O,K)

algebrahomomorphism

ma p

j

Eq

def --::..c:...:.-_> K

I'-----

(8.4.2.4) The elements [ad - q-Ibe], [ad - q-Ieb] E Mat q(2,K) are grouplike with respect to L\q; in case of be = eb, this element is called quantum determinant. (8.4.2.5) The bialgebra {Mat q(2,K),fLq,7]q,L\q,€q} over K does not admit an antipode. Here for convenience the structure maps of Mat q (2, 0, K) are denoted fLq, ... , €q, although they must not be confused with those of K;'o. (8.4.2.6) The tensor algebra T(E4 ) is equipped with the flip T(E4) +-----+ T(E4), such that

7 :

Vp 2: 2 : TP(E4) :;) 72

Xl @ ... @ X p +-----+ T

xp

@ ... @ Xl E

TP(E4), ,

= id(T(E4)), TJ(E4) and T I (E4) being invariant with respect to

map: {a, b, e, d}

--+

7.

The

M at q (2, K), which is defined as

{a --+ dq, b --+ -qbq, c --+ _q-Icq, d --+ aq}, induces a unique homomorphism a : T(E4 ) --+ M at q (2, K) of unital associative algebras over K, which is then combined with the flip 7. The resulting unital algebra-antihomomorphism 70 a vanishes on the ideal I q . Hence one obtains a unique algebra-antihomomorphism

O'q : Mat q(2, K)

--+

Mat q(2, K), such that:

with the unit eq of M at q(2, K). Moreover O'q is surjective, and

[ad - q-Ibc] +-----+[ad - q-Ieb]. Uq

8.4 Transformations of the Quantum Plane

297

(8.4.2.7) The so-called special matrix quantum group over K is defined,

SL q(2,K) := T(K({a,b,e,d}))jideal(Tq U {ad - q- 1 be - e,be - eb}), with the unit e of the tensor algebra T(E4 ). Of course one finds an according homomorphism of unital associative algebras over K:

SL q(2, K)

+-----+

M at q(2, K)jideal( {ad - q-1be - e + 1q, be - eb + 1q}).

Since the structure maps of Mat q(2, K) are compatible with the relations

{ad - q-1be - e, be - cb}, one obtains the Hopf algebra {SL q(2,O,K),j.Lq,7]q,.1 q,€q} over K with the unique antipode (!q, the latter being obtained by factorizing the above antihomomorphism (!q of Mat q(2, K) with respect to the set

{ad - q-1be - e + 1q , be - cb + 1q }. Here the structure maps of SL q(2, K) are denoted just as those of Matq(2, K), for convenience. (8.4.3) Let A be a unital associative algebra over K. Two subsets B, C of A are called commuting, if and only if Vb E B, e E C : be = eb. (8.4.3.1) An A-point of K q is defined as an element

[~]

E

A2, such that

xy - q-lyx = 0. For such an A-point of K q , consider the unique homomorphism v : K q --+ A of unital associative algebras over K, such that

v(xq) = x, v(Yq) = y; then

[~]

is called generic, if and only if v is injective.

(8.4.3.2) Let a : Mat q(2, K) --+ A be the unique homomorphism of unital associative algebras over K, such that

a(aq) = a, ... , a(dq) = d. An A-point of Matq(2, K) is defined as an element that 1m alT.

[~ ~]

= {O}. Obviously, if

[~ ~]

E

Mat(2, A), such

is an A-point of Matq(2, K), then

is also such an A-point. Such an A-point

if and only if a is injective.

[~ ~]

[~ ~]

is called generic,

298

8. Quantum Groups

(8.4.3.3) The quantum determinant of an A-point of Mat q (2, K) is defined as

det q

[~

:]:=

1

ad - q- bc

Here one assumes q2 the set {a, b, c, d}.

i:-

=

da - qcb E A.

-1 K, such that the set {ad - q-l bc} commutes with

(8.4.3.4) Let the A-points

[~:

::] of Mat q (2, K), k = 1,2, commute in

the sense of the sets of their components. Their product

[~:

::]

[~~ :~]

i:-

-IK; then

i:-

-IK, and

is then also an A-point of Mat q (2,K). Furthermore assume q2

[~

(8.4.3.5) Let assume det q with

[~

[~

:] be an A-point of Mat q (2, K), for q2

:]

to be invertible. Then (det q

[~

: ] ) -1

also commutes

:] in the natural sense of components, and the so-called inverse

quantum matrix

'

is an A-point of M atq-l (2, K). Especially [ e~ non-generic point

[e~ e~]

0]-1 [e~ eA0] ,for the

eA

=

of Mat q (2, K).

(8.4.3.6) Consider the injective homomorphism of unital associative algebras over K: Mat(2,A) ---+EndK (A2), which was defined in the foregoing chapter. Let

[~]

be an A-point of K q , and

[~

:] an A-point of M at q(2, K).

If these two A-points commute in the sense of the sets of their components, then

[~ :][~]=[::~]EA2 is also an A-point of K q .

8.4 Transformations of the Quantum Plane

(8.4.3.7) Let an A-point

[~]

299

of K q be generic, and the subset {a, b, c, d}

of A commute with the set {x, y}. Then both are A-points of K q , if and only if

[~

[~ ~] [~]

and

[~ ~] [~]

:] is an A-point of Mat q (2, K).

(8.4.4) Let A be an associative algebra over the commutative ring R, with the unit eA. Consider an element T = L:f=l al ®al' E A®A, with the tensor product of algebras over R. Then the elements T 12 , T 13 , T 23 E A ® A ® A,

[

T 13 := '"', ~ al ® eA ® al" , 1=1

are uniquely determined by T itself. Such an element TEA ® A is called solution of the quantum Yang-Baxter equation (QYBE), if and only if T12T13T23

=

T23T13T12'

(8.4.5) The relations defining M at q (2, K) can be expressed as according main commutation relations (MCR), using an appropriate solution of the QYBE, for q E K \ {O}. (8.4.5.1) Consider the following four-dimensional solution of the quantum Yang-Baxter equation.

Here one also means an isomorphism of unital associative K -algebras:

o 2

AMat (2, A)

3

eAR(q) = R(q)

<--+

Rq

=

eARq

E

Mat(4, A).

300

8. Quantum Groups

(8.4.5.2)

R12(q)R13(q)R23(q) = R23(q)R13(q)R12(q), with the tensor product of unital associative algebras over K. Here, for instance,

R 13 ( q)

=

[~ ~] ® [~ ~] ® [6 ~] + [~ ~] ® [~ ~] ® [~ ~] +

[~ ~] ® [~ ~] ® [q_Oq_1 ~].

(8.4.5.3) Take a, ... ,d E A. The following three statements are then equivalent. (i)

e~ eAo ] ® A [ac

~ e~] ® A [~ ~ ]) ~])([~ ~]®A[e~ e~])R(q).

0 b a 0 c 0 d

b 0 d 0 o a

R(q) ([ = ([

(ii)

Rq

[~

0

~ [~ (iii)

[~ ~]

~ ~] ®A [e~ e~])

c

0 b 0 d 0 0 a 0 c

~][~ ~][~

o

c

([ e

~]

b a 0 0 d cOd 0

~] ~

E M at(2, A) is an A-point of M at q (2, K), and be = cb.

8.5 Transformations of the Quantum Superplane (8.5.1) Consider again the free algebra T(K( {~, 17})) over the set {~, 17}, and assume char K -# 2 for the field K of coefficients. For q E K \ {O}, the quantum superplane of format {I, I} is defined as K~,2:= T(K({~,17}))/~,2, ~,2:= ideal(~,2), ~,2:= {e,172,~17+q-117~}.

(8.5.1.1) The quantum superplane offormat {a,a} is defined as Kg,o.

8.5 Transformations of the Quantum Superplane

301

(8.5.1.2)

Jg,2 = ideal( {v ® w + Pq-1 (w ® v); v, WE K( {~, 7]})}), with the above defined K-linear bijection Pp, p E K \ {O}. (8.5.1.3) 02 C 7]q, ..q7]q C } ) , ..q c2 = 7]q2 = .. Cq7]q K q' = K({ eq,"q,

O,2 C = Jq + q -1 7]q..q = 0,

dimKo,2 = 4, q

with an obvious notation for the equivalence classes of the ideal Jg,2, and the unit eq of Kg,2. (8.5.1.4) Especially for q = 1, one finds the graded-commutative and graded-cocommutative Hopf superalgebra A(K( {~, 7]})), Le, the alternating algebra over K({~,7]}), with the comultiplication ..1 1 , counit C1, and the unique antipode 0"1:

+ e1 ® 6,

6

---+

6

®

6

---+

0,

7]1 ---+ 0,

..:1 1

e1

£1

El

6

7]1 ---+ 7]1 ..:1 1

---+ Ut

-6,

® e1

+ e1 ® 7]1,

7]1 ---+ -7]1· 0"1

(8.5.1.5) The Z2-grading of K( {~, 7]}), which is defined by taking the basis vectors ~ and 7] of odd degree, is uniquely extended to the free algebra over the set {~, 7]} and the field K. Since the elements of the relation Sg,2 are homogeneous, the ideal Jg,2 is graded. Therefore Kg,2 is some unital associative superalgebra over K, with ~q and 7]q being of odd degree. (8.5.1.6) The relations S;'o and S~'?1 are dual in the following sense. Denote Ei := HomK(E2 ,K), E 2 := K({x,y}). Then there is some K-linear bijection: [

(E2 ® E 2 )·

3

'l/J

f---+

if :=

L II ® 91 E E 2 ® E 2, 1=1

such that I

"Iv, wE E 2 : 'l/J(v ® w) =

L II(V)91(W) E K. 1=1

Inserting the dual basis of Ei, such that E2=K({~,7]}), ~(x)=lK, ~(y)=O,

7](x) =0, 7](y)=lK' one easily calculates the following implication. "I'l/J E (E2 ® E 2 )*: 'l/J(x ® y - q-1 y ® x) = 0 ~

if E K

-lin span( {~®~, 7] ® 7], ~ ® 7] + q7] ® 0)·

302

8. Quantum Groups

(8.5.1.6.1) An appropriate dual of the quantum plane K;'o is defined by means of the dual basis of E 2.

K;'o*

:= T(K({~,1]}))/ideal(S;'o*),

with the dual relations

S;'O*:= {e,1]2,~1]+q1]~}, and an obvious isomorphism of unital associative superalgebras over K: K;'o* +-----+ K~'?l' Then the above duality may be rewritten, such that 'V'l/J E (E2 ® E 2)* : 'l/JIS2.0 = 0 {::::::} q

-0 E K

-lin span(S;'o*).

(8.5.2) Let A be a unital associative superalgebra over K. Two subsets B, C of A are called super-commuting, if and only if the super- commutators [b, c] = 0 for all bE B, C E C. (8.5.2.1) An A-point

[~] of K;'o is now redefined as an element E A2, such

that both x and yare even, and xy - q-lyx = O. Such an A-point

[~]

is

called generic, if and only if the subset {x, y} of A is generic in the sense defined previously. (8.5.2.2) An A-point of K~,2 is defined as an element both

~ and 1] are odd, and ~2 =

1]2 =

~1] + q-l1]~ =

[~]

E

A2, such that

O. Such an A-point

[~]

of K~,2 is called generic, if and only if the subset {~, 1]} of A is generic. (8.5.2.3) An A-point of M at q (2, K) is now redefined as an element

[~ ~]

E

M at(2, A), such that a, b, c, d are even, and the subset {a, b, c, d}

of A is an A-point of M at q (2, K). Such an A-point

[~ ~]

is called generic,

if and only if the subset {a, . .. ,d} of A is generic. (8.5.2.4) Let

[~]

be an A-point of

K~~?l'

and

[~ ~]

an A-point of

M at q (2, K). If these two A-points commute in the sense of the sets of their components, then

[a db] [~] C

1]

=

[a~ + b1]] ~ + d1]

is also an A-point of K~~?l'

E

A2

8.5 Transformations of the Quantum Superplane

(8.5.2.5) Let the A-points

[~]

[~]

of K;'o, and

of

303

K~~.?l' both be generic,

and the subset {a,b,c,d} C AD commute with the set {x,Y,~,1]}. Let both

[~

[~] be an A-point of K;'o, and [~ [~ :] is an A-point of Matq(2,K).

:]

Then

(8.5.2.6) Let

[~]

be an A-point of

[~:]

Mat q(2,K), such that

:=

[~

:]

K~~21'

:]

[~]

[~]

and

[~

an A-point of

K~~21'

:] an A-point of

again is an A-point of

K~~21'

Then obviously

f.'1]'

= detq [~

:]

~1],

for be = cb.

(8.5.3) The quantum superplane of format {a, I} is defined as K~,1 := T(K( {x, y} ))jJi,l, Ji,I:= ideal(S~,I), S~,1 :=

{xy - q-l yx , y2}, for q E K \ {O}.

(8.5.3.1) K~,1

= K({eq,yq,x~,x~yq;k EN}), xqyq - q-1 YqXq = Y; = J~,1 = 0,

with the unit eq of KJ,I, and x q := x

+ Ji,l,

Yq:= Y + JJ,I.

(8.5.3.2) Since the relation SJ,1 is homogeneous with respect to the Z2grading, which is due to the basis vectors x being even, and Y odd, the ideal J~,1 is graded. Hence one obtains the unital associative superalgebra K~,I, with the equivalence classes of J~,1 being graded such that x q is even, and Yq odd. (8.5.3.3) The relations SJ,I, and S~,h := {e,~1] - q1]~} C T(K({~,1]})),

are dual in the following sense. Denote

and consider the K-bilinear bijection: [

(E2

12)

E 2 )* 3 'l/J

+----->

.fi; :=

L fz 1=1

12) 91 E

E212) E 2,

304

8. Quantum Groups

such that I

Vv, wE E2 : 1/;(v 0 w) =

L ft(v)gl(w)

E

K.

1=1

With the dual basis of E 2, such that

E 2 = K({~,T7}), ~(x) = lK, ~(y) = 0, 7](x) = 0, 7](Y) = lK, one easily calculates, that V1/; E (E 2 0 E 2 )* : 1/;ls1,1 = q

°<==> ¢

E K - lin

span(S~,h).

(8.5.3.3.1) One therefore defines the dual of the quantum superplane KJ,I,

as an associative superalgebra over K with the unit eq , and with the Z2grading due to the basis vectors ~ being odd, and 7] even, such that ~q is odd, and 7]q even. Obviously there is an isomorphism of unital associative superalgebras over ~ Kl,1 K ·. Kl,h qq'

(8.5.4) The corresponding matrix quantum super-semigroup is constructed as

Ti,l := {ab - q-1ba, ae - q-l ca , bd - qdb, cd - qde, b2 , e2 ,

ad - da For q2

-::F

+ q- 1bc + qcb, ad - da - qbe - q- 1cb}.

-IK, be + cb E K - lin span(Ti,I).

(8.5.4.1) For q2

Mat q (l, 1, K)

-::F

-IK,

= K( {a~b~e~tJ;; k, n E No; l, m = 0, I}),

denoting Vt E T(K( {a, b, e, d})) : t q == [t] := t

+ IJ,I.

(8.5.4.2) The vector space K( {a, ... ,d}) is equipped with the Z2-grading, such that the basis vectors a and d are even, b and e are odd. This Z2-grading is then uniquely extended to the free algebra over the set {a, ... , d}. Since the relations of Ti,l are homogeneous, one obtains some unital associative K-superalgebra Mat q (l, 1, K).

8.5 Transformations of the Quantum Superplane

(8.5.4.3) With the convenient notation the two mappings:

all

305

== a, a12 == b, a21 == c, a22 == d,

are uniquely extended to homomorphisms of unital associative superalgebras over K, and then factorized with respect to the ideal I~,l. Hence one obtains the Z2-graded bialgebra M at q (l, 1, K) over K, which does not admit an antipode.

(8.5.5) Let A be a unital associative superalgebra over K. An A-point of

K~,l

is defined as an element

xy - q-l yx is generic.

[~]

E

A2, such that x E All, Y E AI, and

= y2 = 0; it is called generic, if and only if the subset {x, y} of A

(8.5.5.1) Correspondingly, an A-point of Mat q (l, 1, K) is defined as an element

[~

:] E

M at(2, A), such that a and d are even, band c are odd, and

the subset {a,b,c,d} of A is an A-point of Mat q (I,I,K). Such an A-point

[~

:] is called generic, if and only if the subset {a, b, c, d} of A is generic.

(8.5.5.2) Let

[~]

be an A-point of

K~'l,

and

[~

:] an A-point of

Mat q (I,I,K). If the two subsets {x,y} and {a,b,c,d} of A are supercommuting, then

[~

:]

[~]

is again an A-point of K~,l.

(8.5.5.3) An A-point of K~,h is defined as an element

e

( E A I, "., E AO, and = ("., - q".,( subset {(,,,.,} of A is generic.

(8.5.5.4) Let

[~]

= 0;

be an A-point of

[~]

E A2, such that

it is called generic, if and only if the

K~,h,

and

[~

:] an A-point of

Mat q (I,I,K). If the two subsets {(,,,.,} and {a,b,c,d} of A are supercommuting, then

[~

:]

[~]

is again an A-point of K~,l*.

306

8. Quantum Groups

(8.5.5.5) Let the A-points

[~]

of K~,l, and

[~]

of K~,l., both be generic,

and the subset {a,b,c,d} of A super-commuting with {x,Y,~,1J}, a and d being even, b and c odd. If both

[~

[~]

:]

[~ ~] [~] anA-pointofK~,h,then[~

is an A-point of

K~,l, and

:] isanA-pointofMat q (l,l,K).

(8.5.6) The relations defining Mat q (l, 1, K) can be obtained from according Z2-graded main commutation relations, with an appropriate solution of the Z2-graded QYBE, for q E K\{O}. (8.5.6.1) Mat(l,l,K)®Mat(l,l,K) 3 R(q)

:=[~ ~]@[6 n+[~ n@[~ q~l]+[~ r---+

60 0~ [o 0

q_

~]

°q_l 1

0

q-l

0

=: R q E Mat(4, K),

Here one also means, in the sense of an isomorphism of unital associative superalgebras over R: Mat(l, 1, A)®AMat(l, 1, A) 3 eAR(q) = R(q) r---+

Rq

= eARq

E Mat(4,A).

(8.5.6.2) For q E K\{O}, R(q) fulfills the Z2-graded QYBE, which is defined with the Z2-graded tensor product Mat(l, 1, K)®Mat(l, 1, K)®Mat(l, 1, K). (8.5.6.3) Take a, ... , d E A, and assume q2 ments are then equivalent. (i)

R(q)([~ ~]@A[et e~])([et e~]@A[~ ~]) = ([ eA o eA0]

(ii)

#- -1. The following three state-

Rq

[~

c 0

[~

@

A [acb]) d ([a c

o][a

0 a 0 -b OdO -c 0 d b 0 d 0 o a o c

b

~][~

db]

@

A [eA 0

bOO]

c d 0 0 OOab 0 0 c d

oa ob 0 -c

0]

d

-b 0

o

d

Rq.

0]) R( q). eA

8.6 Transformations of Quantum Superspace

(iii)

ab b2

307

= q- 1 ba, ac = q-1ca, bd = qdb, cd = qdc, = c2 = O,bc = -cb,ad - da = (q - q-l)bc.

In this case, if a and d are even, band c odd, then {a, b, c, d} is an A-point of Matq(l,l,K).

8.6 Transformations of Quantum Superspace (8.6.1) Choose the format {O, ... ,0, I, ... , I} of m even and n odd basis vectors. The quantum superspace K:;"n is constructed over the field K, char K f. 2. Em,n :=

E~,n + E~,n'

E~,n:=

K({Xl,""Xm }),

E~,n:= K({Yll···,Yn})·

Km,n q ''= T(Em,n )/Jm,n q , Jm,n.= q ' ideal(sm,n) q' m n S q'

:=

{-I

XkXI - qkl XIXk, YiYj

-1 + qm+i,m+jYjYi, XkYj

-1

- qk,m+jYjXki

1 ~ k ~ l ~ m, 1:5 i ~ j ~ n}.

Here the choice of deformation parameters is restricted by the conditions, that 't/f'k ~ l, VJ.i ~ j : qkk

= qm+i,m+i = lK,

qklqm+i,m+jqk,m+j

f. O.

For convenience one denotes "'r/fk < l, VJ.i < j, V'{'r, VJ.s : -1

-1

-1

qlk := qkl ' qm+j,m+i:= qm+i,m+j' qm+6,r:= qr,m+6'

and then obtains the following matrix q of deformations parameters. q:= [qklik,l

= l, .. ·,m+n],

~+nk,l: q;/ =qlk,qkk

= lK.

Obviously VJ.j : yJ E K -lin span(S:;"n). (8.6.2) The dual quantum superspace is then defined with respect to the above format.

E;;,n :=K({e, .. ·,~m}), E~n :=K({17 1 , ... ,rt}),

with the dual basis being inserted, i.e,

308

8. Quantum Groups

With this basis one defines the following set of dual relations.

. k I I k sr; ,n* .= {~ ~ + qkl~ ~ ,TJ7r i

.

. i

qm+i,m+jrr17 ,~

k .

rf -

. k.

qk,m+jrr~ ,

1 ~ k ~ l ~ m, 1 ~ i ~ j ~ n}. Obviously Vik : (~k)2 E K - lin span(S;"n*).

K;;"n*

;=

T(E:n,n)fJ;;"n*, J;;"n*:= ideal(s;;"n*).

(8.6.3) The quantum superspace K;;"n is some unital associative superalgebra over K, with the Z2-grading due to that of Em,n, because the ideal J;;"n is graded, with the elements of s;"n being homogeneous. (8.6.4) One immediately establishes an isomorphism of unital associative superalgebras over K: K;"n* ~ K;:m, with the dual matrix q* of deformation parameters defined such that

Vfk,l,Vf:i,j: q7j := q~~i,m+j' q~+k,n+I:= q;/, *

*

-1

*-1

qi,n+1 := ql,mH, qn+k,j:= qk,m+j = qj,n+k' Obviously q** = q. (8.6.5)

V'I/J E (Em,n ® Em,n)* : 'l/Jls""n q

=

°<==> ;f

E

K -lin span(S;"n*).

Here the K -linear bijection:

(Em,n ® Em,n)* ~ 'I/J ~

;f E E:n,n ® E:n,n

is inserted. s;"n is used as subset of Em,n ® Em,n, and s;;"n* as subset of

E;',n ® E;',n' (8.6.6) In the case of m = n = 2, one finds the following relations, which are written conveniently as equations for the corresponding equivalence classes, with the deformation matrix

q12 rll 1 r21 q := !..21 -1 r 21 1 [ r ll -1 -1 -1 r 12 r 22 P12 q12P12 r ll r12r21 r22 # 0. X1 X2 - qll x 2X1 = 0, Y1Y2 + PllY2Y1 = 0, V~k, l : XkYI - r;/Ylxk = 0, YT = 0, 2 2 1 ee + q12ee = 0, rh1 - P1217 17 = 0, ~kTl- rklrle = 0, (~k)2 = 0, e(XI) = 17 k (yz) = Okl, ~k(yz) = 17 k (xd = 0. 1 q-1

806 Transformations of Quantum Superspace

309

(8.6.7)

ror m = 2,n = 0, q =

D

[lq12K

K q2 ,o.

q12] 1 ' K

-1

+--+ y,o,2

liqo .

(8.6.8) The corresponding matrix quantum super-semigroup is defined as some unital associative superalgebra over K. Matq{m, n, K) := T(K( {akl; k, l = 1, ... , m

+ n}))/ I.;n,n,

I.;n,n := ideal(T;',n).

The so-called Manin relations T;',n are equipped with the Z2-grading of basis vectors, such that +

-

-

V'[' nk,l:degree akl:=k+l, kEk:= A

-

{O-1 for 1 < k1-< mk r

-

lor m+

~

~

m+n

0

This Z2-grading is naturally extended to the above tensor algebra, and then applied to M atq(m, n, K), because the elements of T;',n are homogeneous, such that the ideal IJ;,n is graded. The relations of T:;,n are conveniently written as equations of the equivalence classes akl + IJ;,n == [akd· um+n VI t, J, k , l : o

•

[a~k] = 0 for

i + k E I,

[aikail - (-l)(k+lHI+l)qk/ailaik] = 0 for ki

~

-1

iE I

-

[aikail - (-1) qkl ailaik] = 0 for tEO and k

and k < l,

< l,

k E 0 and i < j, = 0 for k E I and i < j,

[aikajk - (-l)iJqijlajkaik] = 0 for

[aikajk - (-l)(i+lHJ+l)qijajkaik] k(+l) -01 [(-1) J aikajl +(-1)3 qklailajk] = [qijl( _l)iJ(( -li(i+l)ajkail [( -l)(k+l)(J+I)aik ajl

+ (-l)iiqklajlaik)]

+ (-l)(J+l)(I+l)q;/ail a jk]

= [qij( _l)(i+lHJ+l)(( -l)(k+l)(i+l)ajk ail

for i

< j and

k

for i < j and k < l,

< l.

+ (-l)(i+l HI+l)q;/ajlaik)]

310

8. Quantum Groups

(8.6.9) These quantum matrices are operating onto elements of the corresponding quantum superspace. V;n+n k : m+n

Em n 3 Xk ,

--+ de!

~ (akl ~

+ I-;,n) ® (Xl + J-;,n)

E

Matq(m, n,K) ® K-;,n.

Here the odd basis vectors Yb ... ,Yn are denoted by X m +1, ... , X m +n . This map is uniquely extended to a homomorphism of unital associative superalgebras over K: T(Em,n) --+ Matq(m,n,K)~K-;,n,which vanishes on the set s-;,n and therefore on its ideal J-;,n. Hence one obtains a unique homomorphism of unital associative superalgebras

am,n q .. Km,n q

--+

Mat q(m "n

K)~Km,n

q'

extending the above map of basis vectors over K. (8.6.10) The relations of the set s-;,n are rewritten as equations of equivalence classes with respect to J-;,n.

~+nk,l: [XkXl - (-l)klqk/XlXk] = 0, hence [X~+I] = ... = [x~+n] = O. The equivalence classes of the dual basis vectors fulfill the relations of the set s-;,n* , denoting 'v'fi : 1Ji = em+i. ~+nk, 1 :

[eel - (-l)(k+l)(i+1)qkle lek] = 0,

hence [e]2 = ... = [e m ]2 = OJ here

for k = 1 e (Xl) = 6kl:= {IK 0 for k f:. 1 . k

(8.6.11) Correspondingly there is a unique homomorphism associative superalgebras over K:

Km,n* q

--+

Mat q(m "n

K)~Km,n*

q'

such that

m+n ~+nk: a;"n*([ekj) =

L

1=1

[akd ® [ell·

a~,n*

of unital

8.6 Transformations of Quantum Superspace

311

(8.6.12) For an arbitrary format and deformation matrix q, one obtains the diagram below. m,n" superalgebra-isomorphism Kn,m Kq • q" an,m

j A mn" Mat q( m,n,K)@K q '

q ..

--- - -

Here an isomorphism of unital associative superalgebras over K is inserted:

Matq(m,n,K)

+-----+

Matq.(n,m,K).

(8.6.13) With these K-linear mappings a,;,n and a,;,n*, both K:;"n and K:;"n* become M atq(m, n, K)-left comodules over K. Moreover these left comodules are Z2-graded in the sense, that both a,;,n and a,;,n* are even with respect to the Z2-gradings of Matq(m,n,K), K:;"n and K:;"m. (8.6.14)

Mat q(O,2,K):= T(K({a,b,c,d})/1g,2, 2,O 1°,2 q .'= ideal(T!,2) q ' T!,2 q = Tq-l, with the notation of the foregoing chapter on the quantum plane.

(8.6.15) Furthermore Matq(m, n, K) is established as some Z2-graded bialgebra over K, with the comultiplication L1 q and counit Cq, for m, n E No.

V{'+n j, k: ajk

~ 8j k E del

m+n

ajk ~

K, 2

L [ajl] 0 [alk] E Q9 M atq(m, n, K).

del 1=1

These two mappings are uniquely extended and then factorized, due to the next diagram. The resulting Z2-graded bialgebra M atq(m, n, K) over K admits an antipode only for special choices of the complex parameters qkl, k, l = 1, ... , m + n.

8. Quantum Groups

312

map de' 1,-' -------->

1\

Matq(m,n,K) ®Matlm,n,K)

~ L1

superalgebra-homomorphism {~k;j,k=

1, ... ,m+nJ

L

I

q

Matq(m ,n,K)

T(K({CJk;j,k= 1, .. , ,m+nJ))

1superalgebra-homomorphism map

de'

K

(8.6.16) Let A be a unital associative superalgebra over K. An A-point of K,:,n is defined as an element

m+n m n k " L..J E Xk m+n E A + , k=l

such that the subset {Xl, ... 'x m+n } of A is an A-point of K,:,n; it is called generic, if and only if this subset of A is generic. (8.6.16.1) Correspondingly, an A-point of K,:,n* is defined as all element of Am+n, such that the set of its components is an A-point of K,:,n*. Such an A-point is called generic, if and only if the set of its components is generic. (8.6.16.2) An A-point of Matq(m, n, K) is defined as an element of Mat(m + n, A), such that the set of its components is an A-point of M at q (m, n, K) j it is called generic, if and only if the set of its components is generic. (8.6.16.3) Let [Xk; k = 1, ... , m + n] be an A-point of the quantum superspace K,:,n, and [akl; k, l = 1, ... , m + n] an A-point of Matq(m, n, K). If the corresponding sets of components are super-commuting, then

[I'

aklXljk

= 1, ...

,m+n]

1=1

again is an A-point of K,:,n.

8.7 Topological Z2-Graded Hopf Algebras

313

(8.6.16.4) If [~k; k = 1, ... , m + n] is an A-point of K:;"n*, furthermore [akl;k,l = l,oo.,m+n] an A-point of Matq(m,n,K), and if these two Apoints are super-commuting, then

[E akl~l;k

= 1, ...

,m+ n]

1=1

again is an A-point of K:;,n*.

(8.6.16.5) Let the A-points x of K:;,n, and ~ of K:;,n*, both be generic. Let the set of components of [akl; k, l = 1, ... , m + n] be Z2-graded, such that

'r{';l+nk, l : akl E A k+i ,

k:=

1

{Q1 for 1~ k ~ m . for m + 1 ~ k ~ m + n

Let the two subsets {akl;k,l = 1,.oo,m+n} and {Xk,~k;k = l,oo.,m+n} of A be super-commuting. If both

[E

aklxl; k = 1, ... ,m + n]

1=1

is an A-point of K:;,n, and

[Eakl~l;

k = 1, ... , m

+ n]

1=1

an A-point of K:;,n*, then [akl; k, l

1, ... , m

+ n]

is an A-point of

Matq(m,n,K).

8.7 Topological Z2-Graded Hopf Algebras (8.7.1) Let R be both a ring and a topological space. R is called topological ring, if and only if the mappings: R x R 3 {r, s} - . r - s E R, R x R 3 {r, s} - . r s E R are continuous with respect to the product topology; then especially R is an Abelian topological group.

(8.7.2) Let 1f. be both a topological space, and a left module over a topological ring R. 1f. is called topological left module over R, if and only if 'H. is an Abelian topological group, and the map:

R x 1f.

3

{r,a}

+-----+

ra

E

1f.

is continuous with respect to the product topology.

314

8. Quantwn Groups

(8.1.2.1) Every continuous 'R.-linear map from a topological 'R.-Ieft modul into such an object is uniformly continuous. (8.1.2.2) Topological 'R.-right modules, and topological bimodules, over commutative topological rings, are defined correspondingly. (8.1.2.3) Two topological 'R.-Ieft modules are called isomorphic, if and only if there exists an 'R.-linear homeomorphism from one of them onto the other. (8.1.3) Assume 'R. to be of Hausdorff type, and denote its unique HausLet 11. be a topological 'R.-Ieft module of Hausdorff dorff completion by type. Then there is, modulo an n-linear homeomorphism, a unique complete Hausdorff topological n-Ieft module ft, called Hausdorff completion of 11., such that 'H is isomorphic with some dense topological subspace of ft. Usually the corresponding injection is viewed as an inclusion, such that 11. itself lies dense in il, il = 11..

n.

(8.1.3.1) Let the 'R.-Ieft module 'H be free over a basis B. Then the unique Hausdorff completion il of 11. is called topologically free over B, and B is called topological basis of ft = 11.. (8.1.4) Let A be an 'R.-submodule of a topological 'R.-Ieft module 11., and ¢ : A ----+ 11.' an 'R.-linear continuous map into a complete Hausdorff topological 'R.-Ieft module 11.'. Then ¢ is uniformly continuous, and there is a unique continuous extension ¢ : A ----+ 'H' of ¢, and moreover ¢ is 'R.-linear and uniformly continuous. (8.1.5) Consider a complete Hausdorff topological bimodule 'H, over a complete Hausdorff topological commutative ring 'R.. Let the 'R.-tensor product 11. ~ ... ~ 11. of finitely many copies of 11. be equipped with an appropriate Hausdorff topology, such that it becomes a topological bimodule over 'R.. The unique Hausdorff completion of 11. ~ ... ~ 'H is called topological tensor product and denoted by 1@ ... ~11. = 11. ~ ... ~ 'H,

if and only if the tensor product map: 'H x ... x 'H continuous. This property is assumed henceforth.

----+

11.

~

...

~

'H is

(8.1.5.1) For instance one may use the strongest topology on 'H ~ ... ~ 'H, such that the above indicated multilinear tensor product map is continuous. For this purpose a subset of 'H ~ ~ 11. is called open, if and only if it is the image of an open subset of 'H x x 11..

8.7 Topological Z2-Graded Hopf Algebras

315

(8.1.5.2) The Hilbert-tensor product ofreal or complex Hilbert spaces is not a topological tensor product. (8.1.5.3) The R-tensor products 1t ® (~1t) and (1t®1t) ® 1t are also proposed as Hausdorff topological 'R,-bimodules. Their Hausdorff completions are then assumed to be isomorphic in the sense of 'R,-linear homeomorphisms: (1t®1t)®1t = (~1t) ® 1t ~ 1t®~1t ~ 1t ® (1t®1t) = 1t®(~1t).

More generally such an associative property of topological tensor products is assumed for finitely many copies of 1t. Furthermore the flip T : 1t ® 1t 3 a ® b ~ b ® a E 1t ® 1t is assumed to be continuous, hence extended uniquely to a homeomorphism T of ~1t onto itself.

(8.1.5.4) Let 1t = 1t0 $1t 1 be the direct sum of topologically closed 'R,submodules. In this case one conveniently assumes, that the usual extension of this Z2-grading to the R-tensor algebra over 1t is compatible with the Hausdorff completion of the tensor product in the sense, that Zl,···,z n E Z 2

thereby establishing an according Z2-grading of the topological tensor product.

(8.1.6) Let 1t be both an algebra and a topological bimodule, over a commutative topological ring 'R,. 1t is called topological R-algebra, if and only if it is also a topological ring. (8.1.6.1) Let 1t = 1t0 $1t 1 be the direct sum of topologically closed 'R,submodules. 1t is called topological superalgebra over 'R" if and only if it is Z2-graded as an algebra, which explicitly means that 'Vi, y E Z2, 'Va E 1t z , b E 1tY : ab E 1t x + y .

(8.1.6.2) Let both 'R, and 1t be complete. The Z2-graded tensor product 1t!i9 .. . !i91t of topological superalgebras over 'R, may be equipped with an appropriate uniform Hausdorff topology, such that the tensor product map becomes continuous. Moreover assume that the Hausdorff completion ~ ... ®1t = 1t!i9 ... !i91t

is compatible with the Z2-grading, such that it is again some topological 'R,superalgebra, which is called topological Z2-graded tensor product of topological 'R,-superalgebras, with the 'R,-subalgebra 1t!i9 . .. !i91t. Again one assumes

316

8. Quantum Groups

an according associative property of topological tensor products, in the sense of isomorphisms of complete Hausdorff topological R-superalgebras. (8.1.6.3) In case of a unital associative complete Hausdorff topological Rsuperalgebra one assumes, that the structure mapping J.L : 1t ¢91t ---+ 1t is continuous, then being uniquely extended to 71 : 1f01t ---+ 1t by continuity. The structure map Tf : R ---+ 1t is obviously continuous. More generally one assumes, that the so-called tensor products of R-linear mappings, for instance T(J.L, J.L), T(J.L, id 1t), T( Tf, id 1t), are continuous, hence continuously extended, in order to obtain commuting diagrams of uniformly continuous structure mappings of complete Hausdorff topological spaces. The canonical embeddings:

1t

:1 a

---+ e1i ¢9 ... ¢9 e1i ¢9 a ¢9 e1i ¢9 ... ¢9 e1i

E

1f0 ... ¢91t

are continuous, because the tensor product map is so by assumption. (8.1.1) The tensor product of formal power series with homogeneous relations is equipped with a uniform Hausdorff topology and completed, thereby inserting implicitly the discrete topology of the coefficient ring R. The corresponding tensor product map is continuous. If the indeterminates are labelled with Z2-degrees, and if the relations are Z2-homogeneous, such that their closed ideal is some Z2-graded subspace of the Hausdorff completion of the free R-algebra over these indeterminates, then this Z2-degree is naturally extended from formal power series with relations to their'topological Z2-graded tensor product, because the involved commutation relations of indeterminates are homogeneous. (8.1.8) Since the relations of quantum algebras, as for instance Uq (sl(2, C)), are inhomogeneous in the powers of generators, their ideal is blown up to the whole algebra by the closure with respect to powers of generators. Therefore formal power series with inhomogeneous relations must be defined artificially, for instance with respect to powers of the deformation parameter h = In q. (8.1.8.1) Consider a unital associative algebra F, for instance formal power series in finitely many indeterminates, over the commutative ring R := R[[h]] of formal power series in h, over a commutative ring R. The ideals

constitute a countable basis of neighbourhoods of 0, which induces the socalled h-adic topology of F. The neighbourhoods U of b E F are defined by the condition, that

3m E No : b + Vm(O)

~

U.

8.7 Topological Z2-Graded Hopf Algebras

317

One thereby establishes the topological 'R.-algebra F. The corresponding tensor product map becomes continuous with respect to the h-adic topology of F ~ F. The algebra structure map: F ~ F ----+ F is continuous, because it is 'R.-linear. Assume

n Vm(O) = {O}; mENo

then F is of Hausdorff type. (8.7.8.2) Let .:J be a closed ideal of F. The factor algebra rt := F /.:J is then equipped with the quotient topology, such that the canonical projection: F 3 a ---+ a + .:J E rt becomes both continuous and open. Since the ideal is assumed to be closed in F, the quotient topology is again of Hausdorff type. If F is complete, then rt is complete too. (8.7.8.3) Of course every 'R.-bimodule M can be equipped with the h-adic topology. An 'R.-linear map of such bimodules: M ---+ M' is continuous. More generally, 'R.-multilinear mappings of such bimodules are continuous with respect to the product topology. (8.7.9) Let rt be a complete Hausdorff topological 'R.-bimodule. rt is called topological coalgebra over 'R., if and only if there are 'R.-linear continuous mappings Ll : rt ---+ 1{0rt and e : rt ---+ 'R., such that the following diagrams of uniformly continuous 'R.-linear maps are commuting.

(i) (ii) 'Va

a E

0

T(Ll, id rt)

Ll = T(id rt, Ll)

0

Ll.

rt:

T(e, id rt) T(id

0

0

rt, e) 0

Ll(a) =

e1(,

~

a,

Ll(a) = a ~ e1(,.

Here the overlined 'R.-linear mappings are defined as the corresponding 'R.linear unique continuous extensions. The topologies of 'R. ~ rt and rt ~ 'R. are identified with that of rt itself, by means of the natural 'R.-linear bijections: 'R. ~ rt f-----+ rt f-----+ rt ~ 'R., and 'R. carries its own topology. (8.7.9.1) Let rtl and rt2 be topological coalalgebras over R.. An 'R.-linear continuous map if; : rtl ----+ rt2 is called homomorphism of topological coalgebras over 'R., if and only if the following diagrams of uniformly continuous mappings commute.

318

8. Quantum Groups

(8.7.9.2) In the case offormal power series, 'R is equipped with the discrete topology. Then the demanded continuity of e means, that 3m EN: e(V m(O)) = {On}. Here the zero-neighbourhood V m(O) by definition contains formal power series with monomials of polynomial degree ~ m. (8.7.10) Let the complete Hausdorff topological space 11. be both an associative unital topological algebra and topological coalgebra over 'R, and assume the structure map J.I. to be continuous. 11. is called topological bialgebra over 'R, if and only if ..1 and c are homomorphisms of unital associative 'R-algebras, which equivalently means that Ii and 1/ are homomorphisms of topological coalgebras over 'R. Here the structure map: (11. 0 11.) 0 (11. 0 11.) ----+ 11. 0 11. is continuous, because the flip is so, and then extended uniquely to the corresponding Hausdorff completions. (8.7.10.1) The topological bialgebra 11. over 'R is called topological Hopf algebra, if and only if there is an 'R.-linear continuous map a : 11. ----+ 11., such that

Ii 0 T(a, id 11.) 0..1 = Ii 0 T(id 11., a) 0..1 = 1/ 0 c. This unique antipode is an antihomomorphism both of algebras and of coalgebras over 'R.

a 0 Ii = Ii 0 TO T(a, a), T(aa) 0..1

= 1'0..1 0 a.

(8.7.10.2) The topological coalgebra 11. over 'R is called Z2-graded, if and only if 11. is the direct sum of topologically closed 'R-submodules, the Hausdorff completion of the 'R.-tensor product of finitely many copies of 11. is compatible with this Z2-grading, and both ..1 and c are even, with respect to the trivial grading of coefficients. (8.7.10.3) The topological 'R.-bimodule 11. is called topological Z2-graded bialgebra over 'R, if and only if it is both some topological superalgebra and Z2-graded topological coalgebra over 'R, and both ..1 and c are homomorphisms of unital assocative 'R-superalgebras, with respect to the Hausdorff completion of the Z2-graded tensor product 11.®11., which equivalently means that Ii and 1/ are homomorphisms of Z2-graded coalgebras over 'R. Here the natural coalgebra structure on 'R is used, and 'R is complete, such that 'R0'R is also complete. 'R

3

r ~ r 0 en ~

E

'R 0 'R, c: = id 'Rj 'R 0 'R

3

r0 s

+-----+

rs

E

'R.

8.7 Topological Z2-Graded Hopf Algebras

319

(8.7.10.4) The Z2-graded topological bialgebra 1t over R is called Z2-graded topological Hopf algebra, if and only if there is an even R-linear continuous map a : 1t ----+ 1t fulfilling the above postulate. This unique antipode a then also fulfills the two antihomomorphism properties, inserting the unique continuous extension 'f of the Z2-graded flip T. (8.7.11) Let 1t be a Z2-graded topological Hopf algebra over R. 1t is called quasitriangular, if and only if there is an element R E 1i01t, from the Hausdorff completion of 1t®1t, such that the following conditions are fulfilled. (i) 3R- 1 E 1i01t : RR- 1 = R- 1R = e1t ® e1t,

with the unit e1t of 1t, the inverse then being unique. (ii)

1i01t®1t:7 R 13 R 23

+----+

T(Ll, id 1t)(R) E (1i01t)®1t,

1i01i01t :7 R 13 R 12

+----+

T(id 1t, Ll)(R) E 1t®(1t®1t),

with the usual notation. Define R 12 denoting 00

R =:

L R~ ® RZ,

+----+

e1t ® R, and,

00

R13 :=

k=O

(iii) Va

R ® e1t, R 23

+----+

L R~ ® e1t ® RZ· k=O

E

1t : 1i01t :7 Ll(a) +-7 RLl(a)R- 1 E 1i01t, T

with the unique continuous extension 'f of the Z2-graded flip

T.

(8.7.11.1) One then finds, that

= e'R, ® e1t,

="".,....,....::-:--,-

T(id 1t, c)(R) = e1t ® e'R,. T(a,id 1t)(R) = R- 1, T(id 1t,a)(R- 1) = R, T(a,a)(R) = R, T(a,a)(R- 1) = R- 1. T(c, id 1t)(R)

Moreover R fulfills the Z2-graded quantum Yang-Baxter equation. (8.7.11.2) The square ofthe antipode a is generated by an invertible element u E 1t, which fulfills similar conditions as in the polynomial case, such that Va E 1t: a 2 (a) = u a u:=

L

U-1j

(_l)Z~Z~ a(RZ)R~, u- 1 =

kENo

Ll(u) a(u)

= ('f(R)R)-l(u ® u), = R~a(RZ), a(u- 1 )

L

kENo

L

(_l)Z~Z~ RZa2(R~),

kENo

=

L

a2(R~)RZ,

kENo

with the degrees z~, z~ of R~ and RZ, respectively. Hence a is bijective. The element ua(u) = a(u)u of 1t is central.

320

8. Quantum Groups

8.8 q-Deformation of sl(2, C) The universal enveloping algebra U(Ad of the Lie algebra 8l(2, C) == A l over C is q-deformed to some quasitriangular topological Hopf algebra Uq(Ad, over the ring R := C[[h]] of complex formal power series in h = lnq. The according perturbation of cocommutativity is described by the so-called universal R-matrix. An appropriate topological basis of Uq(A 1 ) over R is provided by a family of monomials in the generators, the Poincare series of which coincides with that of the universal enveloping algebra U(A 1 ). (8.8.1.1) Using the conventional complex matrix exponential function,

"1M E Mat(n,C): [M]q:= exp(M In q) - e~i(-Mlnq) q- q

-----.

M.

D3q-+l

Here the principal branch of In is used with the cut] - 00,0], and lnq E R for q

> 0; D:= C \ (] - 00,0] U {I}).

(8.8.1.2) Consider formal power series ¢ in finitely many indeterminates over R := C[[h]], with relations; the latter are introduced by factorizing with respect to their smallest closed ideal, with respect to the h-adic topology.

[¢]q :=

q _q- 1 ' q:= q - q-

eh, q

:= exp(h¢).

(8.8.1.3) Let qED, n E N, and assume q as generic, Le., not any root of 1.

[n]q! := [1]q'" [n]q -----. n!, [1]q = 1, [O]q!:= 1. q-+l

/'

n]q [n]q! qn ... qn-k+I [ k .- [k]q![n - k]q! qk . .. q

=

(q2n _ 1) ... (q2(n-k+I) - 1) (q2k _ 1) ... (q2 - 1)

=

[n] n - k q'

[~L [~L:= 1,

[~L=[n~IL [

2n 1 q - = 1 + q2 q2 -1

+ q4 + ... + q2(n-l),

n+ k 1 ] q = [kn _] 1 q + [ nk ] qq2k ,for k = 1, ... , n.

For any two elements a, b of a complex associative algebra A one easily calculates, that "In EN:

8.8 q-Deformation of 8l(2, C)

ba

= q2 ab ==> (a + b)n =

t [~] k=O

321

akbn- k . q

The above expressions may also be understood as complex formal power series in h, e h =: q, then using these series as coefficients of A. (8.8.1.4) Let again qED be generic.

1 - qn 1 2 \In EN: {} n q := - - = + q + q 1- q

+ ... + qn-l

---+

D3q-1

n,

{n}q! := {1}q'" {n}q ---+ n!, {1}q = 1, {O}q!:= 1. q-l

Mn

L -{ } , ---+ exp(M). n 00

\1M E Mat(n, C) : eXPq(M) :=

n=O

q' q---+l

(8.8.1.5) An appropriate q-deformed exponential function is defined, inserting formal power series in finitely many indeterminates over n, with relations.

¢J2

¢In

h

L -{ } , = e'R, + ¢J + -1 +-q +"', q:= e . n 00

eXPq(¢J) :=

n=O

q.

For such objects ¢J, 'l/J one then verifies the following implication.

q¢J'l/J = 'l/J¢J ==> eXPq(¢J + 'l/J) = eXPq(¢J) eXPq('l/J). (8.8.2.1) Consider the unital associative algebra of formal power series in three generators X±, H, over n := C[[h]]. The q-deformation of the universal enveloping algebra of Al == 8l(2, C) is defined by factorization with respect to the smallest closed ideal .Jq, in the h-adic topology, of the set of relations

Sq:= {[H,X±] =f2X±, [X+,X-j - [H]q}, Uq(A 1 ):= n(({X±,H})/.Jq. (8.8.2.2) With the n-linear uniformly continuous structure mappings

.1 : Uq(A 1 ) ---+ Uq(A 1 ) ® Uq(Ad,

€:

Uq(Ad ---+ n,

a: Uq(Ad ---+ Uq(A 1 ) being defined such that:

H ---+H ® I <1

+ I ®H, H ---+0, H ---+ -H, e t7

X±---+X±®q!H +q-!H®X± X±---+O X±---+_q±IX± <1

'

e'

t7

'

with the algebra unit I, Uq(Ad becomes some topological Hopf algebra over with respect to the h-adic topology of formal power series with relations. Here and henceforth the equivalence classes of generators are denoted by the same letters as the generators themselves, for instance shortwriting H instead of H + .Jq.

n,

322

8. Quantum Groups

(8.8.2.3) Using the Baker-Campbell-Hausdorff formula one easily finds, that \:Ie E 'R.:

exp( eH)

--+ exp( ~

eH) ® exp( eH), exp( eH) --+ 1, exp( eH) --+ exp( -eH). ~

u

(8.8.2.4) The so-called classical limit of q --+ 1 is described by an isomorphism of unital associative complex algebras:

onto the universal enveloping algebra of A 1 . (8.8.3) A unital subalgebra of Mat(d, C), dEN, is called finite-dimensional complex representation of the quantum algebra Uq(At}, if and only if it is polynomially generated by three complex matrices X±, H, which fulfill the above relations, thereby inserting a complex deformation parameter qED. (8.8.3.1) Under the special condition of generic q, which by definition means that \:In EN: qn -:F 1, all the finite-dimensional complex irreducible representations of Uq(A 1 ) are obtained by an appropriate deformation of the well-known family of irreducible representations of A 1 • Denote by j

vm ' m

= j,j - 1, ... , -j,

for j

1

3

= 2,1, 2"'"

the Cartesian unit vectors of C 2 j+l, say with the component 1 Of vtn at the position j - m + 1. One then obtains the following irreducible representation on Mat(2j + 1, C).

Hv!n = 2m v!n, m = j,j -1, ... , -j, X±V±j = 0, X±v!n = ([j =f m]q[j ± m + 1]q)1/2t?m±1 for m -:F ±j. In the limit D :;) q

--+

1 one obtains the irreducible representations of A 1 .

(8.8.3.2) Straightforward calculation yields the following Casimir operator

C E Uq(At}. 1

D:= [2(H + I)]q, [D 2 ,X+] = [H]qX+, [D 2 ,X-] = -X-[H]q, C:= D 2

+ X- X+

-

~I,

[C,X±] = [C,H] = O.

For generic values of qED, for j = ~, 1,~, ... , and m = j,j - 1, ... , -j,

. . .. (1)2 1 Cv3m = dm(q)v!n, dm(q):= [j + -2]q - - --+ j(j + 1). 4 q--+l

8.8 q-Deformation of sl(2, C)

323

(8.8.3.3) For qED, the lowest-dimensional complex representation of Uq(Ad, which may be called fundamental one, is provided by the Pauli matrices themselves, because of

[0'3]q

=

0'3.

° 1]

0]

X-._,..-._[O X +._,..+._[O .- v .0' .- v . - 1 0 '

H._,..3._[1 .° -10] . v

.-

(8.8.3.4) For j = 1 and q4 :f: 1, the according irreducible representation is provided by

H := 2

[°1 °° °0] ,X+:= ([2]q//2 [0° °1 0] ° ° -1 °°° 1

,

and the images of X± being mutually transposed complex matrices.

(8.8.3.5) Let qED. The above q-deformed irreducible representations, according to some choice of j E {1, 1,~, ... }, can be constructed, if and only if'v'n E {1,2, ... ,2j}: qn:f: ±l. (8.8.3.6) Without inserting any complex number for the indeterminate h, a finite-dimensional representation of Uq(A 1 ) is defined as homomorphism of unital associative n-algebras

p: Uq(Ad -----. Mat(d,C[[h]]). Obviously p is continuous with respect to the h-adic topology.

(8.8.4) The topological Hopf algebra Uq(A 1 ) over n is quasitriangular with respect to the following so-called universal R-matrix.

R:= q!H®H

f

(1 -

n=O

E

R- 1

Uq(Ad

@

q~2)n q n(yl) (q!H X+ @q-!HX-)n

[n]q.

Uq(A 1 ),

= T(O', id)(R) = q-tH®H

f

n=O

(1 -

q~)n q- n(yl)

(q-!H X+ @q+tHx-)n.

[n]q.

An insertion of this formal power series with relations R into the defining properties of quasitriangularity yields some recursions for the coefficients, which are fulfilled just by the above expression. The equation T(O', O')(R) = R yields some similar expansion of R.

324

8. Quantum Groups

(8.8.4.1) Inserting the Pauli matrices for j following well-known solution of QYBE.

=

~ and qED, one obtains the

0]

1 ®

+-----4

q

q-l/2 [

0

0

0]

0 1 q - q-l o 0 1

0 0

000

[q-l/2 0

0])

ql/2

E M at(4 C)

,.

q

(8.8.5) The Hopf algebra SL q(2, C) may also be considered with respect to coefficients E R := C[[h]], eh =: q. Then the topological Hopf algebras Uq(A 1 ) and SL q(2, C) are dual with respect to the R-bilinear form (I), which is defined by the values

and uniquely extended from polynomials in X±, H to formal power series over R, by continuity with respect to the h-adic topology, the latter being applied also to SL q(2, C). (8.8.6) The structure mappings of Uq(A 1 ) can be rewritten using only polynomials over R, via an appropriate homomorphism A of unital associative R-algebras. Let Eq(Ad be the R-Hopf algebra, which is generated by four elements E, F, K, L, with the following relations and costructure maps.

KL = LK = I, KE = q2 EK, KF = q- 2 FK, q2

EF-FE=--(K-L)' q2 -1

'

E ---. E ® I + L ® E, F ---. F ® K .a .a K ---.K®K, L---+L®L, .a .a €(K) = €(L) = 1, €(E) = €(F) = 0,

+I

® F,

K---.L---.K, E---.-KE, F---.-FL. (j

cr

u

cr

The above indicated homomorphism A : Eq(A 1 ) such that:

---+

E ---. X+q-t H , F ---. x-qt H , K ---. qH, L

Then obviously

..1 0 A = T(A, A) 0..1,

€

0 A=

€, 0'

0 A = A00',

Uq(Ad is established

---+

q-H.

8.8 q-Deformation of 8l(2, C)

325

just not distinguishing between the structure maps of Uq(A 1 ) and Eq(A 1 ) by notation. Note that>. is continuous with respect to the h-adic topology of formal power series with relations. Remembering the classical theory of Lie algebras, the generators of Uq(A 1 ), or of Eq(A 1 ), may be called Chevalley generators. (8.8.6.1) Obviously the above structure mappings can also be used in order to construct a complex Hopf algebra, which may be denoted by Eq(Ad too, for qED. In this case of complex coefficients, complex representations are defined as homomorphisms of unital associative complex algebras. Nevertheless, if one aims at an expression of the universal R-matrix, which works for every finite-dimensional complex representation, one is forced to an appropriate completion procedure, for instance with respect to the h-adic topology. (8.8.7) An involutive ring isomorphism of C[[h]] is provided by: h +-----+ -h, and complex conjugation. Inserting this involution of R, the topological Hopf algebras Uq(Ad and Eq(Ad can be equipped with appropriate star operations, such that:

x+ +-----+ X-, •

H

+-----+

•

Hj K

+-----+ L,

•

E

+-----+ q-l F.

•

These idempotent bijections of Uq(Ad and Eq(Ad are compatible with the corresponding structure mappings, such that: * 0 J.L = J.L

0

*, * 01/ =

1/ 0

*, * 0 L1 = L1 0 *, * 0

€

= € 0 *, * 0

(J'

= (J' 0 *,

with * also denoting the involution of R for the moment. Obviously this star operation on Uq(A 1 ) is uniformly continuous with respect to the h-adic topology of formal power series with relations. Note that these star operations are lifted to Uq(At} 181 Uq(A 1 ) and Eq(A 1 ) 181 Eq(Ad as TO T(*, *) with the flip T, extending by continuity in the former case, and again denoted by *. (8.8.7.1) For generic q and Iql = 1, the irreducible representations of Uq(Ad, or of Eq(A 1 ), by complex matrices yield according representations of these star operations by transposition of their entries. (8.8.8) Correspondingly one defines some star operation on the R-Hopf algebra SL q (2, C), which commutes with all the structure mappings. a*

= a,

b*

= c, c* = b,

d*

= d.

326

8. Quantum Groups

(8.8.9) The topological Hopf algebra Uq(At} over respect to the universal R-matrix 00

R = T(a, a)(R) = ~

(1

[nt!-2)n q

= qtH®H

f

= T(a, id)(R) = L( _1)n 00

(E @ Ft qtH®H

(1 - q-2)n q n(n,-l) (KE@ FLt.

[n]q!

n=O

R- 1

n(yl)

n is quasitriangular with

(1

n=O

= T(a, a)(R- 1 ) = q-tH®H

-2)n - q q- n ( y l ) (KE@FL)nq-tH®H [n]q!

f( -It n=O

(1 - q-2)n q- n(n,-l) (E

@

F)n.

[n]q!

(8.8.10) For qED, an appropriate C-linear basis of Eq(A 1 ) is provided by the following Poincare-Birkhoff-Witt-like family of monomials. Consider the complex Hopf subalgebra Eq(B+) of Eq(A 1 ), which is generated by E, K, L, thereby q-deforming the corresponding Borel subalgebra of AI'

{ErK%jr E No,z E Z}

----+ free over C

Eq(B+).

The complex subalgebra Eq(B-) of Eq(At} is C-spanned correspondingly, just replacing E by F. Hence one obtains a basis {Er F8 K%j r, s E No; Z E Z} of the complex vector space Eq(A 1 ). (8.8.10.1) Denote by Uq(B+) the subalgebra of Uq(A 1 ), which is generated by X+ and H. Then every vector Y E Uq(B+) can be written as formal power series with relations,

Y =

L

c.,.x(X+r H X , with unique coefficients c,.x En.

r,xENo

The vectors of the subalgebra Uq(B-) of Uq(At}, which is generated by Xand H, are similarly expanded uniquely to powers of X- and H. Therefore the set {(X+t(X-)8H X jr,s,x E No} is some topological basis of the topologically free C[[h]]-bimodule Uq(At}, with respect to the h-adic topology. (8.8.10.2) An inductive proof of an according basis property over C[[hlJ was performed by M. Rosso for the q-deformation of Am, mEN, and used in order to construct the universal R-matrix in this more general case. The rather complicated n-linear structure of q-deformation Uq (£.) of an arbitrary simple finite-dimensional complex Lie algebra£. is illuminated by means of an appropriate q-deformation of the corresponding Weyl group, yielding the group of so-called Lusztig automorphisms, which are investigated in one of the following chapters.

q-Deformation of Simple Lie Algebras

327

(8.8.11) The Poincare series of an associative unital algebra A of generators Xi, i E I, with relations, over a field, or more generally over a commutative ring R of coefficients, in the latter case A being assumed to be free over R, is defined as the sequence of dimensions dp,p E No, of homogeneous submodules with respect to the sum of powers of generators. Equipping the generators with the degree 1, one establishes a natural No-grading of A =

EB Ap,

dim A p =: dp ,

pENo

with the degree PI

+ ... + Pn

of monomials X~l ...

Xr.,n, iI, ... ,in E I.

(8.8.11.1) The universal enveloping algebras of simple finite-dimensional complex Lie algebras are q-deformed in such a manner, that their Poincare series remain invariant. (8.8.12) The so-called quantum double construction enables one to construct the universal R-matrix of Uq(.c) from an arbitrary topological basis of the qdeformation Uq(B+) of the Borel subalgebra B+ of a simple finite-dimensional complex Lie algebra.c. Here B+ is generated as complex vector space by the basis, which consists of the positive root vectors and toral generators of .c.

8.9 q-Deformation of Simple Finite-Dimensional Complex Lie Algebras (8.9.1) Let .c be a simple finite-dimensional complex Lie algebra with the Cartan matrix [rkl ; k, l = 1, ... , m], and denote by d k E N, k = 1, ... , m, the integers such that V'tk,l : dk r kl = d1rlk =: B kl , and the greatest common divisor of {d l , ... , dm } is 1, leading to the following table. [Bkl; k,l = 1, ... , m] is called symmetrized Cartan matrix.

.c Am,mEN Bm,m ~ 2 Cm,m ~ 3 Dm,m ~ 4

E6 E7 Es F4

G2

dl 1 2 1 1 1 1 1 1 1

d2 1 2 1 1 1 1 1 2 3

dm 1 2 1 1 1 1 1

1

1

l

dm 1 1 2 1 1 1 1

328

8. Quantum Groups

(8.9.1.1) The q-deformation Uq(.c) of the universal enveloping algebra of .c is defined as the topological Hopf algebra of formal power series in generators

xt and Hk' k = 1, ... , m, over R := C[[h]], with the following relations and costructure maps. Yrk, l : qk := qd k = e hdk , q:= eh , [Hk,Hd = 0, [Hk,XI±j = ±rkIXl±' qHk _ q-Hk [X:' Xl-j = 6kdHk]qk = 6kl k ~l' qk - qk Ifl(_I)j. ,[1- nz]qk!. , (xt)j X ± (xt)l-rkl - j = 0 for k f.l; l j=O [j]qk· [1 - r kl - J ]qk'

Hk

--+ .:1

Hk i8l I

+I

i8l Hk, Hk

--+ 0, f:

Hk

--+

-Hk,

(7

k X k± --+ X± k i8l qk!H + qk-!Hk i8l X± k' X± k .:1

0 X± k

--+, f:

--+ (7

-qk±lX± k.

Here the relations are introduced by factorizing with respect to their closed ideal in the h-adic topology. (8.9.1.2) In the special case of .c := Am == sl(m deformed Serre relations read, Yrk f. l:

X l± (Xt)2 - (q

+ 1, C),

+ q-l)Xt X l±xt + (Xt)2 xt = 0

X: X l± - X l± xt = 0 for Ik -

II

for

mEN, the q-

Ik -II =

1,

~ 2.

(8.9.1.3) Using the involution of R, which is due to: h +-----+ -h and complex conjugation, an appropriate star operation on Uq(.c) can be defined, such that

\fik :

X:

+-----+

•

Xi:, Hk

+-----+

•

Hk,

which is continuous in the sense of the h-adic topology of formal power series with relations. (8.9.2) Quantum matrices and quantum coordinates may also be considered with respect to pairwise commuting indeterminates as deformation parameters. The topological bialgebra Uq(A m ) over R is in duality with the R-bialgebra M at q (m + 1, C) == M at q (m + 1, 0, C), if and only if the deformation parameters of the latter are chosen such that \fi+li, j : qii := e"R., % = qj/

f. 0,

qij := q E C[[h]] for i < j.

The corresponding R-bilinear form is defined such that \fik, \fi+li,j :

q-Deformation of Simple Lie Algebras

329

(XtJaij) = (X; laji) = l5ik15j,k+1, (Hklaij) = 8ik 15jk - 8i,k+1 8j,k+1,

= 8ij ,

(flaij)

(Xtle)

= (HkJe) = 0,

with the corresponding units being denoted by I and e, respectively, and continuously extended by means of the h-adic topology. (8.9.2.1) The complex bialgebra Mat q(m+l, C) can be equipped with some star operation, such that \:f'{'+1i,j : a:j = aji, under the assumption that \:f'{'+1i, j : Iqij I = 1. (8.9.2.2) With the above choice of deformation parameters aiming at duality, and inserting the indicated involution ofn, the star operations on Uq(A m ) and M at q(m + 1, C) are dual in the sense, that "Ix E Uq(Am),a E Matq(m

+ 1, C) : (x*la*)

= (xla).

(8.9.3) The above relations can be transformed to polynomials over n, by means of an appropriate homomorphism of unital associative n-algebras A : Eq(.c) --+ Uq(.c). Let Eq(.c) be the n-Hopf algebra, which is generated by the set {Kk,Lk,Ek,Fkik = 1, ... ,m}, with the following relations and costructure maps. "Irk, I:

[Kk, Kd KkEI

= 0,

KkLk

= qf~1 ElKk,

= LkKk = I, KkFl = qkr~1 FlKk,

q2 8kl~(Kk - Lk), qk -1 (ad+ Ek)1-r~1 (El) = (ad- Fk)1-r~1 (Fl)

[Ek' Fd

Ek

--+ .<1

=

Ek

@

I

.<1

.<1

+------+

u

°

for k

+ Lk @ Ek, Fk --+ Fk @ Kk + I .<1

e(Ek) = e(Fk) = 0, Kk --+ K k @ Kk, Lk --+ Lk Kk

=

Lk, Ek

--+ u

@

@

=1=

l;

Fk,

Lk, e(Kk) = e(Lk) = 1,

-KkEk, Fk

--+ u

-FkLk'

(8.9.3.1) Here the following two q-deformed adjoint representations are used. Denoting: L(X)

Eq(.c)

3

X

--+

Ll(X)

=:

L

1=1

X; @ XI'

E

Eq(.c)

@

Eq(.c),

330

8. Quantum Groups L(X)

Eq(.c)

3

X ~ ad+ X : Eq(.c)

3

L

Y ~

X[Ya(X/')

E

Eq(.c),

de! 1=1 L(X)

Eq(.c)

3

X ~ ad- X: Eq(.c)

3

Y ~

L

X{'Ya- 1 (X[)

E

Eq(.c).

de! 1=1

Obviously "IX E Eq(.c) : ad- X = a-I

Especially one finds, that

0

ad+ X

V'tk, WE

0

a.

Eq(.c) :

ad+Ek(Y) := EkY - LkYKkEk, ad- Fk(Y) := FkY - KkY LkFk· Moreover ad± : Eq(.c) ~ EndR.(Eq(.c)) are homomorphisms of unital associative R-algebras, because

(8.9.3.2) The above indicated homomorphism A is defined such that

Kk

~

Hk Lk qk'

~

qk-Hk , E k

~

X+ k qk-!Hk '

D

r

k

~

V'tk :

Xk qk!Hk .

Then also <10

A = T(A, A)

0 <1,

c: 0 A = c:, a 0 A = A0 a,

with an obvious abuse of notation. Obviously A is continuous with respect to the h-adic topology of formal power series with relations, because it is 'R-linear. (8.9.4) An appropriate q-deformation of the Weyl group leads to PoincareBirkhoff-Witt-like R-linear bases of the q-deformed positive and negative Borel subalgebra of Eq(.c), the latter two being dual as topologically free R-bimodules. The quantum double construction then allows for an explicit computation of the universal R-matrix of the quasitriangular topological Hopf algebra Uq(.c) over'R. (8.9.5) Of course the above polynomial relations and costructure maps of Eq(.c) can also be understood as constituting a complex Hopf algebra. For this purpose one must assume that h = In q, with qED := C\(]-oo, O]U{1}), \fik,l: qk := qd k ,

qf. -:f. ±1 for

1::; j::; 1- rkl and k -:f.l.

8.10 Finite-Dimensional Quantum Double

331

8.10 Finite-Dimensional Quantum Double The so-called quantum double construction is used in order to construct the universal R-matrix of the q-deformed associative envelopes of finitedimensional simple complex Lie algebras. The countably infinite dimension of these quantum groups causes topological troubles, which are overcome by techniques rather specific to these examples, namely using some PoincareBirkhoff-Witt-like basis, and then constructing the corresponding dual Clinear basis. Therefore the algebraic procedure shall be explained for the case of finite dimension, which occurs for finite-dimensional representations by complex matrices. (8.10.1) Consider a finite-dimensional Hopf algebra {ft, /L, T}, L\, e} over the field K, dim ft =: N E N; then the unique antipode a is bijective. The dual Hopf algebra with opposite comultiplication {ft*, /L*, T}*, (L\*)DPP, e*} is denoted by ft°, with the antipode (a*)-l = (a- 1)*. More precisely, ft° is the dual K-Hopf algebra of {ft, /L opp , T}, L\, e}, the latter with the antipode a-I. (8.10.2) The coalgebra D(ft) := 11. ® ft° over K is established as the usual tensor product of coalgebras over K. Obviously the embeddings: ft 3 a

--+

a®I

E D(11.),

ft°

3

f

--+

I ® f E D(ft)

are injective homomorphisms of coalgebras over K. Here the units e1t of ft, and e of ft°, are both denoted by I. The counit of D(ft) is the K-linear mapping: D(ft) 3 a ® f

--+

e(a)f(e1t)

E

K.

(8.10.3) An appropriate multiplication is provided by the so-called double cross product: D(ft) ® D(11.) 3 (a ® f) ® (b ® g) K(b) L(f) --+

L L (abi2l ) ® (t?lg) f?l oa-

1

3

(bill) f?l (bi

))

E D(11.),

k=l 1=1

with the notation: K(bl

11.

3

b

--+

~ (b(l) ® b(2)) ® b(3l E (11. ® 11.) ® 11.

T(~ idlo~ ~ ,

k=l

k

k

k

'

and similarly on ft°, inserting the opposite comultiplication (L\*tPP . The unit of this associative K-algebra is e1t ® e == I ® I. Obviously

Tla E ft, f E 11.0 : (a ® 1)(1 ® f) = a ® f.

332

8. Quantum Groups

(8.10.4) Thereby one establishes the Hopf algebra D(1t) over K, with the antipode: D(1t) 3 a ~

f

---4

(I ~ (O'*)-l(f» (O'(a) ~ I)

E D(1t).

The above embeddings of 1t and 1t0 into D(1t) are homomorphisms of Hopf algebras over K. (8.10.5) This so-called quantum double D(1t) is some quasitriangular KHopf algebra, with respect to the following solution R of the quantum YangBaxter equation. Take any K-basis {bki k = 1, ... , N} of 1t, and the corresponding dual basis {bkj k = 1, ... , N} of 1t*, such that vi' k, l : bk(bz) = Oklo Then the so-called canonical element N

L bk ~ bk E D(1t) k=l

is independent of the choice of basis. Hence also the elements N

R:= L(bk ~ I) ~ (I ~ bk ) E D(1t) ~ D(1t), k=l

N

N

k=l

k=l

R- 1 = L (O'(bk) ~ I) ~ (I ~ bk ) = L(bk ~ I) ~ (I ~ 0'* (b k »)

RR- 1

,

= R- 1 R = (I ~ I) ~ (I ~ I),

are independent of the choice of basis. The quasitriangular properties of Rare proved applying any element E (1t*~1t)~(1t*~1t) onto such E D(1t)~D(1t), using the K-linear bijection: 1t --...+ 1t**.

8.11 Universal R-Matrix of Uq(A m ) (8.11.1) The Hopf subalgebras of the Hopf algebra Eq(A m ) over the ring C[[h)J, mEN, which are polynomially spanned by the sets of generators k {Eki = 1, ... ,m}, {Fkik = 1, ... ,m}, {Ek,Kk,Lkjk = 1, ... ,m}, and {Fk, Kk, Lki k = 1, ... , m}, are denoted by Eq(N±), Eq(B±). One may also consider Eq(A m ) as complex Hopf algebra, choosing q E D\{±i}.

'R :=

8.11 Universal R-Matrix of Uq(A m

)

333

(8.11.2) The q-deformed Serre relations of the q-deformed Borel subalgebra

Eq (13+) read as follows. \fik, l : ad+ Ek(Ed = 0 for Ik - II ~ 2, ad+ Ek 0 ad+ Ek(EI) = 0 for Ik -

II =

1,

the latter relations just meaning that

The unital associative subalgebra E q (13-) is described by analogous Serre relations of Fk, k = 1, ... , m. (8.11.3) The q-deformed adjoint representations are q-deformed derivations in the sense, that "Ix, y E E q (,N+), "Irk, "Ie E C :

LkX = qCxL k ==> 00+ Ek(xy) = ad+ Ek(X)Y + qCx 00+ Edy), and similarly for the restriction of ad- Fk onto Eq(,N-). (8.11.4) The positive roots of Am are strictly ordered according to the longest word w of the Weyl group. w :=

0'1 0 . . .

o 17m 00'1

0 . . . 0 O'm-l 00'1

° ... oO'm_20

... ° 0'1 ° 0'2 ° 0'3 ° 0'1 ° 0'2 ° 0'1' /31 := aI, f32 := 0'1(a2) = a1 + a2,···, /3m := 0'1 0··· 0 O'm-1 (am) = a1 + ... + am, /3m+1 := 0'1 0 · · · 0 O'm(aI) = a2, ... , f32m-1 := 0'1 ° ... o 17m 00'1 ° ... 0 O'm-2(a m-I) = /3p-1 := 0'1 0 · · ·

= a m -1 /3p := p:=

0'1

o 17m 00'1

0 " · 0 O'm-1 0 · · ·

aI,

+ ... + am, . •. ,

° 0'1 ° 0'2 ° 0'3 ° 0'1 (a2)

+a m ,

° ... o 17m 00'1 0 · · · 0 O'm-1 0 · · · ° 0'1 ° 0'2 ° 0'3 ° 0'1 ° 0'2(aI)

!m(m + 1). For instance in the case of m = 3,

/31 =

a2

/32 =

a1 +a2,

/33 =

a1 +a2 +a3,

334

8. Quantum Groups

(8.11.5) For every positive root 13 := Ok

E/3 := ad+ E k 0 ' "

0

+ ... + 01

E 4>+, define

ad+ EI_l(EI), 1 ~ k < l ~ m.

The q-deformed Serre relations of Eq(N+) lead to q-deformed commutation relations of the following kind, calculated quite explicitly by M. Rosso. VI ~ k < l ~ p: O' or

E/3~EI3I - >"E/3IEI3~ =

{

J.LE/3~+/3" 13k ~ 131 ~ 4>+, or vEI3;E/3., k < t < J < l, f3i + f3j = 13k

+ 131 ,

with integers i,j, and non-zero coefficients >",J.L,V, depending on k,l. (8.11.5.1) For instance in the case of m = 3, one obtains the following commutation relations.

ad E 1 (E2) = E/31E/34 - qE/34E/31 = E/32' ad E1(E3 ) = [E/3I'E/36J = 0, ad El(Eol+02) = E/31E/32 - q-l E/32 E /31 = 0, ad El(Eol+02+03) = E/31E/33 - q-l E/33 E /31 = 0, ad El(E02+03) = E/31 E /3, - qE/3,E/31 = E/33' ad E 2(E3 ) = E/34E/36 - qE/36E/34 = EI3" ad E2(E02+03) = E/34E/3, - q- 1E/3,E/34 = 0, E/34E/32 - qE/32E/34 = 0, ad E2(Eol+02+03) = [E/34,E/331 = 0, E/36E/3, - qE/3,E/36 = 0, E132 E /36 - qE/36 E /32 = E 133 , E/36 E /33 - qE/33E/36 = 0, E/3,E/33 - qE/33 E /3, = 0, E/33E/32 - qE/32E/33 = 0, [E/32' E/3,J = (q-l - q)E/34E/33' (8.11.5.2) For any map 1r : N -----+{1, ... ,p}, and kEN, the monomial E/3,,(I) ... EI3,,(~) can be rewritten, via an induction on k and 1r(I), as R-linear combination of monomials of the kind E;: '" E~:, with Tl, ... ,Tp E No. (8.11.5.3) Since the above commutation relations are homogeneous with respect to the root lattice Q:= Z({OI, ... ,Om}), the R-algebra Eq(B+) is Q-graded, such that E;: ... E~: is of degree Tlf31 +... + Tpf3p, the generators K I, ... ,L m being of degree zero by definition. (8.11.6) By induction one calculates, that VI .:1(Eo~+"+ol) = Eod,,+ol ® I 1-1

~

k
~

m:

+ Lk ... LI ® EO~+"+OI

+ (1 - q2) L Eod"+o; Lj+l ... LI ® E O ;+1 +"+01' j=k

8.11 Universal R-Matrix of Uq(A m

(8.11.6.1) Especially for m ..1(E/3~) = E/3~ 0 I

..1(E/35 ) = E/35 0 I

)

335

= 3,

+ L 1 L 2 0 E/3~ + (1 - q2)E1 L 2 0 E2, + L2L3 0 E/35 + (1 - q2)E2L 3 0 E 3,

..1(E/33) = E/33 0 I + L 1 L2L3 0 E/33 + (1 - q2)(E l L 2L 3 0 E/35 + E/3~L3 0 E 3). (8.11.6.2) The comultiplication ..1 preserves the Q-degree in the natural sense.

(8.11.6.3) Let any Q-homogeneous "R.-linear combination of ordered monomials E~~ be zero. An application of ..1 allows for an induction on the Q-degree, in order to prove that all the coefficients of monomials of this polynomial are zero. Obviously E/31' ... , E/31' are "R.-linearly independent.

E;: ...

(8.11.7) The set {E;: ... E~:;rl, ... ,rp E No} is some "R.-basis of Eq(N+). Hence

(8.11.8) The following "R.-linear forms are defined on the complex Hopf algebra Eq(B+). Vii, 'v'rl,.··, rp E No, 'v'Zl,···, Zm E Z:

Z1 (k i±IEr1' /31'. .. Er1 /31 K 1

.= {exp (± lnq .

f

k=l

...

K mZ"')

Zkrik),

if rl

+ ... + r p = 0 '

o otherwise

(fiI Er1' ... E':,1 K Z1... K:n."') := {
/31

1

0 otherwise

+ ... + r p,l3p =

Q'i •

(8.11.8.1) These elements of the unital associative algebra (Eq(B+))* of"R.linear forms fulfill at least the following relations. Vii,j; ktki = kikt = I, hence denote kt [k i , kj ] = [Ii, Ij ] = [k i , lj] = 0,

=ki , ki =kil =Ii;

fik j =qriikjfi, filj =q-riilj!i;

[Ii, Ijl = 0, if Ii - jl ~ 2; li2/j + Ijl; = (q + q-l)fi!J1i, if Ii - jl = 1.

336

8. Quantum Groups

(8.11.8.2) Consider the subalgebra (Eq(B+»O of (Eq(B+»*, which is generated by the set {k;l, Ii; i = 1, ... , m}; it is equipped with comultiplication and counit. Vfli' k±l ---+ k±l ® k±l cO(k±l) 1 . • ~o' ., •

Ii ---+ Ii ® I + k i ® Ii, ~o

= 1'

c°(fi) = O.

(8.11.8.3) Denote by (Eq(B-»OPfJ the Hopf subalgebra of (Eq(ArnWPfJ, which is generated by the set {Kf 1, Fj; l = 1, ... , m}, with the opposite comultiplication and inverse antipode. There is some homomorphism 'l/J both of unital associative algebras and of coalgebras over 'R, such that Vfi : 'l/J(K;l) = k;l, 'l/J(Fi ) = Ii- In order to prove that this homomorphism 'l/J : (Eq(B-»OPP ---+(Eq(B+»O is bijective, one needs some 'R-basis of the latter bimodule. (8.11.8.4)

Vfj : (f(3; IE;: ... E;~ Kt := {({3j) E 'R \

o otherwise

1

•••

K:n"')

{O}, if T1{31 + ... + Tp{3p

= {3j .

ChoosingVfk: k:= (l-q-2)-1,andappropriatevaluesof({3j),I5:j 5:P, for instance <1>(0:1 + 0:2) := <1>1<1>2(1 - q2), one finds, that

"115: k < l 5: m : IOtk+··+OtI = ikIOtk+l+··+OtI - kkIOtk+l+ ..+Otlk;l ik· Hence these 'R-linear forms fulfill the same commutation relations as E{3;, for 1 5: j 5: p. Therefore

(Eq(B+»O

= 'R -

span (

U;:··· I;~k? '" T1,,,.,T p E

k:n"';

No;zll ... ,zrn

E

Z}).

Moreover this set and the above constructed 'R-basis of Eq(B+) are orthogonal with respect to the Q-degree. VT1,"" sp E No, VXll"" Yrn E Z: p (/{3rp

•••

l{3r 1 k X 1 1

1

•••

k mX '" IE{3~Pp ... E{3~l1 Kill 1

•••

KII"') = cbrl&l .. ·6r p 8 p ' m

c E 'R \ {O}. Therefore this set is some 'R-basis of (Eq(B+»o. (8.11.8.5)

Since these elements fulfill the same commutation relations as E{3; , 1 5: j 5: p, one obtains an according 'R-basis of Eq(B-).

8.11 Universal R-Matrix of Uq(A m )

337

(8.11.8.6) Therefore

t/J: (Eq(B-))OPP ..--(Eq(B+))O is bijective. This in turn implies, that the generators k j , f;, 1 ~ j ~ m, fulfill no more relations than the above. One finally establishes an isomorphism of 'R.-Hopf algebras t/J, with respect to the antipode (10, such that

Vij : k j ..-- k)-:-l, Ij 0'0

-+

_kjl f;.

0'0

{Eq(B-), (.1 IEq(B- ))opp, el Eq(B-)' «11 Eq(B_))-l } .

..--.

uomorphum

{(Eq(B+))O, above relations, .1°, eO, (10},

inserting the subalgebra Eq(B-) of Eq(Am ).

(8.11.8.7) The Hopf algebras Eq(B+) and (Eq(B+))O are dual with respect to the above 'R.-bilinear form, which obviously is non-degenerate. Vii,j, Vz E Z : (k;IKj)

(1iIKjI)

= q±Zr;;, =

(k;IEj ) = 0,

0, (fiIEj)

= 1 ~i~_2'

(8.11.9) The so-called quantum double of the q-deformed Borel subalgebra

Eq(B+) is defined as the tensor product of'R.-coalgebras, and as the so-called double cross product of 'R.-Hopf algebras, inserting the restrictions of .1 from Eq(A m ) onto Eq(B±). Note that the comultiplication .10 on (Eq(B+))O must be twisted here.

D(Eq(B+))

:=

Eq(B+) ® Eq(B-).

(8.11.9.1) Vii,j: [Ki ®1,I®KjI] = [Ki- 1 ®1,1®Kfl] =0, (Ki ® 1)(1 ® Fj ) = q-r;; (1 ® Fj)(Ki ® 1), (1 ® Ki)(E j ® 1) = qr;; (E j ® 1)(1 ® K i ), 2

[Ei ® 1,1 ® Fj ] = 6w -l--(1 ® K j - Kjl ® 1). q -1 The coefficients ¢k, k = 1, ... , m, were adjusted with respect to the last relation. Moreover one finds, that V"{'i, j :

and similar relations for the generators F i , i = 1, ... , m.

338

8. Quantum Groups

(8.11.9.2) Hence there is some surjective homomorphism of Hopf algebras over R: D(Eq(B+» ---+ Eq(A m ), such that Vttj:

Kj ® I

---+

(8.11.9.3)

Kj , I ® Kj

---+

Kj , Ej ® I

---+

E j , I ® Pj

---+

Pj .

sp E No, VX1,"" Ym E Z : :I unique c E R \ {O}:

VT1,""

(E;: ... E;~ Kfl ... K~m) ®

(P;: ... PJ: Kfl ... K~m)

~cErp

~

{3p' .•

ErlpSp pS1Kx1+Yl {31 {3p'" {31 1

. ..

KXm+Ym m .

Hence one finds an R-linear basis of Eq(A m ), with the same Poincare series as in the non-deformed case of E(A m ). (8.11.9.4) One then establishes an appropriate 'R-linear bijection:

D(E ( +)) q

B

Eq

+---+

(A ) m

R[K1 , . .. , LmJ ®'d l f {um .. K·L· - I} , t ea 0 Vl J. ]]-

inserting polynomials over 'R in 2m commuting generators K 1, ... , L m . (8.11.9.5) Obviously these polynomial considerations may also be performed over C, choosing q E D\{±i}. (8.11.10) Consider again the topological Hopf algebra Uq(A m ) over 'R, which is generated by the set {Ek, Pk , H k; k = 1, ... , m} in the sense of formal power series with relations. The subalgebras Uq(N±) and Uq(B±) of Uq(A m ) are defined similarly as in the polynomial case, inserting Hk instead of Kt 1 , k = 1, ... ,m. Then the q-deformed Borel subalgebras Uq(B±) are topological Hopf algebras over R. Obviously

with respect to the h-adic topology. Therefore

rp {E{3p

•"

ErlHxl {31 1

...

HXm. m' Tl, ... , X m EN} 0

is some topological basis of the q-deformed Borel subalgebra Uq(B+) in the sense, that every vector Y of the latter can be written uniquely as

""Tl",XmErp ... Erl HXl ... HXm,,_ E 'R {3p {31 1 m , "'"1"'X'" .

Y= rl, ... ,xtnEN o

8.11 Universal R-Matrix of Uq(A m

)

339

(8.11.10.1) Define 'R.-linear forms ~k, TJl3j E

(Uq(B+))*

:=

Homn(Uq(B+), 'R.),

vanishing on every basis vector, except

\fik, Vfj : (~kIHk) (1 :=

m-l

1

6 - 26,

\fik, 1 : hk

= 1,

V2

2 nq

:= -1-(k,

= l.

(TJI3J IEl3j)

k : (k := ~k

(hkIHI) =

-

1

2(~k-l

1 nq

-1-rkl,

+ ~k+l),

TJO: k

1 (m := ~m - 2~m-l.

IE (8+) = q

(1 - q-2)!k.

(8.11.10.2) On the subalgebra of polynomials in E k and H k , k = 1, ... , m,

Ufin(B+) '- 'R.({Erp q .I3 p

•••

rlH xl E 131 1 ... HXm. m ,Tl,···,X m EN}) 0,

2

1m L1lutn(8+)

c

® Uq(B+).

Hence the unital associative algebra (Uq(B+))O of polynomials in the restrictions of 6, ... , TJl3 p onto Urn(B+) is defined, quite similarly as for the dual of any coalgebra.

(8.11.10.3)

VTl,···, Ym E No : (

rp CXmlEsp E S IHI/l •.• HI/m) Yl rl~xl TJI3"' 13'" p "131 1 · · · ..m p 131 1 m

_rrP fir ·s· (1 J

J

q-2)(1 - q-4) ... (1 - q-2rJ ) ( 2 1_ q-) r j

j=1

rr fi x'I/.Xk I m

~

~

oo

k=l

(8.11.10.4) These forms hk and (1 - q-2)-I TJo: k , k = 1, ... , m, fulfill on Utin(B+) at least the same relations as H k and Fk. Hence one obtains some surjective homomorphism of unital associative 'R.-algebras

t/J : Urn(B-) --+(Uq(B+))o, such that \fik: hk --+ Hk, TJO:k --+(1 - q-2)Fk' denoting by urn (B-) the corresponding restriction to polynomials over 'R..

340

8. Quantum Groups

(8.11.10.5) Since the monomials TJp: ... e~"', Tt, ... ,Xm E No, are "Rrlinearly independent, due to the above orthogonality, 1/J is bijective. Then the unique continuous extension of 1/J to formal power series with relations yields some isomorphism of topological Hopf algebras

inserting an appropriate comultiplication, counit, and antipode, on the unique Hausdorff completion of the right hand side.

(8.11.10.6) The resulting "Rrbilinear form: Urn(B-) x U!in(B+) ---+ 'R can be continuously extended to formal power series with relations, with respect to the h-adic topology. (8.11.10.7) The quantum double D(Uq(B+)) := Uq(B+) ~ Uq(B ) is then constructed by unique continuous extension of structure mappings. One obtains some surjective homomorphism of topological Hopf algebras:

such that Vik:

Hk

~

I

---+

Hk, I

~

Hk

---+

Hk, Ek

~

1---+ Ek, I

~

(8.11.10.8) The completion with respect to the h-adic

Fk

---+

Fk·

topolo~

also yields

(8.11.11.1) In order to write down explicitly the so-called universal Rmatrix of V, remember the following R-bases:

{E? ···H;""'jTll ... ,xm E No} "

{TJp"· .. e;""'jTll ... ,XmENo} "

---+ (Uq(B+))o, Iree ot'er 'R.

{F? ... H;""'j Tll ... , X m E No} "

(8.11.11.2)

---+ Urn(B+)j I ree Ot/er 'R.

---+ U[in(B-). I ree Ot/er 'R.

I

'"

'U[in(B-),

Universal R-Matrix of q-Deformed Simple Lie Algebras

341

(8.11.11.3)

Therefore VI S k

< l S m : (( Ctk + ... + Ctl)) -1

1) (

= ( 1 - q2

1 )I-k - q2

Vij: ¢(F{3j) = (/3j)1]{3jIU!i"(B+)"

(8.11.11.4) The universal R-matrix of the quantum double is defined as the following formal power series with relations over R.

Rv:=

t

I:

rl.···,xmENo

(E~:

1 p. (1 - q- 2 j Xl!"'X m !(II(1-q-2) ... (1-q-rj))

... H:,.,m (1)

3=1

@

(I@¢-I(1];:) ... ¢-I(~~m)) E V@V.

The corresponding quasitriangular properties are proved by orthogonality of the inserted topological R-bases. (8.11.11.5) The homomorphic image of Rv is the universal R-matrix of Uq(A m ). j

(_ 2)rj(I->'j). (II r~o '"' (1(1_-q-2)2r q E q-2) ... (1 - q-rj) 1

R=

00

j=p

r

, @

{3j

. F r ,) {3j

m

exp(lnq

I: (r- 1)kI H k @HI)' k.I=1

where the product is ordered according to decreasing j = p, ... , 1. The length of the root /3j := Ctk + ... + Ctl is defined by Aj := l - k + 1, 1 S k S l S m. (8.11.11.6) Inserting h = In q for generic qED, one obtains thereby the Rmatrix of an arbitrary finite-dimensional complex representation of Uq(A m ).

8.12 Universal R-Matrix of q-Deformed Simple Lie Algebras The automorphisms Tk, k = 1, ... , m, of a simple finite-dimensional complex Lie algebra [, of rank m, can be q-deformed to automorphisms of the unital associative algebra E q ([,) over R, which in the sequel are again denoted by Tk. These automorphisms can be used conveniently, being composed along the longest word of the Weyl group, to construct the q-deformed positive and

342

8. Quantum Groups

negative root vectors, which in turn are inserted into the ordered monomials of some Poincare-Birkhoff-Witt-like R-linear basis of E q (£). (8.12.1) Let £ be a simple finite-dimensional complex Lie algebra of rank m, and denote q := eh, qk := qd k = ehdk .

There are automorphisms Tk of E q (£), k = 1, ... , m, in the sense of unital associative R-algebras, such that V'f'k, l:

for rkl = 0, Tk(KI) = K l , TdEz) = El, Tk(FI) = F l ;

for

nl =

-1,

Tk(Kz) = KkKI, Tk(EI) = q;1/2(EkEI - qkEIEk) = q;1/2ad+ Ek (Ez) , Tk(Fl)

for

rkl =

= _q;3/2(FkFl -

qkFIFk)

= _q;3/2ad- Fk(Fl);

-2,

Tk(KI) = K~KI, Tk(EI)

= _[I_j-[Ek,EkEI- q~EIEkj = _[I_]-ad+Ek oad+~k(Ez),

Tk(FI)

= ~[1] [Fk , FkFl

~2ft

qk 2 qk

~2ft'

-

q~FlFkj = ~[lj ad- F k 0 ad- Fk(Fl). qk 2 qk

(8.12.1.1) The inverse automorphisms of E q (£) are operating, such that V'f'k, l: Tk-1(Kk) = Lk' T;;l(Ek ) = -KkFk , T;;l(Fk) = -EkLk;

for rkl = -1, Tk-1(Kz)

= KkKI,

T;;l(EI) = -q~/2(EkEI - q;l EIEk), T;;l(Fl) = q;1/2(Fk FI - q;l FlFk);

for

rkl =

-2,

T;;l(KI) = K~KI, T;;l(EI) =

[~:k [Ek,Ek E I-q; 2EIEk], 1

Tk-1(FI ) = -[2] [Fk,FkFI- q; 2FIFkj. qk qk

Universal R-Matrix of q-Deformed Simple Lie Algebras

(8.12.1.2) For rkl = -3, which occurs only in the case of k = 1, l = 2, the automorphism T l is acting, such that

:=

G2, for

= K~K2'

T l (K2) = Tl- l (K2 ) Tl (E2) =

.c

343

q3/2~3]q! [El , [E l , [El , E 2]q3 ]q]q-I = q3/2 [3]q! (ad+ EI)3(E2), 1

-1 -1 _ 3 T l (F2) = q9/2[3]q! [Fl , [Fl , [Fl , F2]q3]q]q-1 = q9/2[3]q! (ad Fd (F2), -1 _q3/2 Tl (E2) = -[-]-, [E l , [El , [E l ,E2]q-3]q-l]q, 3 q. l T l- (F2) = q3/2~3]q! [Fl , [Fl , [Fl , F2]q-3]q-l]q,

with the q-deformed commutators [X, Y]p:= XY -pYX, X and Y E Eq(.c), for q and pER. (8.12.1.3) These automorphisms, which were introduced by G. Lusztig for the cases of Am, mEN, and D m , m ~ 4, fulfill the following braid-like relations. Vik, l:

T k 0 TI

= TI 0 T k for

r kl

= 0;

Tk

0

TI(E k ) = E I , T k 0 1/(Fk) = FI,

Tk

0

T I 0 Tk =

1/ 0

= -2 or Ftk

for rkl

=

Tk

0 1/

for rkl = r lk = -1;

-2,

Tk

0

TI

0

Tk(Et} = EI, Tk

01/ 0

Tk(FI) = FI, Tk 0 TI

Tk

0

TI

0

Tk

0 1/ 0

Tk;

for rkl

0 1/

= TI

0

Tk

0

Tk(KI) = KI,

= -3, which means k = 1, l = 2, in the case of .c := G2,

(8.12.1,i) In the case of vectors

E{3 := ad+ Ek

0···0

.c

:=

Am, mEN, the q-deformed positive root

ad+ EI-l(EI),

13:= ak + ... + ai,

1~k


~ m,

can be constructed, inserting the Lusztig automorphisms into the longest word of the Weyl group. For instance in the case of m = 3, one finds the six q-deformed positive root vectors with complex coefficients below.

E l =: E{3p Tl (E2) = q-l/2E{3~, Tl oT2(E3) = q- 1 E{33' Tl oT2 OT3(EI) = E 2 =: E{34' Tl oT2 oT3 o Tl (E2) = q-l/2E{3s' T l 0 T 2 0 T3 0 Tl 0 T2(El ) = E 2 =: E{36.

344

8. Quantum Groups

(8.12.2) Let (7k l 0 •.. 0 (7k p ' k1 , ... , kp E {I, ... , m}, be the longest word of the Weyl group of £, and I3j,j = 1, ... ,p, the positive roots, which are ordered such that ~j : I3j :=

(7k l 0 ... 0 (7kj_l (Ok j ),

131

:=

0k 1·

Inserting the q-deformed positive and negative root vectors

E(l3d F(l3d

:= :=

Ek l , E(132) := Tk l (Ek~),"" E(l3p ) := Tk 1 0 · · · 0 Tkp_l (Ek p), Fkl' F(132) := Tk1 (Fk~),"" F(l3p ) := Tk l 0 · · · 0 Tkp_l (Fk p)'

one obtains the Poincare-Birkhoff-Witt-like 'R-linear basis:

{E(l3p t p ... E(l3d rl Kfl ... K:"m; Tl, ... , T p E No; ZI, .. " ---+

free over 'R.

Zm

E Z}

Eq(B+),

of the Hopf subalgebra Eq(B+) of E q(£), which is polynomially spanned by the set of generators {Ek,Kk,Lk;k = 1, ... ,m}, and correspondingly some 'R-linear basis of the q-deformed negative Borel subalgebra Eq(B-), just replacing Ek by Fk, k = 1, ... , m. (8.12.3) In order to construct the universal R-matrix, one needs the following 'R-linear bases of Utn(B±), which are defined as the subalgebras of Uq(£) consisting of polynomials in the generators {E k , Hk; k = 1, ... , m} and {Fk , Hk; k = 1, , m}, respectively:

{E(l3p t p

E(131t l Hfl ... H;'m j Tl, ... , X m E No}

---+

free over 'R.

U!in(B+),

and similarly of Utn(B-), replacing Ek by Fk, k = 1, ... ,m. (8.12.4) The universal R-matrix of the quasitriangular complex topological Hopf algebra Uq (£) is calculated explicitly, using the quantum double construction. (8.12.4.1) The Hopf algebras Eq(B+) and (Eq(B-no pp , the latter being equipped with the restricted opposite comultiplication of E q (£), are dual with respect to the following 'R-bilinear form, which is described conveniently in terms of Hk instead of Kk and Lk, k = 1, ... , m. Vf'k, l :

Bkl

:=

dknl

=

-

Blk, Bkl:= rkldl m

1-

=

Blk,

(ihIHl ) = Okl, ih:= lnqL(.B- 1hl Hl, 1=1

Universal R-Matrix of q-Deformed Simple Lie Algebras

345

(8.12.4.2) One then calculates by induction, that ~i,j:

(F(,Bi)IE(,Bj») = 1

8·· lJ -2'

- qkj

,Bj := (Jk 1

(Jk j_1 (Ok j )'

0··· 0

,Ym E No:

along the longest word of the Weyl group, and generally TIT!,

F(,Blr 1 iIfl ... iI~Tn jE(,Bp)8 p ••• E(,Bd 81 Hfl m p [T'] , = 8 8 J qk j ' -rj(rj-l)/2 . XkYk Xk .. rj8j (1 _ -2)rj qkj k=l J=l qk j

(F(,Bprp

•••

II

H~Tn)

'II

(8.12.4.3) The above Poincare-Birkhoff-Witt-like bases can be used, as was shown explicitly in the case of Am, mEN, in order to prove that this nbilinear form, and similarly the corresponding bilinear form:

UJin(rr) x UJin(B+) -

R,

are non-degenerate. Therefore again the quantum double construction can be used to construct the universal R-matrix of Uq(.c), for any simple finitedimensional complex Lie algebra .c. (8.12.4.4) The surjective homomorphism of topological Hopf algebras over R, which is defined with respect to the h-adic topology on the quantum double:

D(Uq(B+» such that

:=

Uq(B+) 0 Uq(B-)

=

Utn(B+) 0 Ulin(B-)

-+

Uq(.c),

vr k:

Hk01 -

Hk, Ek 01 -Ek, 10Fk - Fk ,

Hk, 10Hk -

then yields the universal R-matrix of Uq(.c), as image of that of the quantum double.

R

=

L

E(,Bpr p

•••

H~Tn 0 F(,Bprp

•••

iI~Tn

rt, ... ,xTnENo

II (1 - qk p

1 Xl!'"

Xm ! j=l

-2)rj j

q~j{r;-1)/2.

[Tj]qkj!

j

Hence one calculates, that R

=

(11 f: j=prj=O

(1

~qk/r q~~(r,-1)/2(E(,Bj)

[J]qk j

0 F(,Bj)rj)

'

m

exp(lnq

L

(iJ-l)kIHk 0 HI)'

k,l=l

the product being ordered according to decreasing j = p, ... , 1.

346

8. Quantum Groups

(8.12.5) One may also use the Chevalley generators, which are defined such that "Irk:

Ek := KkEk = -O'(Ek ),

Fk := FkLk

= -O'(Fk )·

(8.12.5.1) Along the longest word O'k 1 o· .. OO'k p of the Weyl group, one then defines the q-deformed positive and negative root vectors. ~2j: E({3)·) := T k-1 1 0 · · · oTk~l (Ek.), F({3)·):= T k-1 1 0 )-1)

.. ·

oTk~l (Fk)' )-1

The ordered monomials of these q-deformed root vectors, together with k = 1, ... ,m, yield again 'R-linear bases of Eq(B±).

Kt 1 ,

(8.12.5.2) The above defined 'R-bilinear form yields just the same values, inserting the new q-deformed root vectors, ordering their products with increasing j = 1, ... ,p, and placing the toral generators Hk, k = 1, ... , m, on the left hand side of the involved monomials. Hence one obtains the expression m

T(O',O')(R) = R =exp(ln q

IT t

j=lrj=O

L

(iJ-1)kIHk ® HI)

k,l=l

(1 ~qk/t

q~~(rj-l)/2(E({3j)

® F({3j)tj,

[)]qk j

(8.12.5.3) An application of T(O', 0') on one of these two expressions of R yields the other one. (8.12.5.4) The above expressions of R are independent of the choice of the longest word of the Weyl group, due to the braid-like relations of Lusztig automorphisms. (8.12.6) Denote by R k an appropriate embedding of the universal R-matrix of the topological Hopf algebra Uqk (Ad, which is polynomially spanned over and Hk, and then completed and factorized in the sense of formal 'R by power series with relations, with respect to the h-adic topology. "Irk:

xt

With these truncated universal R-matrices factorization formula

Rk, k =

1, ... , m, one obtains the

8.13 Quantum Weyl Group

R

= exp(lnq

f:

347

(iJ-lhIHk ® HI)R,

k,l=l p

R = Rk l

II (Tk~l ® Tk~l)

0··· 0

(Tk~~, ® Tk~~, )(Rk;)

j=2

8.13 Quantum Weyl Group Let I:- be any simple finite-dimensional complex Lie algebra of rank m, and consider the topological Hopf algebra Uq(l:-) over R := C[[h]]. (8.13.1) The Lusztig automorphisms Tk of Uq(I:-), k = 1, ... ,m, in the sense of unital associative R-algebras, can be reconstructed from an appropriate extension Uq(l:-) of Uq(l:-) by means of invertible generators W l , .. ·, W m , which fulfill the following commutation and braid-like relations. ~k,W E Uq(I:-): Tk(Y) = WkYWkl.

Here one defines, that

~k,l: WkWI = WIWk, rkl = 0;

WkWIWk = WIWkWI,

nl =

-1;

WkWIWkWI = WIWkWIWk, rkl = -2; WkWIWkWIWkWI

= WI Wk WI WkWI Wk ,

rkl

= -3.

(8.13.2) In the case of A l , including the new generator W with the relations WEW- l

= -FL, WFW- l = -KE, WKW- l = L,

one obtains some topological Hopf algebra Uq(At} over R, with the structure mappings: W

7

R-lq,H~H (W ® W), e(W) = 1,

and the antipode being determined by its defining property, just as that of V below. Obviously, denoting

348

8. Quantwn Groups

The coassociative property just means, that R fulfills the QYBE. Moreover the topological Hopf algebra Uq(Ad is quasitriangular with respect to R. Here the corresponding algebraic completion Uq(Ad of Uq(A 1), such that WH = -HW, is inserted. (8.13.3) With the universal R-matrix denoted by R =: L:~o R k ® R'(., the square of the antipode of Uq(Ad is generated by the following invertible element U. 00

00

k=O

n=O

U:= Lu(R'(.)Rk = q-tH~ L(-l)n 00

00

U- 1 = LR'(.u2(R k) = qtH~ L

(1

n=O

k=O

(1

-2)n - q , q_n(ys) pnEnL n ,

[n]q. -2)n

-nq I []q.

3

q n('2+ } pnEnKnj

WE Uq(Ad : u 2(y) = UYU- 1, u 2(U) = U.

Similarly one calculates, that 00

u(U)

00

= LRku(R'(.),

u(U- 1)

= Lu2(Rk)RI:.

k=O

k=O

Applying T(u,id) on L1(V) = R-1(V ® V) yields, that U

= u(V)V,

u(U)

= V u(V).

Both V2 and UL belong to the centre of Uq(Ad. Moreover u(UL) = UL, hence (UL)2 = Uu(U).

(8.13.4) Since the universal R-matrix of Uq(A 1) fulfills the equality (V ® V)R(V ® V)-l = r(R),

one finds the following central and group-like element Z of Uq(Ad. Z:= (UL)-lV 2, L1Z = Z ® Z.

(8.13.5) For qED, the complex representation of Uq(A 1), such that = u±, reads:

x±

_[00 0

E V

ql/2]

,F

=

[0 00] ,H = [10 -10] ' W = [01 -q] 0'

= q-l/4 [~ _~-l],

ql/2

U

= q-l/2 [~ q~2]'

q

u(U)

-1

= q-l/2 [q~2 ~].

More generally for generic q, consider the irreducible complex representations of Uq(A 1) on C 2j +l with highest weights

8.13 Quantum Weyl Group

349

.. 1 3 2), )=2,1'2'···' m=j,j-1, ... ,-j. Then

vV'! = (_1)j-m qm- j (j+1)t?-m' ) m UL = q- 2"( ))+1) h)+1, Z = (-1) 2·)12)+1,

hereby denoting by Vj the corresponding representation of V. These representations of Z indicate, that one may identify Z2 with the unit of Uq(A 1 ). The action of Vj on is normalized, such that one obtains the usual tensor product of complex representations of the algebraic extension Uq(A 1 ).

v;

(8.13.6) Again for a simple finite-dimensional complex Lie algebra'c of rank m, the commutation relations of W k , k = 1, ... , m, with the generators of Uq(,c), are defined by their inner automorphism property. The resulting extension Uq(,c) becomes some topological Hopf algebra over 'R, the so-called quantum Weyl group, with the structure maps such that Vik: Wk

---+R;lqtHk~Wk(Wk L1

® Wk )

= R;1(Wk ® Wk),

c(Wk )

= l.

(8.13.6.1) Obviously H (Wk ® W k )Rk(W;1 ® W;1)q;!Hk0 Hk = q;!Hk0 kr (Rk),

inserting the flip T of the tensor product. The comultiplication is compatible with the braid-like relations, and also with the inner automorphism property of the Weyl elements W k , k = 1, ... , m, which means that W E Uq(,c) :

(8.13.7) The universal R-matrix of the quasitriangular topological Hopf algebra Uq(,c) over 'R is factorized such that m

R = exp(lnq

L

(iJ- 1)kI H k ® HI)R,

R = (Wmaz ® W maz ).1

(W;;~z),

k,l=l

with the symmetric matrix iJ E Mat(m,Z) such that iJkl := rkldl1, and W maz := Wk 1 ••• Wk", along the longest word ak 1 0·· .oak" of the Weyl group.

350

8. Quantum Groups

8.14 q-Deformation of Oscillator Algebras (8.14.1) Let H q (l) be the unital associative C[[h]]-algebra of formal power series in the generators a, at, N, with the following relations.

[N,a] = -a, [N,a t ] = at, [a, at] = [N

+ I]q - [N]q

=

qN+! ql/2

+ q-N-! + q-l/2

(8.14.1.1) The last relation could be gained from the relations aat _ q±lata = q~N,

but the latter are not compatible with the following counit. The unital associative C[[h]]-algebra with these stronger relations is denoted by Hq (I).

(8.14.2) With the uniformly continuous structure mappings indicated below, H q (l) becomes some topological Hopf algebra over R:= C[[h]], with respect to the h-adic topology.

..1(at) = e-!9. at ® q!(N+!) ..1(N) = N ® I

+ e i9'q-!(N+!) ® at,

'0)

1 - ;n~ I +I ® N + (2

(12 - iO

e(a) = e(a t ) = 0, e(N) = -

® I,

z

lnq ) ,

a(a) = _q-l/2 a , a(a t ) = _ql/2 a t , a(N) = -N _ 2 (~ _ iO z ) , 2

with an angle Oz := ~ + 21l'z, ei9• = i, e2i9• = -1,

Z

lnq

E Z, and

at denoting

both a and at.

(8.14.3) These structure mappings can be rewritten, via an appropriate homomorphism of unital associative R-algebras, which is also compatible with ..1,e, and a, in terms of polynomials over R.

K

---4

qN+!; ..1(K) = -iK ® K, e(K) = i, a(K) = _K- 1;

K±1/2 ---4 q±!(N+!); ..1(K±1/2) =

e~!9. K±I/2 ®

K±I/2,

a(K±I/2) = ±iK~I/2; ..1(a t ) = at ® e-!9. K 1/ 2 + e!9. K- 1/ 2 ® at, e(a t ) = 0,

e(K±1/2) =

e±~9.,

the antipode acting on at as indicated above; [a,a

t] _

K

+ K- 1

_

- ql/2+ q -l/2' Ka-q

-1

t _

t

aK, Ka -qa K.

8.14 q-Deformation of Oscillator Algebras

(8.14.3.1) The R-linear map T(L1, id)

0

351

L1 is acting such that:

(8.14.3.2) With respect to these polynomials, one could also insert a complex parameter qED:= C\(] - 00,0] U {I}). (8.14.4) One easily establishes some homomorphism of unital associative R-algebras:

Uq(Ad

---4

Hq(l) ® Hq(I),

which is called oscillator or boson representation of the former: X+

---4

at ® a2, X-

---4

al ®

at H

---4

N 1 ® 12 - II ® N 2.

Unfortunately this representation is not compatible with the involved comultiplications. Of course, this tensor product over R describes two noninteracting q-deformed harmonic oscillators. (8.14.5) The two relations, which are not compatible with the counit, obviously imply aa t

= [N + Il q, ata = [N]q.

The Hamilton operator of this q-deformed harmonic oscillator may be defined by 1

H:= -(aa t +ata) 2

1

= -([N +Il q + [N]q). 2

(8.14.6) Consider the non-generic case of qP = 1, for p E {3, 4, ... }. Then [Plq = a implies, that there is some representation of H q(l) on CP, with at being the transposed of a, such that the complex matrix H is symmetric. Vf.k,l:

akl = ark = Ok+l,I([k]q)1/2, N kl = Okl(k -1), ([N + I]q)pp = [P]q = O. In this case aP = (at)P = O. (8.14.6.1) For instance take p = 3. ,:= ([2]q)I/2 = (q

q3

= 1; a =

[~0 0~ 0~], a2 = [~0 0~ 0~], a

3

+ q-l)I/2 j

= 0, N =

[~00~ ~2]'

352

8, Quantum Groups

(8.14.7) For any formal power series that



over R, the relations of H q (l) imply

(8.14.8) '
= qn/2(Kl/2atta + (Kl/2att-l K 1/ 2(Mo + ... + Mn-d),

kK+ -kK-l '
(8.14.9) The q-deformed harmonic oscillator algebra H q (l) is quasitriangular, with the following universal R-matrix R E H q (I)®Hq (I). (8.14.9.1) Similarly as in the case of Uq(Ad, starting from the ansatz

+ d 2 (N ® I + I

R :=exp(dlN ® N 00

L Cn (K

X

/

2a t ® K-I//2 a)

® N)

+ d3 1 ® I)

n

,

n=O

T(.1, id)(R) = R13R23 determines the parameters db d2, d 3, anq x = 1; T(id,.1)(R) = R 13 R 12 yields also d 1,d2,d3, and y = 1. d 1 = lnq, d 2 =

~2 lnq -

iB z , d 3 = d 2

(~2 -

liB z ) nq

.

Moreover each of these two equations implies, that

\.Jk I N ' v,

E

CkCI _ 0.--

CHI

[k

+ l] k

q-l/~

_ -kl/2 [k -q

+ llql/~!

_ " ' CO-I.

[k]ql/~.[llql/~.

(8.14.9.2) R.1(K) = TO .1(K)R = .1(K)R yields no further conditions on the coefficients. R.1(a) = TO .1(a)R yields the following recursion formula. Cl = i(1

+ q-l)CO,

\.JvnE N

:-=~

C2 = -(1

+ q-l)CO, n-l

Therefore

Cn Cn-l

'(I +q -1) -[] q-r -. n

ql/~

8.14 q-Deformation of Oscillator Algebras

353

(8.14.9.3) Hence one obtains

R= eXP(lnqN®N+

LCn (K

(~lnq-iBz).1(N») n

00

1 2 t / a

®K- 1 / 2

a) .

n=O

(8.14.9.4)

R- 1 = T(a, id)(R) =exp (-ln q N ® N 00

L(-i)n(1 +qt n=O

(~lnq -

iBz) .1(N»)

n q-In(n-l) I (K-tat ®Kt a) . [n]ql/2.

(8.14.10) Inserting the involution of'R, which is due to: h ~ -h and complex conjugation, the star operation on the topological Hopf algebra Hq (1), such that:

N-=N,

K~K-l, a~at,

-

-

is compatible with the comultiplication in the sense that:

.1 0 * = TO T( *, *) 0 .1. (8.14.10.1) With respect to the polynomial version, an according star operation can be defined for qED and Iql = 1, using complex conjugation. (8.14.11) The q-deformed harmonic oscillator can be reconstructed by factorizing an appropriate quantum double, from the 'R-Hopf algebras, which are generated by the sets {K±1/2, a} and {K±1/2, at}, with the corresponding restrictions of structure mappings. (8.14.11.1) An appropriate 'R-bilinear form is constructed, such that the Hopf algebras.A:= {K,a t ;IL,1],.1,c} and {K,a;IL,1],.1 ow ,c}, with the antipodes a and a-I, are dual with respect to it; here IL, 1], .1, c, a denote the restrictions of the corresponding structure maps of H q (1). Denote

(K 1/ 2 IK 1/ 2 ) =: "', (ala t ) =: (K±1/2IK'f 1/ 2)

= '!'e ill .,

Q:j

(K- 1/ 2 /K- 1/ 2 )

= -"',

'"

(K±1/2IK±l) = (K±1IK±l/2) = ",2e'fill./2, (K±lIK±l) =

_",4,

(K±1IK'f 1)

= ~j

'"

(ulKa t - qat K) = (.1°W(a)IK ® at - qat ® K) = 0 ~

",4

= q.

354

8. Quantum Groups

(8.14.11.2) The quantum double is constructed, using this non-degenerate R-bilinear form.

D(A):= {K,a t ;jl,7],.1,c} ® {K,a;jl,7],.1,c}, with the usual tensor product of "R-coalgebras, and the double cross product, inserting the restrictions of .1 from H q (1) into the latter. (8.14.11.3) In order to prove that this "R-bilinear form is non-degenerate, one needs some "R-bilinear bases of the Hopf algebras, which are generated by {K±1/2,a} and {K±1/2,a t }, respectively. Take for instance:

{a P K z / 2;p E No,z E Z}

--+

free over "R.

"R - alg span{K±!,a},

and similarly for at and K±1/2. (8.14.11.4) Calculating the double cross product one finds, that

[K®I,I®K] = 0; (1 ® K)(a t ® I) = q(a t ® 1)(1 ® K), (K ® l)(a t ® I) = q(a t ® I)(K ® I), (I ® K)(1 ® a) = q-l(1 ® a)(1 ® K), (K ® 1)(1 ® a) = q-l(1 ® a)(K ® I);

[I ® a, at ® I] = q-l/2 CXK K 1/ 2 ® K 1/ 2 _ ~K-l/2 ® K- 1/ 2. K

(8.14.11.5) Inserting the parameters ._ K .-

. 1/4

±tq

,

._ CX.-

=fq1/4 ql/2 + q-l/2 '

one obtains

[I ® a, at ® I] =

K 1/ 2 ® Kl/2 + K- 1/ 2 ® K- 1/ 2 ql/2 + q-l/2

Hence the polynomial version of the q-deformed harmonic oscillator is reconstructed as Hopf algebra over "R, factorizing this quantum double with respect to the relation {K 1/ 2 ® I - I ® Kl/2}.

8.14 q-Deformation of Oscillator Algebras

355

(8.14.12) Concerning systems with finitely many fermionic degrees of freedom, since the generators of different degrees of freedom are assumed to be graded-commuting, it suffices to consider one degree of freedom. Choose the following relations and costructure maps of odd generators at == a or at, and an even generator N. Na - aN

= -a, Nat - atN = at, (a t )2 = 0, 1

K+K._ N+!. a a + aa - ql/2 + q-l/2' K.- q , t

t_

L1(a t )

= e-!e.at ® K 1/ 2 + e!e. K- 1/ 2 ® at,

L1(N)

=N

®I

+I

®N

+ e(a+) = 0, e(N) = -

a(a)

'0) I ® I,

1 - ~~ +(2

(12 - iO

z

lnq ) ,

= -q-!a, a(a t ) = -q!a t , a(N) = -N - 2 (~-

denoting Oz := L1(K)

::~),

I + 21l'Z, Z E Z. Then obviously

= -iK ® K, e:(K) = i, a(K) = _K- 1 .

(8.14.12.1) The last one of the above relations could be obtained from aa t

+ q±lata = q±{N+ll,

hence ata = [N + I]q, aa t = -[N]q,

but these relations are not compatible with the above counit. The homomorphic image of (1), which is due to these stronger relations, is denoted by H-/q (1).

H!

(8.14.12.2) With these R.-linear structure mappings one establishes some (1), which is quasitriangular with the Z2-graded topological Hopf algebra universal R-matrix

H!

R = exp (InqN ® N (I ® I

+

C~q -

iOz) L1(N))

+ i (1 + q-l) K!a t ® K-!a).

As in the bosonic case, R- 1 is obtained from R by the substitutions: q +-----+ q-l, which means: h +-----+ -h, and: Oz +-----+ -Oz'

(8.14.12.3) Again one finds the following graded star operation on N*

= N, K +-----+K- 1 , a+-----+a t . *

*

i: .'

...

::,'

i~t .1;·';' !.

\.

DER r:;;\ ~'.;;:. r'\1

~I

,

H! (1):

356

8. Quantum Groups

(8.14.12.4) For qED, inserting the complex representation:

_[01] t_[OO]

_[-10] 0 0'

a- OO , a - 10 , N -

one easily calculates the according representation:

o

°

-iq-!

o 8.15 Oscillator and Spinor Representations of q-Deformed Universal Enveloping Algebras (8.15.1) Consider m commuting q-deformed harmonic oscillators, using for instance the tensor product of algebras over 'R := C[[h]] , with the natural embeddings:

Hq (l)

:3

X

---+

I ® ... ® I ® X ® I ® ... I m

placing X at the position k = 1, ... ,m, m = 2,3 .... The homomorphism of unital associative 'R-algebras: -b Uq(A m ) ---+ Hq(m + 1),

such that

V]'k : Xi;

---+

alak+1> Xi:

---+

akal+l' Hk

---+

Nk - Nk+1>

is called oscillator or boson representation of Uq(A m ), mEN.

HI

(8.15.2) The system (m) of m graded-commuting q-deformed fermionic oscillators is defined by the following relations of even generators Nk, and odd generators ak and at. k = 1, ... , m. 'v"lk, l:

[Nk,Nd- = 0, [ak' ad+ = [al, aiJ+ = [ak' alJ+ = 0 for k a~ = (al)2 = 0, [Nk,ad-

= -Oklal,

t

[ak, akJ+ =

qNk+! ql/2

[Nk,alJ-

+ q-Nk-! + q-l/2

= Okl a:'

Il,

Oscillator and Spinor Representations

357

Here the super-commutator is denoted explicitly by

[a, b]±

:=

ab ± 00, for a, bE

H[ (m),

the anticommutator occurring if and only if both a and b are odd. The last relation is implied by demanding, that Yrk:

akat

+ q±lal ak = q~Nk,

hence akat

= [Nk + I]q,

= -[Nk]q,

atak

but these relations are not compatible with the counit below. Nevertheless the latter relations are useful for an establishment of the subsequent spinor representations. Including these stronger relations, the resulting unital associative 'R.-superalgebra is denoted by (m).

iI!

H!

(8.15.2.1) One establishes (m) as some topological Z2-graded Hopf algebra over 'R, with the structure mappings such that Yrk: it 1/2 . -1/2 t tO L1(at ® ak , k ) = e-'I°'a k ® K k + e • K k

L1(Nk) = Nk ® I

+ I ® Nk +

(~ - ::~) I ® I, (~ - ::~) ,

e(ak)

= e(at) = 0,

e(Nk) = -

a(ak)

= -q- 1/2 ak,

a(a kt ) = -q 1/2 atk , a(Nk)

= -Nk -

2

(12 - iO

z

lnq ) ;

here one again denotes K~/2:= q!(Nk+!), such that L1(K~/2) = e-to. K~/2 ® K~/2, e(K~/2)

with an angle

Kkak

=q

(}z := ~

-1

+ 21rz, t

akKk, Kk ak

= eio.,

a(Kt!2)

= iKi:1/2,

eiO• = i, Z E Z;

t

= qakKk,

t

t

ak ak + akak

Kk

+ K;l

= q1/2 + q-1/2·

(8.15.2.2) Obviously there is an isomorphism of unital associative 'R-superalgebras, of (m) onto the skew-symmetric tensor product of m copies of

H-/q (1).

iI!

(8.15.3) The homomorphism of unital associative 'R-algebras:

Uq(A m ) -----+ H-/ q (m + 1), such that Yrk:

X k+ -----+ akt ak+1,

x-k -----+ -a tk+1ak,

H k -----+ N k - N k+1J

is called spinor or fermion representation of Uq(A m ), mEN.

358

8. Quantwn Groups

(8.15.4) The homomorphism of unital associative R.-algebras: Uq l / 2 (B m

) -----+

iIt (m),

such that um-1k x t H k-----+ N k - N k+l, VI : X+ k -----+a t k-----+-ak+lak' k ak+1> t XX m+ -----+ am' m

-----+ am,

Hm

-----+

2Nm

+ ( 1±

27ri) I,

lnq

is called spinor or fermion representation of the involved q-deformed universal enveloping algebra, m ~ 2. Note that, for instance,

(8.15.5) An oscillator or boson representation of Uq(Cm ), m as the homomorphism of unital associative R-algebras:

~

3, is defined

-b

Uq(Cm ) -----+ Hq(m), such that

(8.15.6) One also establishes an appropriate spinor or fermion representation for m ~ 4:

Uq(D m ) -----+ H-/q (m), such that

Hm

-----+

Nm -

l

+ Nm +

(1 ± 1~7rq)

I.

Main Commutation Relations and Matrix Quantum Semigroups

359

8.16 Main Commutation Relations and Matrix Quantum Semigroups (8.16.1) Let A R be the unital associative algebra over a field K, which is generated by the set {aij j i, j = 1, ... , d}, and factorized by the main commutation relations (MeR) with respect to an invertible element

R E Mat(d, K)

~

Mat(d, K),

such that V1i,k,p,q: d

d

~

~

LJ (~jklajpalq - aklaijRjplq) = 0, R =:

j,I=1

LJ ~jklEdij ~ Ed~ , i,j,k,I=1

with the components of E~j being zero, except the entry 1 in the i-th column and j-th row. (8.16.2) Via an injective homomorphism of unital associative K-algebras AR ---+ A, the above quadratic relations can be rewritten in terms of the tensor product over A. a :

RA([a] ~A IA)(IA ~A [a]) = (lA ~A [a])([a] ~A IA)RA E Mat(d,A)

~A

Mat(d,A),

RA := eA ~ R, IA:= eA ~ ld, with the units eA of A, and ld of M at(d, K), and suppressing conveniently the notation of a, [a]:= [a(aij)ji,j = I, ... ,d]. Here

Mat(d,A):=

A~

Mat(d,K),

which with respect to the R-linear structure means an A-left module over R, which is A-free over the set {E~j j i, j = 1, ... , d}. (8.16.3) With the structure mappings of so-called matrix quantum semigroups, such that d

ud· . v1l,) :

aij

~

~

LJ aik k=1

10. 'C>I

akj, aij

c ----;+ Uij,

AR becomes some bialgebra over K.

360

8. Quantum Groups

(8.16.4) Assume that R satisfies the quantum Yang-Baxter equation (QYBE), namely [

R12 R 13 R 23 = R23 R 13 R 12, R 13 :=

L R l ~ IN ~ R l',

[

R =:

1=1

L R l ~ R l', 1=1

and with similar meaning of R 12 and R23. Using the flip P on Mat(d,K) ~ Mat(d,K), every solution R of QYBE can be rewritten as an according representation of the braid group B4 on K d ~Kd ~Kd. 812823812 = 823812823, 8:= P R; "Ix, y E K d : P(x ~ y) = y ~ x, with the corresponding embeddings of 8 into the threefold tensor product of M at(d, K). Obviously the components of 8 are calculated from those of R as d

d

8 = "L..J 8ijkl E dij ~ Edkl , 8ijkl = Rkjil, P:= " L..Jk Edl ~ Edlk . i,j,k,l=1 k,l=l (8.16.5) Let! be an arbitrary non-constant polynomial in one indeterminate over K. The free K-algebra over the set {Xl, ... ,Xd} may be factorized with respect to the relations below and then denoted by XR(f). This so-called quantum K-vector space, associated with Rand !, can be established as some AR-Ieft comodule over K: d

XR(f) 3 xk

L akl ~ Xl E AR ~ XR(f); de! 1=1

--+

d

V1i, k:

L

!(8)ijklXjXl = 0, j,1=1 due to the commutation relation 2

[eA~8,~)o:(aij);i,j=1, ... ,d]] =0. An according Z2-graded version of the above MeR, and of corresponding relations of so-called quantum super-coordinates, will be presented later on the occasion of duality considerations.

(8.16.5.1) Inserting especially the relations of Mat q (2, 0, K) == M at q (2, K), which for q2 f. -1 can be rewritten as the main commutation relations associated with the R-matrix below:

o

0

0 o 1 q _ q-1 0 1

o

0

q

R+--+

[

o

0] o

[q0

o

0,8+--+

0 0 01

1 q _ q-1

q

0 0

o

8.17 Fundamental Representation of Uq (A 2 )

361

the polynomials f(t) := t - q or t + q-1, assuming char K =1= 2 in the latter case, yield the quantum plane K;'o or superplane K~i~' with the relations

{X1X2 - q-1 X2X t} or {x~,X~,X1X2 + qX2Xt} , respectively.

8.17 Fundamental Representation of U q (A 2 ) (8.17.1) The system of positive roots, and the longest word 0'max of the Weyl group are

4>+ =

{0'1,0'1(0'2) = 0'1

+ 0'2, 0'1 0 0'2(0'1) =

0'2},

(8.17.2) Let q E D\ {±i}. Starting from the complex representation of Uq (A 2 ) on C 3 , such that:

[~o ~0 ~], xi 0

xi =

t (Xn =

HI =

[~o ~10 ~], H [~0 0~ -1~], 0 2

= (Xi)t =

[~0

o o

o

0] 1

,

0

=

one finds the following complex representation of generators of E q (A 2 ):

K1=

0 0q q-1 [o 0

0] 0 , K 2 = [10 0q 00] . 1 0 0 q-1

The q-deformed positive and negative root vectors are represented by the following complex matrices. /31 := 0'1,

132

:= 0'1

+ 0'2,

/33 := 0'3;

E(/3t}:= E 1, F(/3t} := F1,

E(/32):= T 1(E2) = q-1/2(E 1E 2 - qE2Et}, E(/33):= T 1 OT2(Et} = E2' F(/32) := T 1(F2) = _q-3/2(F1F2 - qF2Ft}, F(/33) := T 1 0 T2(Ft} = F2; E(/32) = q1/2

0 0 1] [00 00 00

=

(F(/32))t;

vb: (E(/3j))2 =

(F(/3j))2 = O.

362

8. Quantum Groups

(8.17.3) Inserting the Cartan matrix r, one obtains the corresponding complex representation of the universal R-matrix.

[ 2 -1] ,r 31[2 1]

r=

-1

eXP(ln q

-1

2

t

1 2 ;

=

q!(2Hlfl)Hl+2H~fl)H~+Hlfl)H~+H~fl)Htl

(r- 1)kI H k ® HI) =

k,l=l +----+

q2/3(EJI

+ Eg s + E~9)

+ q-l/3(E~2 + E3 3 + E~4 + Eg6 + E~7 + E~8), 0

0 0

0

0

0

q _ q-l

0 0

0

0

0

0

0 0

q _ q-l

0

0

0 0 0

1

0 0

0

0

0

0 0 0

0

q

0

0

0

0

0 0 0

0

0 1

0

q _ q-l

0

0 0 0

0

0 0

1

0

0

0 0 0

0

0 0

0

1

0

0 0 0

0

0 0

0

0

q

q

0 0

0 1 0 0 0

R(q)

+----+

q-l/3

1

(8.17.4) The two Weyl elements are calculated easily from their commutation relations. WI

=C

[

0 _q-l

o

0 01 0] 0 ,W2 = c[1 OO 0 1 0 _q-l

Wm •• = WI W,WI

~ W,WIW, ~ c'

[J, -i-

I

~],

with a non-zero complex number c. (8.17.5) The element, which generates the square of the antipode, reads

oo ] q-4

,a(U)

= q-2/3 [ q0-4 0

8.18 Fundamental Representation of U q (B2)

363

V~k : UEkU-1 = q2 Ek, UFkU-1 = q-2 Fk , UKkU-1 = Kk.

Note here that the representation of U can be calculated from the universal R-matrix, namely representing

U

= Ii 0 T(<1, id) 0 r(R),

inserting the flip

<1(U)

= Ii oT(id, (1)(R),

of tensor product, and the algebra structure map Jl..

T

(8.17.6) Inserting R(q) into the MCR yields the complex Hopf algebra SL q (3, C), which is in duality with the Hopf algebra E q (A 2 ) with respect to the complex-bilinear form, which assigns to the elements of E q (A 2 ) their representatives. Denoting the above fundamental representation of the unital associative complex algebra E q (A 2 ) by

p: E q (A 2) -----+ Mat(3, C), W E Eq(A2),V~k,l: (Ylakl) = p(Y)kl.

8.18 Fundamental Representation of U q (B 2 ) (8.18.1) Consider the complex Lie algebra B 2 and its Weyl group.

r

=

[2 -1] -2

2

'

B:=[rkldl1jk,l=1,21=[!1
n.

~1],B-1=[~

{/h:= 01, (32:= <11(02) = 01 +02, (33 := <11

0

<12(01) = 01 + 202, (34 := <11

0

<12

0

<11(02) = 02},

along the longest word of the Weyl group,

(8.18.2) For qED and q4 =I- ±1, consider the following q-deformed complex representation:

xt= Xl

[H H o o

= (Xt)t,

0 0 0 0 0 0

X:;

o] o

,xt = ([2]q)1/2

-1

o

= (xt)t, with the index

[00 0 0 0

0 0 0 0

01 0 0 0

o -1

o o

0]0 0

0 0

t denoting transposition,

,

364

8. Quantwn Groups

o1 HI =

0 [o

o

0 -1 0 0 0

[00 02 00 00 0] 0 0 0 ,H2 = 0 0 0 1 0 0 0 0 0 -1 0 0 0

00 0 0 0

00 0 -2 0

(8.18.3) The positive and negative q-deformed root vectors are

E(/31)

:=

E1=

:=

q

o

0 0 0

o o

:=

] , F(/3d := F 1 = Ef,

0 0 0 -q 0 0 0 o

T 1(E2) = q-l(E1E2 - q2E 2E 1)

=

F(/32)

~

0 0 0 0 [

E(/32)

0 0

0

([2]q)1/2

[H H o o

0 0 0 0 0 0

T 1(F2) = -q-3(F1F2 - q2 F2Fd

=

([ 2] )1/2 q2

[~q3 0~ 0~ 0~ 0~] , 0

q

0 0

o E(/33)

:=

T1 0 T2(Ed

=

T2- 1(Ed

=

=

0 -1

[~q [E(f32), E 21

[H ~ ~q ~] o o

F(/33)

:=

T1 0 T2(Fd

=

-1

T2 (Fd

=

0 0 0 0

0 0 0 0

o , o o

0 0

-1

[2]q [F(/32) , F2] 1

=-

0 0 0

q [ _q2

o

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0

8.18 Fundamental Representation of Uq (B2)

~

,= T , a T, a T , (E,)

E(!i.)

F(!i.),~

~ ~

T , aT, aT, (F,)

[~

= ([21,)'/'

E,

F,

([21,)'/'

0 0 0 0 0 0 0 q 0 0

[~

0 1 0 0 0 0 0 0 -1 0

~]

0 0 -q 0 0 0 0 0 0 0

365

,

~]

(8.18.4)

-

-

q2 F I F2]1 = q-

1 T 2(E I )

= q[2]q [E2, E2E I

T 2(FI)

= 3[2f [F2, F2FI

1

q

0 _q2 0 0 0 0 0 0 0 0 0 o 0 0 0 o 0 0 o 0 0 -1 0 o 0 0 q2 o 0

0 0 0 q2 E I E2] = q ~ 0 0 o 0

q

[0

[0

~]

,

~]

(8.18.5) E~

= Ff =

E~

= Fi = (E(f32»3 = (F(f32)3 = (E(f33»2 = (F(f33»)2 = O.

(8.18.6) The two Weyl elements, which generate the Lusztig automorphisms, are

WI =

W,

[0 1 CI

_q-2 0

o o

~ ~ [~

W_

~

0 0 0 0

0 0 0 _q-2 0

W , W,W, W,

0 0 0 0 1 0 0 0 0 q-2 0 0 _q-2 0 0

~

~]

-1 0 0 -1 0 0 0

,

~]

W,W , W,W,

~ clq [ ~ q-6

0 0 0 q-4 0

0 0 q-4 0 0

0 q-2 0 0 0

~]

366

8. Quantum Groups

with non-zero complex numbers

CI, C2.

(8.18.7) The universal R-matrix of Uq (B 2 ) is then represented by

R(q)

E

Mat(5, C)

~

Mat(5, C)

+-----+

Mat(25, C),

which is calculated straightforward. An explicit expression of the fundamental representations of Uq(.c), for the series .c := Am, m ~ 1, ... , D m, m ~ 4, and of the corresponding universal R-matrices, is presented later.

(8.18.8) The element, which generates the squared antipode, is calculated from the universal R-matrix.

oo ] o o q-2

o o o

q-8

[1 0 0 0 0 0

q-4 0 0 0

o o

q-8

o o

o o o

q-8

o

J.,]

o oo ] .

o q~12

8.19 q-Deformation of Basic Classical Lie Superalgebras Let the finite-dimensional complex Lie superalgebra .c be basic classical, of rank r, and exclude the case of A(1, 1). Denote by s the number of simple roots;

s = r, if and only if .c

~

{A(n,n);n = 2,3, ... };

for .c := A(n, n), n = 2,3, ... , s = r

+ 1.

(8.19.1) The unital associative superalgebra of formal power series with relations Uq(.c) is constructed by an appropriate q-deformation of the defining relations of.c itself, over the commutative ring := C[[k]].

n

8.19 q-Deformation of Basic Classical Lie Superalgebras

367

(8.19.1.1) The Chevalley generators Ek and Fk are assumed to be even for k E {I, ... , s} \ T, and odd for k E T. One can choose card T = 1, which is proposed henceforth. The Chevalley generators H k , k = 1, ... , s, and correspondingly the generators Kk and L k defined below, are assumed to be even. Let r := [rkl; k, I = 1, ... , s] be the Cartan matrix of £., which depends on the choice of the index set T ~ {I, ... ,s}. It can be normalized, such that

rkk = 2 for k E {I, ... , s} \ T, rkk = 0 for k E T and £. ¢. {(B(O, n); n EN}, nk = 2 otherwise. The so-called symmetrized Cartan matrix is defined by

B := [Bkl := dkrkl = dllik; k, I = 1, ... ,s], with the integers dk E Z \ {OJ chosen such that the greatest common divisor of {dk;k = 1, ... ,s} is equal to 1, and 'Vk E T: dk > o. For instance in the case of A(m, n), (m, n) E No x No \ {{O, O}, {I, I}}, s = m + n + 1, choose T

:= {m

+ I};

then d 1 = ... = dm+1 = 1, dm+2 = ... = ds = -1.

(8.19.1.2) The angular momentum-like relations are deformed, such that

q := e h , 'V~ k, I : qk := qd k = ehdk , [Hk' Hz] = 0, [Hk, Ed = rklE1, [Hk' Fd = -rkIFl, 2

[Ek' Fl] = 8kl /k 1 (q:k - qk Hk ). qk The Serre-type relations are q-deformed such that (ad+ Ek)l-hl(EL)

"If k =II:

= (ad- Fk)l-hl(Fl) = 0,

Ef = Ff = 0 for rk k = O. Here the modified Cartan matrix f is obtained from r, substituting -1 for the positive components in the rows with the diagonal component O. (8.19.1.3) In case of £. := A(m,n) with mn ~ 1, choosing additional fourth order relations are q-deformed such that

ad+ Em +1 (ad+ Em (ad+ E m+1(Em+2 )))

T

as above, the

= 0,

ad- Em+1(ad- Fm(ad- Fm+1 (Fm+2 ))) =

o.

The more complicated additional relations in other cases of s ~ 3 are correspondingly q-deformed, in order to obtain a Poincare-Birkhoff-Witt-like topological basis of Uq (£.), which consists of similar monomials as in the undeformed case.

368

8. Quantum Groups

(8.19.1.4) Here one conveniently uses the following q-deformed adjoint representation.

Vik,Vy

E

Z2,Y E U!(i:-): ad+Ek(Y):= EkY - (-1)1I%LkYKkEk, ad- Fk(Y) := FkY - (-1)11% KkYLkFk,

z := +1 for k ¢. defined below.

T,

-1 for k E

T,

and inserting the Chevalley generators

(8.19.1.5) \.18k l . K k·'- qHk .- q-Hk V1 , . k ' L k·k ' K k L k -- I ,

kl I:" _ K k±1EI -- q±r k E I K±1 k' K±1 k.C'1 (8.19.1.6) Correspondingly one defines

q~rkl

k

l:"

.C'I

K±1 k .

V~k,l:

_ + tHk -tHk [+ _] Kk - L k X k := Ekqk ,Xk := Fkqk ,Xk ,XI = 0kl -1' qk - qk (8.19.2) The Z2-graded topological Hopf algebra Uq(i:-) is established, in the sense of formal power series with relations over 'R . V~ k:

Hk - ~ Hk iZI I

+I

Ek - ~ Ek iZI I

+ Lk iZI Ek,

Fk -Fk iZI Kk ~

+I

iZI Hk, Hk -

E

0, Hk - 0' -Hk,

Ek -E0 , Ek - 0' -KkEk,

iZI Fk, Fk - 0 , Fk E

0'

-FkLk'

(8.19.2.1) In terms of Chevalley generators {Ek' Fk, Kk, Lk; k = 1, ... , s}, one obtains some Z2-graded Hopf algebra Eq(i:-) of polynomials, and an according homomorphism of unital associative 'R-superalgebras:

which is also compatible with .1, e, and

(J.

(8.19.3) Inserting the involution of 'R, which is due to: h +-----+ -h and complex conjugation, one obtains some graded star operation on Uq(i:-) , which is uniformly continuous in the sense of the h-adic topology. In terms of the Chevalley generators, V~ k:

X:

+-----+

•

Xi;, Hk +-----+ Hk· •

(8.19.3.1) Substituting generic q E C for the indeterminate h = In q, one obtains the complex Z2-graded Hopf algebra Eq(i:-). For Iql = 1, and using complex conjugation, one finds an appropriate graded star operation on Eq(i:-).

8.19 q-Deformation of Basic Classical Lie Superalgebras

369

(8.19.4) For instance consider the case of £, := C(2) == osp(2,2). Choose := {2}, such that

T

0 ' d [-22 -1]

r=

l =

2,

d2 =

[4 -2 -2] o ' r=r.

1, B =

The Serre-type relations read E~E2 - (q2

+ q-2)E l E 2El + E2E~ =

F't F2 - (q2

+ q-2)Fl F2Fl + F2F't

Due to

r 22 =

~ [~

E1 =

E2 =

= O.

0 one finds, that E~ = F? =

(8.19.4.1) For qED, q4 mental representation:

HI

0,

0 0 0 0

0 0 1 0

0 0 0 0

0 0 0 0

[~ [1

0 0 0 0

~

:f:.

o.

±1, one finds the following q-deformed funda-

] , H,

-1

~ [~

0

0 -1 0 0

0 0 -1 0

~]

0 0

0 0 0 0

,

6o0] = Fl , t

0

i.fiq 0 0

~] ~ [~

i

, F,

~

~] o . 0

Since the entries of H l and H 2 are ±1, being composed to ±a3 , the qdeformation does not become manifest by the fundamental representation of

xt, k = 1,2. Here the usual representation, which yields H 2 =

_ [

~2

is transformed for convenience.

~3l

(8.19.4.2) The roots are written in terms of the Cartesian unit vectors ei, i = 1,2, ofR2.

L1

= {al := 2e2,a2 := el - e2},

{3l := al, {32 := al

+ a2,

!li~

= {ad,

=

{a2,al

+ a2}'

(33 := a2·

E/31 := E l , E/3~ := ad+ E l (E2) = E l E 2 F/31 := F ll

!li~

q2 E 2 E l , E/33 := E 2. 2 F/3~ := ad- Fl (F2) = F1 F2 - q F2Fl, F/33 := F2· -

370

8. Quantum Groups

(8.19.4.3) With respect to the complex Hopf algebra E q (£) one finds, that qED, q4 =I- ±1

==}

E~~

= FJ~ = O.

(8.19.4.4) The following Poincare-Birkhoff-Witt-like R-linear bases of the q-deformed subalgebras

Eq(B+) := R - alg span({Ek' Kt 1 ; k = 1, 2}), Eq(B_) := R - alg span({Fk, Kt\ k = 1, 2}), can be constructed: {E~:E~:E~:K:IK~~jXl E Nojx2,x3 E {O,I}jZl,Z2 E Z} ---4

free over R.

Eq(B+),

and similarly for Eq(B_) with Fk, k = 1,2, instead of E k . Hence one constructs an according topological basis of Uq (£) over C[[h]]. (8.19.4.5) Choosing qED, q4 =I- ±1, one finds according C-linear bases of the q-deformed subalgebras Eq(B±) of the unital associative complex superalgebra E q (£). (8.19.5) Consider the case of £ := A(m, n), m =I- n, m and n E No, such that s = r, and choose T := {m + I}, such that

r

= [(1

+ (_I)b k,"'+l )Okl - Ok,l+l -

(-I)b k,... +10 k+1,I; k, l = 1, ... ,r],

r = m + n + 1; d 1 = .. , = dm +1 = 1, dm +2 = ... = dr = -1.

t

[rkkOkl - Ok,l+1 - Ok+1,lj k, l = 1, ... ,r]. L1 = {Ct1 := e1 - e2, ... , Ct r := er - er+1}, =

inserting the Cartesian unit vectors ei, i = 1, ... , m + n + 2, of Rm+n+2. The sets of even and odd positive roots are

w~ = {Ctk +

+ Ctlj 1 ~ k ~ l ~ m, m + 2 ~ k ~ l ~ r},

w~

+ Ctl j 1 ~ k ~ m + 1, m + 1 ~ l ~ r}.

=

{Ctk +

(8.19.5.1) According to the set of positive roots W+ = w~ uw~ = {Ctk + ... +Ctljl ~ k ~ l ~ r}, the following q-deformed root vectors are defined.

'VIk : E ak

:=

E k , FOlk

:=

Fk ·

VI ~ k < l $ r : E Olk +"'+ OlI := ad+ Ek 0 ' " 0 ad+ EI- 1 (El), FOlk +" +Olj := ad- Fk 0 · · · 0 ad- FI- 1 (Fl ).

8.19 q-Deformation of Basic Classical Lie Superalgebras

371

(8.19.5.2) One then finds, that V{J E lPl : E~ = FJ = 0, if q E C is chosen to be generic, or if q E C[[h]]. (8.19.5.3) One establishes the following C-linear basis of complex polynomials in Ek, Fk, Kk, Lk, k = 1, ... ,r, for generic q:

Y Z Xt ... EX" {E/3t /3" FYt /31'" F/3""K 1 l

...

KZr. r ' XI, ••. , Yp EN· 0, Z 1, ... , Zr E Z}

----+ E q (£),

c

with Xj and Yj actually E {O, I} for {Jj E lPr.j = 1, ... ,po Here the set of positive roots lP+ =: {{Jl,' .. ,{Jp} is strictly ordered such that "11

:5 i < j :5 p : {Ji = ak + ... + aI, (Jj = as + ... + at, with k < s, or k = s and I < t.

(8.19.5.4) Of course one could also use the h-adic topology, in order to construct an according topological basis of Uq (£) over C[[h]]. (8.19.5.5) In case of £ := A(O, 1) with F=

[~1

;],

T

:= {I},

d 1 =1, d2 =-1; ..1={al:=cl- c2,a2:=c2- C3},

lP~ = {{J3 := a2}, lPl = {{Jl := al,{J2 := al + a2}' E/32 := ad+ E 1 (E2) = E 1 E 2 - q-l E 2El, F/32 := ad- F1 (F2) = F1 F2 - q-l F2F1 . The Serre relations read E~El - (q

+ q-l)E2E 1 E 2 + EIE~

FiFl - (q

+ q-l)F2FIF2 + F1 Fi

= 0,

= 0.

Since E? = Ff = 0, also E~2 = F~ = 0. For qED, q2 tal representation is calculated easily:

=I -1, the fundamen-

372

8. Quantum Groups

(8.19.6) In case of B(I, 1) := osp(3, 2) with r=

[~2

T

:= {I}, such that

;], d1 =2, d2 =-I, L1={01:=c2- C1l 0 2:=cd,

!Ii~ = {02,2(01 +(2)}, !Ii~ = {01,01 +02,01 +202}, inserting above the Cartesian unit vectors ci, i = 1,2, of R 2 , the subsequent fundamental representation of Chevalley generators of Uq (B(I, 1)) does not show up any q-deformation, for qED and q2 :f -1.

HI = diag[O, 0,1,1,0], H 2 = diag[l, 0, -1, 1, -1],

xt

0 0 0 0 0 0 0 00] 0 = (Xd = 0 0 0 1 0 , [ 00000 00000

xt

1 0 0]0

[00 o o 0 0 = (Xi)t = 0 o 0 0 0 o 0 0 0 000

0 1 0

.

(8.19.7) In case of B(O, 2) := osp(l, 4) with T := {2}, the roots being located in R 2 with Cartesian unit vectors C1 and C2, such that

o

-

!Ii+ = {01,01 +202,2(01 +(2),202}, !Iii = {01 +02,02},

one easily finds the following fundamental representation of Uq(B(O, 2)), for qED and q2 :f -1. ~ := ([2]q)l/2.

HI = diag[O, 1, -1, -1, 1]' H 2 = diag[O, -2,0,2,0],

xt =

[~ ~ ~ ~ Oo~], xt o

0

Xi

0

0

0

-1

0

e

0]

~ 1{ ~ ~ ~ ~ [

=

[eoq

xt = (Xi)t,

n~],

0 0 0 0

8.20 Universal R-Matrix of Uq (B(O, 1))

373

8.20 Universal R-Matrix of Uq (B(O, 1» (8.20.1) The topological Z2-graded Hopf algebra Uq (B(O, 1)) over the ring 'R := C[[h]] is defined by the following relations and costructure maps. HE - EH H E

---+ .:1

H

---+ .:1

E

@ @

= 2E, I I

+I

@

---+ 0,

H, H

H

E

=

+-1 2

q -

(qH _ q-H).

-H,

---+

E

+ q-H @ E,

EF + FE

= -2F,

HF - FH

U

---+ 0,

E

---+

E

_qH E,

U

F---+F@qH +I@F, F---+O, F---+-Fq-H . .:1

E

U

Here the generators E and F are odd, whereas H is even. Note especially that E2 and F 2 are non-zero, due to the Cartan matrix r = 2. (8.20.2) This q-deformed universal enveloping superalgebra over 'R is quasitriangular with respect to the following universal R-matrix. 00

R

=L

en(E n @ Fn)q!H®H,

co:= 1,

Cl

= C2 := -(1

_ q-2),

n=O

"In EN: en:= (-It(1- q- 2tq!n Cn-l)

n

(11 (k)q) -

1

,

k=l

(k)q

(_I)k+1 qk + q-k :=

q + q-l

,(I)q

= 1.

(8.20.3) Let qED, q2 =I- -1. Inserting the fundamental representation:

H = diag[O, -2,2],

q 0] ,F=(q+q-l)' [0- 1 00 0 q] , E~ (q+q-I)I [~ 00 o

0

0

0 0

such that E 3 = F 3 = 0, and an isomorphism of unital associative complex superalgebras: M at(3, C)®Mat(3, C) --.+ M at(9, C), such that 'V~i, i, k, l: E~j

@

E~l

--.+

(_1)U+3)k E;Ci-l)+k.3 Cj -l)+I,

1:= 0, 2 = 3 := 1,

one calculates the following representation R(q) of Ron C9:

374

8. Quantwn Groups

R(q)~

0 0 1 0 0 1 0 0 0 0 0 0 0 qTl 0 0 0 0

1 0 0 0 0 0 0 Tl

0 Tl :=

0 qTl 0 1 0 0 0 0 0

-q(1 - q-4), T2 := (1

0 0 0 0 q2 0 0 0 0

Tl

0 0 0 0 q-2 0 T2 0

+ q2)(1 _

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 q-2 0 0

0 0 0 0 0 0 0 0 q2

q-4).

8.21 Duals of Quasitriangular Z2-Graded Hopf Algebras Consider a Z2-graded Hopf algebra {1t, fL, 71, .1, e} over a field K, chaT K = 0, which is quasitriangular with respect to the universal R-matrix R E 1t®1t, the latter being even by definition. (8.21.1) Let p : 1t --+ Mat (d,K) be a representation of 'H. on K d , in the sense of unital associative superalgebras over K. The Z2-grading of these K -endomorphisms is arranged such that V{'k:

E3 even, v:trk: E3 odd,

m

+n =

d,

denoting by E~ the Cartesian unit vectors of K d • (8.21.2) The corresponding representation of R is denoted by: d

R

---+

T(p,p)

Rp

L

=:

~jklEY @E~l E Mat(d,K)®Mat(d,K).

i,j,k,l=l

(8.21.3) Calculations of examples may be done conveniently using an isomorphism of unital associative superalgebras over K: Mat(d,K)®Mat(d,K) 3 Ey @E~l ~ (_1)(i+3)kE~~-1)d+k,(j-l)d+1E Mat(d 2 ,K),

denoting

k := 0 for

1 ~ k ~ m, and 1 for m

+1 ~

k ~ d.

Duals of Quasitriangular Z2-Graded Hopf Algebras

375

(8.21.3.1) Denoting the components of A,B E Mat(d, K)®Mat(d, K) similarly as above those of R p , one easily finds that d

\.Id· vlt,P, k ,q.. (AB) ipkq --

~ (l)(J+p)(k+I)A L..J ijkl B jplq' j,l=l

(8.21.4) The matrix quantum super-semigroup AR p == A(R; p) is defined as the unital associative K-superalgebra, which is generated by even generators akl for 1 ~ k, l ~ m and m + 1 ~ k, l ~ m + n, and odd akl for the remaining choices of 1 ~ k, l ~ m + n, and factorized with respect to the homogeneous main commutation relations (MCR) RA ([a] ®A lA) (lA ®A [aJ) = (lA ®A [aJ) ([a] ®A lA) RA E Mat(d, A)®AMat(d, A),

suppressing an explicit notation of an injective homomorphism a of unital associative K-superalgebras: A(Rj p) --+ A, Le. inserting d

[aJ:=

L

a(akz) ®E;l E Mat(d,A):= A®Mat(d,K),

k,l=l

and denoting lA := eA ® ld, RA := eA ® R p • Note that here the graded tensor product A®Mat(d,K) is constructedj one could also use the ungraded one. The latter was used for transformations of finite-dimensional quantum superspace. (8.21.5) In terms of components these MCR explicitly read, such that Vti, k,p, q : d

L

(~jkl

ajpal q( _l)(J+p)(k+tl) - (-l)(J+P)(k+l)aklaijRjPlq) = O.

j,l=l

(8.21.6) Obviously A(Rj p) becomes some Z2-graded bialgebra over K, such that

3

(8.21.7) The representative R p solves QYBE on @Mat(d,K), due to the quasitriangular properties of R itself.

376

8. Quantum Groups

(8.21.8) Inserting the Z2-graded flip "Ix E 'H, Vyi,j, k, l : (T(p, p)

0 T 0

T

on 'H®'H, one easily finds that

L1(X))ijkl = (_l)(i+J)(k+i) (T(p, p)

0

L1(x)hlij.

(8.21.9) Writing without loss of generality 'H as some free K-algebra factorized by certain homogeneous relations, one then establishes 'H and A(R; p) as dual Z2-graded bialgebras, with respect to the K-bilinear form such that Vtk,l: 'H x A(R;p) 3 {x,akl} -

(p(x)hl E K.

This K-bilinear form is compatible with the relations of 'H, because p is some representation of 'Hj it is compatible with the MCR too, due to the above identity involving the representatives of L1(x) and TO L1(x), x E 'H. (8.21.10) Consider the matrix quantum semigroup Matq(m

+ 1,0, C) == Matq(m + 1, C), mEN,

which is defined by the following Manin relations. Vf+li < j, k < l : qkl E C\{O},

= qkl-1 ailaik, aikajl + qklailajk = aikail

= qij-1 ajkaik, qi/(ajkail + qklajlaik),

aikajk

-1 ai/ajk = -qij ( ajkai/ - qkl - ajlaik 1 ). aikajl - qkl

(8.21.10.1) Especially let qkl := q for 1 ~ k < l ~ m

+ 1,

and assume

q2 =F -1. Then

Vf+li

< j,k < l:

ajkail = ai/ajk, al/akk - akkal/ = (q - q-1)aklalk.

Inserting the fundamental representation of the universal R-matrix of Uq(A m ) for qED, the resulting MCR are equivalent with the Manin relations. For instance for m ~ 2,

= 0 for q12 = q, q131a13allixt Xi) = 0 for q13 = q.

(alla12 - qlla12alllXi) (alla13 -

(8.21.10.2) The MCR of the fundamental representation of Uq(A m ) allow for an appropriate definition of the so-called quantum determinant detq[akl; k, l = 1, ... , m

+ 1] == d:=

L PEP",+l

(_q-1)'~P alii··· am+l,j"'+ll

Duals of Quasitriangular Z2-Graded Hopf Algebras

denoting the permutations

[~

Jl

~ + 1]

=:

Jm+1

377

P, and by Ap the length,

Le. minimal number of transpositions of P. This quantum determinant is both central and group-like. vr+1k, l : akld

= dakli

d --+ d @d, c(d)

a

= 1.

The factor algebra SLq(m + 1, C), which is obtained from the MCR with the additional relation d = e := unit, becomes some complex Hopf algebra, because L1 and c are compatible with the new relation, with the antipode (1 such that vr+1k,l:

m+l

L

m+l

akj(1(ajl) =

j=1

L

p.= [1 ... k-l, .

(1(akj)ajl = 6kl e,

j=1

jl .. ·jl-l

k+l ... m+l] jl+1" ·jm+1 .

This complex Hopf algebra SLq(m+ 1, C) may be called special matrix quantum group. (8.21.10.3) Let again qED, q2 :f= -1. The complex Hopf algebra Eq(A m ), consisting of polynomials in the Chevalley generators, is in duality with the complex Hopf algebra SLq(m + 1, C), with respect to the C-bilinear form, such that Vik, vr+1i,j:

(Eklaij) = (Fklaji) = ql/26ik6j,k+1l (Kklaij)

= q6ik6jk + q- 16i ,k+1 6j,k+1'

(8.21.10.4) Correspondingly the topological Hopf algebra Uq(A m ) over the ring'R := C[[hll, consisting of formal power series with relations in X;, Hk , k = 1, ... , m, is in duality with SLq(m + 1, C), with respect to an 'R-bilinear form such that Vrk, vr+1i,j:

(Xtlaij) = (X;laji) = 6ik 6j,k+1, (Hklaij) = 6ik 6jk - 6i ,k+1 6j,k+1'

Here SLq(m + 1, C) is considered with respect to coefficients E 'R. The representatives of these Chevalley generators of Uq(A m ) are not qdeformed, because [(13]q = (13. The universal R-matrix is represented by

378

8. Quantum Groups m+l

qmh R(q) = q

L E~k+l i8l E~k+1 + L k=1

+ (q -

q-l)

E~\1 i8l E~+1

19~I=:;m+l

L

E~+1 i8l E~+1'

l=:;k
(8.21.10.5) The resulting MCR are compatible with the following commutation relations of quantum coordinates, in the sense of an A(R; p)-left comodule over R or C, respectively. \:ft+lk


: XkXl = q-l xIXk .

(8.21.11) Consider the representation p of the q-deformed harmonic oscillator H q (l) on C3:

a=

00 01 0] ~ , N = [00 01 0] 0 , q:= [00000 2

J3' -21 ± i T

q3 = 1,

the representatives of a and at being mutually transposed, and denoting ~ := (q + q-l)I/2. The resulting representation of the universal R-matrix R yields some matrix quantum semigroup A(Ri p), the MCR of which are compatible with the following commutation relations of quantum coordinates. Denoting by P the flip of Mat(3, C) i8l Mat(3, C), V'~i, k : 3

L

3

(PR - RllllI)ijkl XjXI =

j,I=1

L

Rkjil XjXl -

R l l l l XiXk = O.

j,l=1

The corresponding factor algebra, which is obtained from the free complex algebra over the set {Xl,X2,X3} with respect to the relations X2Xl = -iql/2XIX2, X3Xl = -qXIX3, X3X2 = iql/2X2X3,

qx~ = XIX3, 1 + q + q2 = 0,

does not collapse; here one chooses

~

= i.

(8.21.12) Inserting the Z2-graded flip P of Mat(d, K)®Mat(d, K), such that V'1i, j :

P(E~ i8l E~) = (_l)iJ E~ i8l E~,

d

P:=

L

(_l)iE;l i8l E~k,

k,l=1

the above introduced MCR can be rewritten as the commutation relation

[eA i8l Sp, ®A[akli k, l = 1, ... , dJ] = 0,

Sp := PRp =: SijklEY i8l E;I.

Duals of Quasitriangular Z2-Graded Hopf Algebras

379

Then the following quadratic relations of quantum super-coordinates Xk, k = 1, ... ,d = m + n, namely such that for an arbitrary non-constant polynomial f in one indeterminate over K, Vti, k: d

2: f(Sp)ijkIXjXI (_1)3 k = 0,

Sijkl = (_I)i 3+ik+3k Rkjil,

j,l=1

are compatible with the MCR, in the sense of an A(R; p)-left comodule over K, the latter being Z2-graded in the following natural sense. Consider the generators Xk as even for 1 ~ k ~ m, odd for m + 1 ~ k ~ d, such that the above coordinate relations are homogeneous, thereby establishing the unital associative superalgebra X(R; p) over K. Then the structure map: X(R; p) ~ A(Rj p) ® X(R; p) is even. For instance inserting the polynomial f(t) := t - A, A E K, one finds the relations, Vti, k: d

2: Rkjil xjxI(-I)i3 = A XiXk· j,l=1

Of course these MCR and quantum super-coordinate relations may also be understood via an injective homomorphism of unital associative Ksuperalgebras Q into a common pool A, such that

using the super-commutator. Then the above relations of quantum supercoordinates and their transformation can be short-written as (eA ® f(Sp)) ([x] ®A [xl)

= 0,

[x']

= [a][x],

the latter explicitly meaning that d

Vtk : Xk :=

2:( -l)(k+i)iakIXI' 1=1

Note here that R p , P, and therefore also Sp are even elements. (8.21.13) Some of the above considerations can also be performed over the ring n := C[[h]], using complex formal power series in h as matrix elements. As an example consider the q-deformed fermionic oscillator over n. Denoting

a and d even, band c odd, x even, y odd, the MCR read ab

= i q l/2ba,

ac

= i q l/2 ca ,

bd = -iq l/2db, cd

be = -cb, ad - da = +i (ql/2 + q-l/2) be.

= _i q l/2dc,

380

8. Quantum Groups

The corresponding coordinate relations, choosing f(t) := t->., are non-trivial only for two values of>. E 'R; otherwise x~ = x~ = X1X2 = X2X1 = O. One just inserts the representation of R on 'R 4 , together with the 'R-linear bijection:

P +---+

1 0 0 0 1 0 [ o 0 0

0] 0 0 -1

OOl

>. := R llll = exp (

\

" := q

-lR1111

w2k l . E k

,v1"

irr :t1 lnq +"2 -

~ X

2

2

= Y =

2

to. '01

(J2)

ln~

E l +---+ 2

(_l)ki E(k-1)2+1

4'

.

~ xy = tq

1/2

yx.

0 ,xy = tq . -1/2 yx.

8.22 Deformed Tensor Product of Matrix Quantum Super-Semigroups (8.22.1) Consider a Z2-graded bialgebra {1tj /-L, 11j Ll, e} over K, char K = 0, and let 0:,13 be representations of the unital associative superalgebra 1t on KP and Kq respectively. The tensor product of these two representations is defined, with an obvious notation of degrees:

1t

:3

~

L(x) ~

I

"T(Q,{3)

x ---+ L..J x>. ~ x>.

---+

+---+ T~(o:,13)(x)

:

>'=1 L(x)

KP ~ Kq

:3

1.£

~V

---+

(_l)X~U

L o:(x~)(1.£) ~ 13(x~)(v) E KP ~ Kq. >'=1

In the special cases of K(Q) over a group Q, such that Ll : Q :3 9 ---+ 9 ~ g, and of a Lie superalgebra .c, such that Ll : .c :3 x ---+ x ~ e + e ~ x, with the unit e of E(.c) in the latter case, one finds the usual tensor products of representations of such objects. (8.22.2) Let especially 1t be a Z2-graded Hopf algebra, which is quasitriangular with respect to the universal R-matrix R. One then obtains, according to the proposed representations, two matrix quantum super-semigroups A(R; 0:) and A(R; 13), with the generators aij and bkl respectively, for 1 ~ i, j ~ p, 1 ~ k, l ~ q. Now consider the Z2-graded bialgebra A(Rj 'Y), 'Y +---+ T~(o:, 13), with the generators Cr~, 1 ~ r, s ~ pq. One easily calculates that

Deformed Tensor Product of Matrix Quantum Super-Semigroups

381

L(x)

(XIC(i-l)q+k,(j-l)q+l) =

L (x~laij)(x~lbkl)( _1)(i+])k,

).=1

with the above notation of L1(x). Moreover, denoting by R~;kf) the components of T(o:, (3)(R) ,

0= T(o:, (3)(RL1(x) L(x)

=

q

Jl

L .~ L

(

T 0

L1(x)R)

R~;kf)(x~ @x~lajm @bln)(-l)(}+m)(k+n)

>'=1 1,1,m=1 k,l,n=1 _(_l)(}+m)(k+i) (X'>.

10. VY

X">. Ibkl

.0,

VY

a11.)R(.Ci,{3)) E Jlim l m1n

.0,

VY

E qkn ,

and similarly by an application of T({3, 0:), with corresponding components

({3,Ci) R klij

(8.22.3) These identities, although the involved bilinear forms may be degenerate, are hints to the subsequent statement. There is some homomorphism of unital associative K-superalgebras:

A(R; /,)

3

C(i-l)q+k,(j-l)q+l

----+

aijbkl( _l)(t+J)i

E

A(R; o:)®A(R; (3),

into an appropriately deformed tensor product of unital associative Ksuperalgebras, both of which were defined by main commutation relations. It is constructed from generators aij and bkl, 1 ~ i, j ~ p, 1 ~ k, l ~ q, with respect to the following homogeneous relations. The generators aij among themselves, and bk1 among themselves, fulfill the MeR-type relations of A(R; 0:) and A(Rj (3), respectively. Moreover the mixed relations below are proposed. Jl

q

~~ (R(~,{3)a. bl (_l)(}+m)(k+n) - (-l)(}+m)(k+i)bkla .. R(Ci,{3)) = 0

L..J L..J

11k1

1m n

11

lmln

'

j=11=1

(8.22.3.1) Using copies into an associative unital superalgebra A, denoting

R<;,(3) := eA

@

R(Ci,{3) , R(Ci,{3):= T(o:, (3)(R) ,

and similarly for T({3, o:)(R) , these mixed main commutation relations read as follows.

R<;,(3) ([a] @A IA)(IA @A [bJ) = (IA @A [bJ)([a] @A IA)R<;,{3) ,

R~,Ci)([b] @AIA)(IA @A [aJ) = (IA @A [a])([b] @A IA)R~'Ci).

382

8. Quantum Groups

The proof of this theorem is performed by means of the following lemma. Rb,-Y) := T(-y, "()(R)

<----+

T(a, (J, a, (J) 0 T(L1, .1)(R) _ R(a,{3)R(a,a) f}({3,{3) f}({3,a) -

14

13

H24

A

"23

.

Here the lower indices describe embeddings into an appropriate four-fold tensor product of matrices over K. One then easily shows, that the above representation of R on KP 0 Kq 0 KP 0 Kq serves for an MCR-type intertwining property of [a] 0A [b]. Denoting R1'-Y) <----+ eA0Rb,-Y), one calculates that R1'-Y)([a10A [b] 0A IA)(IA 0A [a]0A [bJ) = (fA 0A [a] 0A [bJ)([a]0A [b]0A I A )R1'-Y)·

8.23 Deformed Tensor Product of Quantum Super-Vector Spaces (8.23.1) Let P, Q, S, T be the graded flips on

KP 0 KP, Kq 0 Kq, KP 0 Kq, Kq 0 KP, and denote eA 0 P =: PA, ... . Here again K denotes a field of char K = O. P

q

P

q

ki 0 E ik T := ""(_l)k E ik 0 E ki S := ""(_l)i E q,p L..J L..J p,q' L..J L..J p,q q,p' i=1 k=l

i=l k=l

TS

= id(KP 0

KP), ST

= id(Kq 0

Kq),

involving matrices over K with p rows and q columns, and conversely; for instance, (E:'~)lj = DklDij' The mixed MCR-type relations of the foregoing chapter can be rewritten, using again faithful copies into an associative unital superalgebra A, as

SAR~,{3)([a] 0A [bJ) = ([b] 0A [aJ)SAR~,{3), TAR~,a)([b] 0A [aJ) = ([a] 0A [bJ)TAR~,a). Therefore the subsequent relations of Xi and Yk, 1 compatible with their transformation according to [x'] = [a][x], [y'] = [b][y]. For fJ.,V E K,

SAR~,{3)([x] 0A [yJ) = fJ.([yJ) 0A [xJ), TAR~,a)([y] 0A [xJ)

= v([x] 0A

[yJ).

~

i ~ p, 1 ~ k ~ q, are

Deformed Tensor Product of Quantum Super-Vector Spaces

383

(8.23.2) This invariance property induces the following construction. Let X(R; a) and X(R; (3) be the unital associative superalgebras of quantum super-coordinates, which are due to the relations (PAR<;'O:) - OA)([X] ®A [x])

= 0,

(Q A R<.:.,{3) - T/IA)([Y] ®A [y])

= 0,

" T/ E K, denoting

R<;'O:) := eA ® R(O:,o:) , R(O:,o:):= T(a, a)(R),

and similarly for the representation (3 of the quasitriangular Z2-graded Hopf algebra ?t. Moreover, for>. E K, define quantum super-coordinates Zr, 1 ~ r ~ pq, with respect to the relations

(P AR<;"'Y) - >.IA)([Z] ®A [z]) = 0, "I ~ T.a(a, (3), P A := eA ® P, denoting by P the graded flip on

Kpq ® Kpq ~ KP ® Kq ® KP ® Kq. pq P:= E~ ® E;~( -1)§

L

r,8=1

P

q

~ '"' '"' Eij ® E kl ® Eji ® E 1k ( -1) (i+.i)(k+i)+.i+i ~ P ® Q, L..J L..J P q P q r~3 i,j=1 k,I=1

S = 3+[ for s = (j - l)q + l, where T23 denotes the indicated embedding of the graded flip T of matrices over K into the involved four-fold tensor product. (8.23.2.1) On the other hand, a deformed tensor product X(R; a)®X(R; (3) is constructed with respect to the following relations of quantum supercoordinates. The coordinates Xi among themselves, and Yk among themselves, 1 ~ i ~ p, 1 ~ k ~ q, fulfill the relations of X(R; a) and X(Rj (3) respectively.

Moreover they are assumed to obey the mixed relations above with parameters Il, v E K. Choose>' = ''T/Ilv. Then the subsequent theorem holds. There is a homomorphism of unital associative K -superalgebras: X(Rj "I) 3 Z(i-l)q+k

--+

XiYk E X(R; a)®X(R; (3),

which can be short-written as [z]

--+

[x] ®A [y].

(8.23.2.2) The proof of this theorem is due to the following lemma.

P Rb,"Y) ~T23(P ® Q)R~~,{3) R~~'O:) ~,(3) R~~'O:) = T23 ((P ®

Q)R~~,{3) R~~'O:) R~,(3) (T(R))~~,{3))

_ (T(R))({3,o:) P R(o:,O:)Q

-

23

13

13

Q({3,{3) Q({3,o:)

24~"24

~"23

One then calculates, that

(P AR<;""Y) - >.IA)([X] ®A [y] ®A [x] ®A [y]) = O.

.

384

8. Quantum Groups

(8.23.3) As an easy example consider the two-fold tensor product of complex representations of Uq(A 1 ), qED, q2 f. -1 for odd coordinates, such that: - ± H - 3 R -1/2 X ± -u, -u, +---+q

Choosing for instance J.I. =

v = ql/2,

[

q 0 0

0 1 0

q - q-l

0 1

0] 0 O'

o

0

0

q

one obtains the mixed relations

One thereby obtains the following relations of these new quantum supercoordinates [z], inserting even or odd quantum coordinates Xk and Yk, for k = 1,2.

Case 1. ~ ZlZ2

= T/ = ql/2, >. = q2 j

= q-2 Z2 Z l,

ZlZ3

qXlX2

= q-2 Z3 Z1,

= X2Xl,

qYlY2

Z2 Z4

= q-2 Z4 Z2,

XlX2

=

= Y2Yl; Z3 Z4

= q-2 Z4 Z3,

ZlZ4 = q-4 Z4Z l, Z2Z3 = Z3Z2·

Case 2. ~

= T/ =

2 = X22

Xl

2_2_2_2_0 Z2 -

ZlZ2

=

q X lX2

ZlZ2

=

Z3 -

Z2Z1

=

Z4 -

= X2 X l, =

2

Z3

_

,ZlZ4 -

Z4 Z l, Z2 Z3

=

=

= Z3Z1 = Z2Z4 = Z4Z2 = T/ = _q-3/2, >. = -1;

ZlZ3

= ql/2,

2 = Z22

Zl

-qX2Xl, YlY2

=

-qY2Yl,

= Yl2 = Y22 = O',

Zl -

Case 3. ~

>. = q-2;

_q-3/2,

YlY2

=

-qY2Yl,

Z3 Z 2, Z3Z4

= Z4Z3 = O.

2 0 =2 Y2 = ;

Yl

2 0 = ,ZlZ3 =0 Z3 Z l = Z2 Z 4 = Z4 Z 2 = ,

Z4

-Z2Zl, ZlZ4 = _q-2 Z4Z l, Z2Z3

=

_q-2 z3z2 , Z3Z4

=

-Z4Z3.

Covariant Differential Calculus on Quantum Superspaces

385

8.24 Covariant Differential Calculus on Quantum Superspaces Let A be an associative superalgebra over an algebraically closed field K of characteristic O.

(8.24.1) Let r = rO EEl r i be a graded A-A-bimodule over Kj the latter assumption especially means, that 'V"( E r, 'Va, b E A : (a"()b = abb); moreover the module-multiplications are compatible with the Z2-grading in the natural sense. Let d : A ---+ r be an odd K-linear map, which obeys the Z2-graded Leibniz rule, Le.,

'Vx, fi E Z2, 'Va E A~, bE AY : d(ab) = (da)b + (-lY'a db E

r

X

+II +l.

The pair {r, d} is called first order graded differential calculus over A, if and only if . n

'V"( E

r: 3n E N, 3al,b1, ... ,an,bn E A: "( =

Lakdbk' k=l

(8.24.1.1) For instance let A be unital, and consider the Z2-graded differential algebra H(A), which is obtained from the unital universal differential envelope n(A) +-----+ H(A) by identifying the unit of A := K EEl A with the unit eA of A itself. Denoting the graded structure map of A by

A: AI8iA 3 al8ib ---+ (-l)lI a b E A

for bE AY,fi E Z2,

one easily constructs the following first order graded differential calculus {ker A, b} over A: A 3 a ~ eA I8i a - (-l)X a I8i eA E ker de!

A for a E AX, x

E Z2'

Here ker A becomes a graded A-A-bimodule over K, with respect to the module-multiplications, such that 'Va, b, c E A:

(a I8i b) 0 c = a I8i be, a 0 (b I8i c) = ab I8i c. (8.24.2) Let {r, d} be a first order graded differential calculus over A, and moreover assume {A, fL, 1/,..1, e} as Z2-graded bialgebra over K. This calculus is called left or right covariant, if and only if n

r

3

L akdbk k=l

n

= 0 ==> L

..1(ak) (T(id A, d)

0

..1(bk))

= 0,

k=l n

==> L ..1(ak) (T(d, id A) 0 ..1(bk )) = 0, k=l

or

386

8. Quantum Groups

respectively. In these two cases one then defines the K-linear mappings:

n

-----. L ~(ak) (T(d, id A) ~R

0

~(bk» E

r @A.

k=1

Here one denotes, inserting the comultiplication images of a, b E A, k

~(a) (T(id A, d)

0

~(b» :=

I

L L( _l)'rklllakb; @ al:db;', k=II=1

denoting by Xk and Yl the degrees of al: and b;, respectively, and similarly from the right. In these two cases of left or right covariance one thereby finds the commuting diagrams: ~L

0

d = T(id A, d)

T(~, id

T(id

r)

0

~L =

r,~)

0

~R

o~,

or ~R 0 d = T(d, id A) o~;

T(id A, ~L)

0

= T(~R, id A) 0

~L,

or

~R.

(8.24.2.1) One thereby establishes some A-left or right comodule over K. If both ~L and ~R are well-defined, then, modulo ordering of tensor products as just above, these comodule-comultiplications moreover obey the ~commuting diagram:

in this case the first order graded differential calculus is called bicovariant.

(8.24.2.2) The above defined calculus {ker

fl,o} is bicovariant.

(8.24.3) Let 8 E M at(p, K)®Mat(p, K) be a solution of both the graded braid relation and the graded Hecke condition, Le., 812823812

= 823812823,

8

2

= Ip @Ip + (q -

q-l)8, q =I- 0, q2 =I- -1,

with the grading format according to p = m + n. For instance one may insert 8 := PR p , with the flip P, and the usual defining representation of the universal R-matrix of Uq(Ap-d, p 2:: 2, for qED, q2 =I- -l. Then the quantum superspace of even quantum coordinates Xll . .. ,X m , and odd x m + 1, ... , X m + n , m + n = p, is extended to the following graded differential calculus over itself.

q;

Covariant Differential Calculus on Quantum Superspaces

387

(8.24.3.1) Let ns(Xs) be the unital associative superalgebra over K, the even generators Xl,· .. , Xm, Ym+l,···, Yp, and odd Xm+l,···, xp, YI,···, Ym of which are subject to the homogeneous relations

S.A. ([x] @.A. [xJ) = q ([x] @.A. [xJ) , S.A. ([y] @.A. [xJ) = q-I[X] @.A. [y], S.A. ([y] @.A. [yJ) = _q-l[y] @.A. [y], S.A. := e.A. @ S, with the convenient notation of skew-symmetric tensor products over A, suppressing an injective homomorphism of unital associative K-superalgebras into the pool of copies A. With the odd derivation d, such that vf)

Vi k : Xk

d

---+

Yk

d

---+

0,

one thereby establishes the Z2-graded differential algebra ns (Xs) over K, using the graded Heeke condition on S in order to factorize d.

(8.24.3.2) For the quantum plane one easily calculates the following relations. Let q E C\{O}, q2 f. -l.

S,~ [~

!._:._, ~];

.x,x, ~ x,x,;

Xkdxk = q2(dxk)Xk' (dXk)2 = 0, k = 1,2; dXldx2 = -qdX2dxI; x l dx2 = q(dX2)XI' x2dxI

+ q-Ixldx2 =

q(dXI)X2

+ q2(dx2)XI.

(8.24.4) Under the same conditions on S as above, the matrix quantum super-semigroup As of generators akl, k, l = 1, ... ,p, which obey the graded main commutation relations (MeR) due to the solution PS of the graded QYBE, with the graded flip P, is extended to the following Z2-graded differential K-algebra ns(As) , again using the graded Hecke condition.

S.A. ([a] @.A. [aJ) = ([a] @.A. [aJ) S.A., [a] @.A. [da] = S.A. ([da] @.A. [aJ) S.A.,

[da] @.A. [da] = -S.A. ([da] @.A. [daJ) S.A..

388

8. Quantum Groups

(8.24.4.1) The comultiplication ..1 of the Z2-graded K-bialgebra As, such that Vik, l: n

akl -Li- "LJ akj ® ajl, akl -E- Okl E K, j=l is then extended uniquely to a homomorphism of Z2-graded differential algebras: ii(As) - - ns(As)@ns(As), from the corresponding unital universal differential envelope into the involved skew-symmetric tensor product of Z2graded differential algebras over K, and then factorized by the above relations of ns(As). Hence one obtains some homomorphism of Z2-graded differential K-algebras

..1. : ns(As) - - ns(As)@ns(As). (8.24.4.2) Constructing also c. : ns(As) - - K as an appropriately factorized extension of the counit c, such that c. 0 d = 0, one obtains the Z2-graded bialgebra ns(As) over K. (8.24.4.3) The restriction of ..1. to the As-As-bimodule AsdAs over K yields some bicovariant first order graded differential calculus over As, such that Va,b E As: ..1.(adb)

(8.24.4.4) Let again q

= (..1 L + ..1 R )(adb).

:f:

0, q2

:f:

-1. The relations of the bialgebra

M at q (2, K) are consistently supplemented by the following ones.. alldall = q2(dall)al1' allda 12 = q( da 12)al1'

allda21 = q(da 21 )all' allda 22 = (da22 )all'

+ (q - q-l)allda 12 , a12da12 = q2(da 12 )a12' a12 da 21 = (da2da12 + (q - q-l)al1 da 22, a12da22 = q(da22)a12' a21dall = q(dall)a21 + (q - q-l)allda 21 , a21da21 = q2(da 21 )a21' a21 da 12 = (da 12 )a21 + (q - q-l)al1 da22, a21da22 = q(da 22 )a21 , a22da l1 = (da ll )a22 + (q - q-l)(a12da21 + (da 12 )a21), a12dall = q(da ll )a12

a22da22

= q2(da22)a22, + (q q(da2da22 + (q -

a22da 12 = q(da 12 )a22

q-l)a12da22,

a22da21 =

q-l)a2 1da22.

(da l1 )2 = (da 12)2 = (da 2d 2 = (da 22 )2 = 0, (da l1 )(da 22 ) = -(da 22 )(da ll ), (da ll )(da 12 ) = -q(da 12 )(da ll ), (da ll)(da 21 ) = -q(da 21 )(da l1 ), (da12)(da2d + (da21)(da12) = (q - q-l)(da l1)(da 22 ), (da 12)( da 22) = -q(da22)(da 12 ), (da21)(da 22 ) = -q(da22)(da21)'

FUndamental Representations

389

For q = lone obtains the anticommuting differentials of four pairwise commuting even generators. (8.24.5) The transformation of quantum superspaces by matrix quantum super-semigroups is compatible with the above odd derivations in the sense of the following left covariance. The homomorphism of unital associative Ksuperalgebras: p

Xs 3

Xk --+ Lakl

~XI

E As®Xs,

1=1

which is extended to a unique homomorphism of Z2-graded differential algebras, on the non-unital universal differential envelope:

can be factorized by the relations of Ds(Xs), in order to obtain a unique homomorphism of Z2-graded differential K-algebras, which might be called induced transformation of differential forms on the quantum superspace Xs: Ds(Xs) 3

dXk p --+ L

(da kl ~XI + (-l)k+i akl ~dXI)

E Ds(As)®Ds(Xs).

1=1

Here

k + i describes the Z2-degree of akl, defining

k := 0 for 1 ~ k ~ m, k:= 1 for m + 1 ~ k ~ m + n =

p.

This left covariance, which is due to compatibility of the relations of Ds(Xs) and Ds(As), can be short-written, using matrices over a common pool superalgebra A:

[a] ~ [x] ~[da] ~ [x] + [a] ~ [dx].

8.25 Fundamental Representations of q-Deformed Universal Enveloping Algebras (8.25.1) With the conventions: Am

= sl(m + 1, C),

B m := {Y E gl(2m + 1, C); KmY + y t K m = O}, em := {Y E gl(2m, C); JmY + y t J m = O}, Dm := {Y E gl(2m,C);L mY + ytL m = O},

390

8. Quantum Groups

000 000

Sn :=

001] 010

for n EN,

...

[ 010 100

000 000

inserting above the transposed matrices yt of y, one finds the following defining representations of these complex Lie algebras £. (8.25.1.1) For Am, mEN, Vfk, vr+1i,j : (X:)ij = Oik Oj,k+1, (Hk)ij = OikOjk - Oi,k+1 0j,k+1' (8.25.1.2) For B m , m ~ 2, vr- 1k , v~m+li,j : (X:) ij = Oik Oj,k+1 - Oi,2m+1-k Oj,2m+2-k, (Hk)ij = Oij (Oik - Oi,k+1 (X;t;)ij =

+ Oi,2m+1-k -

V2 (Oim Oj,m+1 -

Oi,2m+2-k) ,

Oi,m+1 0j,m+2), (Hm)ij = 20ij (Oim - Oi,m+2)'

(8.25.1.3) For Cm, m ~ 3, vr-1k, v~mi,j: (Xin ij = OikOj,k+l - Oi,2m-k Oj,2m+l-k,

+ Oi,2m-k - Oi,2m+l-k), (Hm)ij = Oij (Oim - Oi,m+l).

(Hk)ij = Oij (Oik - Oi,HI (x;t;)ij

= Oim Oj,m+1J

(8.25.1.4) For D m , m of Cm' ~mi,j:

~

4, 1

~

k

~

m -1, the representation looks like that

(x;t;)ij = Oi,m-IOj,m+1 - Oim Oj,m+2, (Hm)ij = Oij (Oi,m-l

+ Oim -

Oi,m+l - Oi,m+2)'

In all four cases, X: and Xi: are represented by mutually transposed real matrices, for k = 1, ... , m. (8.25.2) Choose qED such that all the coefficients occurring in the relations are non-zero complex numbers. The fundamental representations of the corresponding q-deformed universal enveloping algebras yield the same complex matrices as above for q = 1, except in the following case. For B m , m ~ 2,

v~m+1i,j: (X;t;)ij

= (X~)ji = ([2]q)1/2 (OimOj,m+l

- Oi,m+IOj,m+2),

all the other generators being represented by the same real matrices as for q

= 1.

Fundamental Representations

391

(8.25.3) For qED, let

R(q) E Mat(n, C)

~

Mat(n, C), n

= m + 1, 2m, 2m + 1,

be the corresponding representation of the universal R-matrix of Uq (£). Then R(q) is some complex matrix solution of QYBE. (8.25.3.1) For Am, mEN,

L

q~ R(q) = q

E kk ~ E kk + l$k~l$m+l

l$k$m+l

+ (q -

L

q-l)

E kl ~ Elk,

l$k
(8.25.3.2) For Cm, m

R(q) = q

~

3, and D m , m

L E kk ~ E kk + L E

l$k$2m

+ q-l

~

4,

L

1$k~l$2m,k+I~2m+l

E kk

2m + 1- k ,2m+l-k ~

L

+(q_q-l)(

E kk ~ Ell

Ekl~Elk

1$k
± 1$k$m,m+l$1$2m

L

qk-l E kl

~ E 2m +1- k'2m+l-I) ,

1$k
with the upper signs for Cm, and lower ones for D m .

R(q)

L

= q2

E kk ~ E kk

+ Em+l,m+l

~ em+l,m+l

1$k$2m+l,k~m+l

+

L L

E kk ~Ell

1$k~I$2m+l,k+I~2m+2

+ q-2

1$k$2m+l,k~m+l

E 2m +2- k ,2m+2-k

~

E kk

392

8. Quantum Groups

+ (q2

L

_ q-2) (

L L

E kl ® Elk

1::;k<19m+l

q2(k-l) E kl ® E 2m +2- k,2m+2-1

1::;k
® E 2m +2- k,2m+2-1

1::;k::;m,m+2::;1::;2m+l

- L

q2(k-m)-1 Ek,m+l ® E 2m +2- k,m+l

l::;k::;m

_ L

q2(m-I)+3 E m+1 ,1

® E m+l,2m+2-1).

m+2::;1::;2m+l

Occurrence of coefficients q2 instead of q in the last expression is due to the choice of dk = 2 for 1 :::; k :::; m - 1, dm = 1. (8.25.4) Let qED as above. Denoting by P the flip on CP®CP, p:= m+1, 2m, 2m + 1, the complex matrix R( q) := PR( q) fulfills the braid relation R 12(q)R23(q)R 12(q) = R23(q)R12(q)R23(q), with the usual notation of an embedding into the three-fold tensor product. Denote I p ® I p =: I. (8.25.4.1) For the series Am,m (S(q»2 = 1+ (q -

q-l)

E

N, the Heeke condition

S(q), S(q) := qrnhPR(q),

holds. S(q) is diagonalizable, if and only if q2 (8.25.4.2) For B m , m

~

1= -1.

2, S(q) := PR(q) fulfills the condition

(S(q) - q21) (S(q) + q-21) (S(q) - q-4m I) = O. S(q) is diagonalizable, if and only if (1 + q4) (1 + q2-4m) (1 - q-4m-2) (8.25.4.3) For Cm,m (S(q) - qI) (S(q)

~

3, S(q):= PR(q) fulfills the equation

+ q-l 1) (S(q) + q-2m-l 1)

=

o.

S(q) is diagonalizable, if and only if (1 + q2) (1 + q-2m-2) (1 - q-2m) (8.25.4.4) For D m , m (S(q) - qI) (S(q)

~

1= O.

4, S(q)

:=

+ q-l 1) (S(q)

-

1= O.

PR(q) fulfills the condition ql-2m I) =

O.

S(q) is diagonalizable, if and only if (1 + q2) (1 + q2-2m) (1 - q-2m)

1= O.

Generic Representations

393

8.26 Generic Representations of q-Deformed Universal Enveloping Algebras Let £ be a finite-dimensional complex siPlple Lie algebra, and assume the deformation parameter q E C\ (] - 00,0] U {I}) to be generic, Le. not any root of 1. Then quite analogously to the non-deformed case of q = 1, the following statements hold, for a representation of the unital associative algebra E q (£) of polynomials in the generators Ek, Fk, Kk, and Lk = Ki: 1 , k = 1, ... , m, on a finite-dimensional complex vector space V. Of course V may be viewed as an Eq(£)-left module over C, with the module-multiplication such that "IX E Eq(£),v E V: Xv:= p(X)v.

(8.26.1) The endomorphisms p(Ek) and P(Fk)' k = 1, ... , m, are nilpotent. The representatives p(Kk), k = 1, ... ,m, can be diagonalized simultaneously. (8.26.2) A homomorphism of unital associative algebras A : Tq called weight of p with the weight space V>., if and only if V>. := {v E V; 'Vik : p(Kk)V = A(Kk)V}

=1=

---+

C is

{O}.

Here Tq denotes the complex subalgebra of the generators Kk and Ki: 1 , k = 1, ... , m, of Eq (£). Any two different weight spaces have only the zero vector in common.

(8.26.2.1) If p is irreducible, then V is the direct sum of all the weight spaces. Of course, inserting the adjoint representation of a simple Lie algebra £, one finds the root space decomposition of £ itself. (8.26.3) A non-zero vector Vmax E V is called highest weight vector with the highest weight Amax , if and only if Amax is some weight of p such that

'Vik : p(Kk)Vmax

=

Amax(Kk)Vmax, p(Ek)Vmax = O.

There is always at least one highest weight vector.

(8.26.4) The Eq(£)-left module V over C, or equivalently the representation p, is called cycliC, if and only if V = {p(X)V max ;X E Eq(.c)} for some highest weight vector Vmax with the highest weight Amax . In this case

V = C - lin span( {vmax}U {p(Fil ... Fjp);jl, ... ,jp E {I, ... , m},p EN}), and the corresponding weights are easily calculated from the q-deformed commutation relations, such that Vfk :

394

8. Quantum Groups

Every weight of p is of this type. Moreover dim V.x maz

= 1.

(8.26.5) If p is irreducible, then there exists only one highest weight, and therefore only one complex ray of highest weight vectors. (8.26.6) If the irreducible representations p and a of E q (£), on V and W respectively, are of the same highest weight Amax , then they are equivalent in the usual sense of a C-linear bijection 'T : V +-----+ W, such that

"IX E E q (£) : a(X) =

'T 0

p(X)

1 0 'T- .

(8.26.7) The irreducible representations of Eq(AI) are classified by their dimension 2j + 1, j = 1, ~, ... , with the highest weights Amax (K) = ±q2j , and the corresponding weights ±q2m, m = j, j - 1, ... , - j.

!'

(8.26.8) The highest weight of any finite-dimensional irreducible representation of E q (£) is of the type Amax(Kk)=Skq~k, nkEN,

skE{+I,-I}, k=I, ... ,m.

(8.26.9) The notion of Verma modules is generalized in order to study representations of E q (£). For A := {Ak E C\{O}; k = 1, ... , m}, the Verma module M.x is defined as the factor algebra

(8.26.9.1) Cyclic highest weight representations are called Verma modules too. Such a representation p of E q (£) on V is reducible, if and only if

:J 0 =I Vsing E V : Vsing =I Vmax , V{'k : p(Ek)Vsing

= 0,

p(Kk)Vsing

= J.LkVsing,

with non-zero complex eigenvalues J.Lk; then Vsing is called singular vector of p. In this case the representation p of E q (£) can be restricted to a cyclic highest weight representation on the linear subspace {P(X)Vsing; X E E q (£)} of V, with the highest weight vector Vsing, and the highest weight J.Lmax such that

(8.26.10) Every finite-dimensional representation of E q (£) is completely reducible.

Generic Representations

395

(8.26.11) The so-called tensor product of two representations Q and {3, on finite-dimensional vector spaces V and W over a field K respectively, of an associative unital superalgebra, which moreover is a Z2-graded bialgebra 1i over K, is defined such that:

1i

3 x --+ --+ +--+ T~(Q,{3)(x) E EndK(V ~ W). ~ T(o:,13)

(8.26.12) For instance in the case of Eq(Ad for positive q =I- 1, with the notation == Ij, m) of normed eigenstates, such that

vtn

Klj, m) = q2mJj,m), m = j,j - 1, ... , -j, X±lj, ±j) = 0, X±lj, m)

= ([j

=f m]q[j ± m

+ l]q)I/2 Ij , m ± 1)

for m =I- ±j,

these irreducible representations, j = !' 1,~, ... , are combined by means of the following q-deformed Clebsch-Gordan coefficients. (8.26.12.1) In the lowest case of jl

= j2 = !' one easily calculates that

[~]

1 1 1 1 1 1 '2' 2' 1, 1) = 12 ,2) ~ 12 ,2) +--+ ~

,

I~ _~) 1/21~2' -~) I~ ~)) I ~2' ~2' 1, 0) = ([2] q )-1/2 ( q -1/21~~) 2' 2 ~ 2' 2 + q 2 ~ 2' 2 +--+

1 1 12,2' 1, -1)

([2]q)- 1/2

[q-~/2] q~2'

1 1 1 1 ,-2) = 12 ,-2) ~ 12

I ~2' ~2' 0, 0) = ([2] +--+

q

)-1/2 (

q

+--+

[~]~

,

1/21~~) I~ _~) _ q -1/21~2' -~) I~ ~)) 2' 2 ~ 2' 2 2 ~ 2' 2

q~2

] ([2]q)- 1/2 [ _q~I/2'

inserting the Pauli spinors

[~] , [~]

for I!, ±!). Hence one finds:

396

8. Quantum Groups

X

+ ----+

0 q-l/2 ql/2 0 0 0 0 0 [0 000

q-l/2

o ]

1 2

q~

+----+

tran8po8ition

+--

_ X .

Obviously these eigenvectors are orthonormal, because q is positive. (8.26.12.2) One obtains the following q-deformed Van der Waerden formula.

j]

=

m

h

jl [ ml

I' . . m ),

q )1,)2,),

j] m q

m2

+ h - j]q![h + j - h]q![j + jl - h]q!) 1/2 [jl + j2 + j + l]q! q!Ul+h-j)(it+h+j+l)+jlm2-hml ([2j + l]q)I/2

._ 0 ([jl .- ml+m2,m

([jl + ml]q![jl - ml]q![j2 + m2]q![h - m2]q![j + m]q![j - m]q!)1/2 ~

(_1)8 q -8(jl+h+J+l)

L..J [S]q![jl + j2 - j - S]q! 8~O 1

[jl - ml - S]q![j2

+ m2 - S]q![j - h + ml + S]q![j -

jl - m2

+ s]q!'

denoting [O]q! := 1, and restricting the sum such that jl + h

- j-

jl - ml -

S ~

S ~

0,

0, h

+ m2 -

S ~

0, j - h

+ ml + S

~

O.

(8.26.12.3) Since q is positive, these eigenvectors are orthonormal, and the q-deformed Clebsch-Gordan coefficients are real. Hence one obtains, for Ijl - hi 5: j 5: jl + j2 and m = -j, ... ,j, that it

Ijl,i2,j,m)

=

L

Cyclic Representations

397

(8.26.12.4) Due to the transformation of orthonormal bases one enjoys the following relations. j1

h

l:

l:

m2=-)2

ml=-31

l:

[ )1 ml

12 m2

t. [;:1

)] [)1 12

m2

13I-hl~j~3t+h m=-)

)2 m2

m q m1

.,

~,

]= q

Ojj,Omm"

)] [)1 12 1 ~L

m q m'

m~

= Oml m; Om2m;'

(8.26.12.5) Both symmetry relations and recursion formulae for ClebschGordan coefficients can be generalized to these q-deformed coefficients. For instance one finds the following ones.

((j =f m]q[j ± m

+ 1]q)I/2

[;:1

:2

m

~ 1] q

= qm 2 ((jl ± m1]q[j1 =f m1 + 1]q)1/2 [ )1 1 12 m1 =f

+ q-m

1

((j2 ± m2]q[j2 =f m2

+ 1]q)1/2

[ )1 ml

m2

)2 m2=f1

8.27 Cyclic Representations of q-Deformed Universal Enveloping Algebras In the so-called non-generic case of the deformation parameter q being a root of unity, the finite-dimensional representations of q-deformed universal enveloping algebras differ essentially from those of the non-deformed ones. Let £. be a simple finite-dimensional complex Lie algebra with the Cartan matrix Fkl , 1 ~ k, l ~ m, and dk such that dkFkl = dlFlk, and denote qk := qd k • Let

q:= exp(i7rnjN) , n E N, N

~

3, 'V'{'k: N > dk,

the latter assumption concerning the exceptional case of G2 with F 12 = -3 and d 2 = 3. Here the smallest common divisor of n and N is assumed to be equal to 1. Then qN = ±1, and therefore [N]q = O. (8.27.1) In the non-generic case, the centre of the complex Hopf algebra E q (£.), with the generators Ek, F k , Kf1, k = 1, ... ,m, contains certain powers of these generators.

398

8. Quantum Groups

(8.27.1.1) By induction one calculates the following commutation relations. For 1 ~ k, l ~ m, \In E N: K I-n K- 1 n-l D1l] [ Ek,rk

[]

kqk

D1l-1

=qknqkrk

- k qk -1' qk - qk

En p ] _ [1 Kkq~-n - K k q'k- E n- 1 [ k, k - qk n qk 1 k' qk - qk nF1 - q-nrkl F1 K n nEl - k qnrkl E lk' K n Kk Kk l-k lk' kl l:"nK EnK K pn _ q-nr K k E In -- qnrkl k I k, k I - k rl k· 1

1

(8.27.1.2) Moreover the Serre relations imply that Vf'k, l:

- k q-rklNElk' EN p kN F1l - k qrklN F1pN E kNEl lk' (8.27.1.3) Hence one finds in the case of even n and odd N, such that qN = 1, that Ef, FI:, Kt N , k = 1, ... , m, belong to the centre of Eq(£). In case of odd n, such that qN = -1, E~N, FfN, Kt 2N , k = 1, ... , m, are contained in this centre. Moreover in the case of B m , m::::: 2, qf rkl = 1 for 1 ~ k, l ~ m, such that Ef, FI:, K;N, k = 1, ... ,m, are central for Eq(Bm ), \In E N. (8.27.1.4) Due to the construction of q-deformed positive and negative root vectors by means of Lusztig automorphisms one finds, that the corresponding powers of E({3j), F({3j), j = 1, ... , p, along the longest word ak 1 q ... 0 ak p of the Weyl group, are also central for E q (£). (8.27.2) The ideal of E q (£), which is generated as

Zq(£):= ideal ({E({3j)N, F({3j)N, K I±2N -I; l = 1,oo.,mj j=1,oo.,p}), with the unit I of E q (£), fulfills also the coideal conditions

..1(Zq(£))

~

Eq(£) ® Zq(£)

+ Zq(£) ® Eq(£), c(Zq(£))

= {O}.

Moreover

a(Zq(£))

~

Zq(£),

such that one obtains the finite-dimensional complex Hopf algebra

E;ed(£) := Eq(£)/Zq(£), with the C-linear basis

{E({3pt p ... E({3d r1 F({3p)Sp ... F({3d s1 Kt 1 T1,"" T1

sp

E

Noj Zl,""

Zm

E

•••

K:n"';

Zj

< N,oo.,sp < Njlz11 < 2N,oo.,lzml < 2N}.

Cyclic Representations

399

(8.27.2.1) Of course this ideal is also generated by the set

{Ef, F't, K I±2N

-

ljl = 1, ... ,m}.

(8.27.3) Also in case of odd n, the Nth powers of non-toral generators are factorized to zero, in order to construct a universal R-matrix for u;ed(£). On the other hand, the Nth powers of toral generators are not factorized to unit, in order to include for instance boson and fermion representations. (8.27.3.1) Using the homomorphism of unital associative C[[h]]-algebras: E q (£) ---+ Uq (£), which describes the change of generators from Ek, Fk, Kt 1 , k = 1, ... , m, to Hk, one can also factorize the topological Hopf algebra Uq(£) with respect to the smallest closed ideal Zq(£) of the above relations, which is obtained as the closure of the image of Zq(£). Since the latter fulfills the conditions

xt,

e(Zq(£)) = {O}, I1(Zq(£)) ~ Zq(£),

.1 (Zq(£)) ~ Uq(£) 0 Zq(£)

+ Zq(£) 0 Uq(£),

one thereby constructs the topological Hopf algebra

u;ed(£)

:=

Uq(£)jZq(£).

Here one uses the closure with respect to the unique Hausdorff completion of the tensor product of formal power series with relations, with respect to the h-adic topology.

(8.27.3.2) The reduced quantum algebra u;ed(£) is quasitriangular with respect to some universal R-matrix Rred. The latter is obtained formally from the universal R-matrix of Uq (£) itself by omitting the powers;::: N of q-deformed positive and negative root vectors. (8.27.4) The irreducible finite-dimensional representations of Eq(Ad, which are highest weight representations, are classified modulo equivalence in the following way. Denote by d the dimension, and by >. E C the highest weight of the representation, the latter not being written explicitly as homomorphism for convenience.

K'l/Jl

= q>"-21'l/J1,

X+'l/Jl = ([l]q[>' X-'l/Jl

+ 1 _l]q)1/2 'l/Jl-l,

= ([l + l]q[>' _l]q)1/2 'l/Jl+l, l = 0,1, ... , d -

l.

Here X± are inserted as generators of Eq(Ad instead of E and F, for convenience. Obviously X+'l/Jo = OJ moreover d ~ N. One also demands, that

400

8. Quantum Groups

(8.27.4.1) For d < N, demanding [A - d + l]q = 0 means, that N

A = d - 1 + Z - , zEZ. n (8.27.4.2) For d that A

= N,

demanding [A -l]q

:f. 0 for l = 0, ... , N

- 2, means

~ {l + Z ~ ; Z E Zj l = 0, ... , N- 2}.

(8.27.4.3) Only for d < N one obtains K2N = [d. Of course one always finds, that (X±)d = O. Therefore in case of d < N the above representation can also be used for E~ed(Ad. (8.27.4.4) The Casimir operator

takes the eigenvalue ([!(A + 1)]q)2 - ~. Note that this eigenvalue does not suffice to determine one of the above representations. The dimension d depends not only on the highest weight A, but also on the choice of Z E Z, for d
(8.27.4.5) Including the trivial representation with d

= 1,

N A = Z-, n

Z

E Z, X±

= 0,

one finds that every complex number A can be used as highest weight.

(8.27.5) The non-highest weight representation of Eq(Ad for q4 = 1, such that:

X

+

---+

1 1 _ /20' ,X

---+

-i 2 /20' ,K

---+

q

(13

•

,q = ±l,

inserting the Pauli matrices, is cyclic in the sense that

Cyclic Representations

401

(8.27.6) Choose complex parameters A, a, {3, such that a{3 =I- O. The following irreducible N-dimensional representation of Eq(Ad is cyclic in the sense that (X±)N = C±IN, with complex numbers c± depending on N and the parameters. KtPI = q>'-2I tP1 , 0 ~ l ~ N - 1; X+tPI = ([l]q[A

1 ~ l ~ N - 1;

X-tPI =

0 ~ l ~ N - 2;

X+tPo

+ 1 -l]q + a{3)1/2 tP1 _ lt ([l + l]q[A -l]q + a{3)1/2 tP1 +l'

= {3tPN-l,

X-tPN-l

= atPo·

Obviously K N = qN>'IN . Note that here the non-toral generators are also represented by invertible complex matrices.

(8.27.7) The centre of Eq(Ad consists of all the complex polynomials in C,EM,FM,K M , M:= N for even n, M:= 2N for odd n. (8.27.8) Finite-dimensional representations of E q (£) can be constructed, which are reducible, but indecomposable in the sense, that they cannot be written as direct sum of irreducible representations. Consider for instance the following indecomposable representation of Eq(A 2). Let J1. + A = -1. X1tPkl = tPk+l,l, 0 ~ k ~ N - 2, 0 ~ l ~ N - 1; XttPkl = [k]q[A - k + l

+ l]qtPk-l,l, 1 ~ k ~ N [l + l]q[J1. -l + k]q'¢k,l+lt 0 ~ k ~ N -

1, 0 ~ l ~ N - 1;

XitPkl =

1, 0 ~ l ~ N - 2;

XttPkl = tPk,l-l, 0 ~ k ~ N - 1, 1 ~ l ~ N - 1; X1tPN-l,1 = XttPOI = XitPk,N-l = X:}tPkO = 0, 0 ~ k,l ~ N -1; K1tPkl = q>'-2k+l tPkl , K 2tPkl = ql-'-21+k tPkl , 0 ~ k,l ~ N-1.

Obviously this N-dimensional representation is irreducible for complex numbers J1. such that [J1. - l + k]q =I- 0 for 0 ~ k, l ~ N - 1.

(8.27.8.1) Choose for instance n = 2, N = 3, A = 1. Then the above representation is reducible, but indecomposable, because there is an invariant subspace spanned by the vectors tPkl, 0 ~ k,l ~ 2, {k,l} =I- {0,2}. For A = 2 a similar situation occurs, excluding the indices {O, I}, {O, 2}, {I, 2}. (8.27.8.2) For n = 2, N = 5, and A = 2, the corresponding 25-dimensional representation is indecomposable, with the invariant subspace, which is spanned excluding the indices {O, 3}, {O, 4}, {I, 4}.

402

8. Quantum Groups

°

(8.27.8.3) Consider again the case of n = 2, N = 3, A = 1. Since xt1P20 = for k = 1,2, one can factorize the above 8-dimensional reducible representation of E q (A 2 ), in order to obtain an irreducible 7-dimensional representation on the vector space C( {lPk/; k, l = 0,1,2; Ik - II ~ 1}). The above 6-dimensional representation for A = 2 is irreducible.

(8.27.9) Consider a representation of Eq(AI) on Cd, which is spanned by the eigenvectors of K, such that

K'l/J/=Ko/'l/J/, l=O, ... ,d-1. The quantum dimension of this representation is defined by dq := K.o

+ '" + Kod-l·

For the irreducible highest weight representations of dimension d < None finds, that d-l

" >'-2/ = [A+ 1]q = [d+z-]q, N dq = 'L...Jq Z E Z. /=0

n

For d = N, the irreducible representations are characterized by d q = 0. Representations with the quantum dimension dq = 0 are called such of type I, whereas those with non-zero quantum dimension are representations of type II. (8.27.10) Let x, y be elements of an associative algebra over C, with the commutation relation yx = q2 xy , q:l ±1.

Then qN = ± 1 implies, that (x+y)N =x N +yN.

Now consider the matrix quantum semigroup Matq(m, 0, C), and choose the parameters qk/ := q2, 1 ~ k < l ~ m, and assume q4 :I ±1. Then the Nth powers of generators a{:f, 1 ~ k < l ~ m, are central, and moreover

Cyclic Representations

403

(8.27.10.1) With this choice of parameters qkl, 1 :::; k < l :::; m, one can define the so-called quantum determinant, which is both central and grouplike, by

detq[akl; k, l = 1, ... ,m]

=d:= L

(_q-2)'~P aliI' .. ami",;

PEP",

Vik, l : akld = daklj L1(d) = d ® d, e(d) = Ij here Ap denotes the length of the permutation P :=

[1

]1

~] . One

Jm

then calculates, that

d N = det[afzjk,l = 1, ... ,m]:=

L

(-I) AP af;1 .. . a::'i",·

PEP",

(8.27.10.2) For m = 2 one finds the following N-dimensional representation of Mat q(2, 0, C) : all'l/Jk

= 0'(1- q-4k)1/2'l/Jk_1' 1:::; k:::; N -1, all'l/Jo = OJ

a22'l/Jk = 8(1 - q-4k-4)1/2'l/Jk+1, 0:::; k :::; N - 2, a22'l/JN-1 = OJ a12'l/Jk

0'8 =

= {3q-2k-d'l/Jk, -{3"1 ::I 0, s + t

a21 'l/Jk

= "Iq-2k-t'l/Jk,

= 2j 0', ... , t E C.

0 :::; k :::; N - Ij

9. Categorial Viewpoint

Investigations of structures, which later on, following V. G. Drinfel'd (1987), were called quantum groups, started from at least four different points of view, as was pointed out in the foregoing chapter, due to the pioneering work by M. Jimbo (1985), S. L. Woronowicz (1987), Yu. I. Manin (1989), N. Yu. Reshetikhin, L. A. Takhtadzhyan, L. D. Faddeev (1990), J. Wess and B. Zumino (1990), and in particular the contribution by V. G. Drinfel'd to the Proceedings of the International Congress of Mathematicians at Berkeley, California, 1987. Therefore the question, what maybe new kinds of categories could be used in order to classify the amazing variety of new mathematical tools, is by far non-trivial. On the purely algebraic level, forgetting about topological and differential aspects, one is concerned with some generalization of rigid tensor categories. Monoidal categories and their coherence properties were discussed for instance by S. Mac Lane (1971). In particular Tannakian categories as a special kind of tensor categories were investigated extensively by N. R. Saavedra in his thesis (1972), and by P. Deligne, J. S. Milne (1982). Their use for the categorial understanding of quantum groups is describ,ed explicitly by A. Joyal and R. Street (1991). The symmetry constraint, which as some functorial isomorphism defines a tensor category as special kind of a monoidal category, is weakened to a socalled quasisymmetry constraint, thereby establishing quasitensor (braided tensor) categories. The notion of rigidness in terms of the internal H omfunctor, and the so-called evaluation map, are generalized straightforward from tensor categories to the braided case. Of course the involved categories should be Abelian, and the quasitensor bifunctor is then assumed to be biadditive. Finite-dimensional representations of quasitriangular Hopf algebras were identified as the objects of rigid quasitensor categories by S. Majid (1990), who in the sequel used such categories for the study of so-called braided groups and braided matrices (1993). Abelian tensor categories and their generalization to quasisymmetry constraints are described in the Montreal lectures (1988) and an introductory overview by Yu. I. Manin (1991). An advantage of quasitensor categories is their categorial view of quasitriangular quasi-Hopf algebras; the latter were

406

9. Categorial Viewpoint

introduced by V. G. Drinfel'd (1990), aiming, among other applications as for instance invariants of knots, at the construction of gauge transformations in models of conformal field theory. In particular the system of equations by V. G. Knizhnik and A. B. Zamolodchikov (1984), which is satisfied by the correlation functions in the Wess-Zumino-Witten model, provides an example of associativity constraint, which is due to some quasi-coassociative comultiplication. A rather restrictive notion of quantum category is due to J. Frohlich and T. Kerler (1993), namely as an Abelian semisimple finite rigid quasitensor category. Topological properties are moreover proposed for the morphisms of so-called C· -quantum categories. The latter are used as an abstract framework for the study of quantum group symmetries of local quantum field theories in 2 and 3 space-time dimensions. Topological aspects are also included in the definition of monoidal W·categories by M. S. Dijkhuizen in his thesis (1994), in order to classify compact quantum groups, and to establish the Tannaka-Krein duality. The categorial viewpoint is presented in detail in the monographs by St. Shnider and S. Sternberg (1993), V. Chari and A. Pressley (1994).

9.1 Categories and Functors (9.1.1) A category Cat is defined by the following conditions. (i) To every pair of objects A, B from the object class Ob(Cat), a set Mor(Cat; A, B) := Mor(A, B) of morphisms from A to B is assigned, such that VA, B, C, DE Ob(Cat) : A"I C or B"I D

===}

Mor(A,B) n Mor(C,D) =

(ii) To every triple of objects A, B, C morphisrns is assigned:

E

0.

Ob(Cat), an associative product of

Mor(B,C) x Mor(A,B) 3 {,B,a} ---+,Ba E Mor(A, C),

such that Va E Mor(A,B),,B E Mor(B,C),-y E Mor(C,

(iii) VA,B

D) : -y(,Ba)

= b,B)a.

E Ob(Cat): 3eA E Mor(A,A),eB E Mor(B,B) : Va E Mor(A,B) : aeA = eBa = a.

This so-called identical morphism eA of A E Ob(Cat) is then unique. Morphisrns a E Mor(A, B) are conveniently denoted by a : A ---+ B. The class Mor(Cat) of morphisms of the category Cat is defined as the class of objects a, such that 3A,B E Ob(Cat): a E Mor(CatjA,B).

9.1 Categories and Functors

407

(9.1.2) Let a E M or (A, B); a is called monomorphism, if and only if VC E Ob(Cat), V'Yl, 'Y2 E M or(C, A) : a'Yl

= a'Y2 =} 'Yl = 'Y2;

a is called epimorphism, if and only if

a is called bimorphism, if and only if it is both a monomorphism and an epimorphism. Moreover, define

isomorphism a {::::::} 3{3 E Mor(B,A) : {3a

= eA,

a{3

= eB;

then (3 is unique, hence denoted by a-I. Every isomorphism a is some bimorphism. A morphism a : A --+ A is called endomorphism of A. An isomorphism a : A --+ A is called automorphism of A. An isomorphism a E M or( A, B) is also denoted by a : A ~ B.

(9.1.2.1) Two morphisms of a category ¢k : Ak --+ Bk, k = 1,2, are called isomorphic, if and only if there exist isomorphisms a : Al --+ A 2, {3 : B 1 --+ B 2, such that (3¢1 = ¢2a. (9.1.3) The dual category Cat* with respect to a category Cat is defined, such that

(i) VA, BE Ob(Cat) = Ob(Cat*) : Mor(Cat; A, B) = Mor(Cat*; B, A);

(ii) with a{3 denoting the product of the category Cat, Mor(Cat*; B, C) x Mor(Cat*; A, B) 3 {{3, a}

--+

a{3 E Mor(Cat*; A, C).

(9.1.4) Let Catl, Cat2 be categories. A covariant (contravariant) functor from Catl into Cat2 is defined as a pair of maps

such that the following conditions are fulfilled.

==}

Mor(Catl;A1,Bd 3 al

--+

F mor

a2 E Mor(Cat2;A2,B2)

(Mor(Cat2; B 2, A 2».

(ii)

Al

--+ Fob

A2

==}

eAI --+ eA~' F rnor

408

9. Categorial Viewpoint

(iii) If Mar(Catlj All B 1 ) 3

01 02 F rnor

MOr(CatliBl,Cd 3 {31 -

Frnor

E Mar(Cat2; A 2, B 2),

(32 E Mar(Cat2;B2,C2)'

then

Obviously the composition of functors yields again some functor. For instance, contravar. functar

0

contravar. functar = covariant functar.

(9.1.5) A subcategory Cat' of Cat is defined by the following conditions. 'VA, BE Ob(Cat') ~ Ob(Cat) : Mar(Cat'; A, B) ~ Mar(Cat; A, B); 'VA E Ob(Cat'): cA E Mar(Cat';A,A); 'Va E Mar(Cat';A,B),{3 E Mar(Cat'jB,C): {30 E Mar(Cat'jA,C)j

here the product (30 is defined in the sense of Cat. The product of morphisms in the sense of Cat' is then defined as that of Cat, and cA E Mar(Cat'j A, A) is also used as the identical morphism of A in the sense of Cat'. The subcategory Cat' of Cat is called full, if and only if 'VA,B E Ob(Cat'): Mor(Cat';A,B)

= Mar(Cat;A,B).

(9.1.6) Let A, B E Ob(Cat). The covariant and the contravariant representation functor, Mar(Catj A, -) and Mar(Cat; -, B), both from Cat into the category Set, the latter with sets as objects and maps as morphisms, are defined: Ob(Cat)

3

Mar(Cat;A,X) or Mar(Cat;X,B)

X -

Mar(CatjX, Y)

3

eA or eB

e-

E

Mar(Set),

such that in the covariant case: Mar(Cat;A,X) 3

a-eo E Mar(CatjA,Y), eA

or in the contravariant case: Mar(Catj X, B) 3 {3e ~ (3 E Mar(Catj Y, B),

eB

respectively.

E

Ob(Set),

9.1 Categories and Functors

409

(9.1.7) The product Catt x Cat2 of two categories Catk,k = 1,2, is defined in the following way. Let ak E Mar(Catk; Ak, B k ), 13k E Mar(Catk; B k , Ck), k = 1,2. Then

Ob(Catl x Cat2)

3

{A I ,A2}

--+ {B I {Qt,Q2}

,B2} E Ob(Catl x Cat2)

and {{3I, {32} E Mar(Cat l x Cat2; {B I , B 2}, {CI, C2}) are composed according to {{3I,!32}{al,a2}:= {{3lal,{32 a 2}, and C{A t ,A 2}:= {cAllCA2}'

The product of finitely many categories is defined correspondingly. (9.1.8) A functor F : Catl x Cat2 --+ Cat, also denoted as bifunctor, is called (a) co-covariant (b) contra-contravariant (c) co-contravariant (d) contracovariant, if and only if the following conditions are fulfilled. Let Fob, Fmor act as:

{Al,A 2}

--+

{al,a2}

--+

A, {B I ,B2} --+ B, {Al,B 2}

--+

c, {B I ,A2} --+ D,

a, {{3l,{32} --+ (3, {al,{32} --+" {{3I,a2} with the morphisms ak, 13k, k = 1,2, denoted above. Then

--+

b,

a E Mar(Cat;A,B), {{3lal,{32a2} --+ (3a; a E Mar(Cat;B,A), {{3lal,{32a2} --+ a{3; (c) a E Mar(Cat;C,D), {{3lal,!32a2} --+ b,; (d) a E Mar(Cat; D, C), {{3lal,!32a2} --+ ,b. (a)

(b)

Moreover one demands, that Fmor

: C{At. A 2} --+ CA.

(9.1.8.1) The product F I x F2 of two functors Fk : Catk --+ Catk' k = 1,2, is defined as some bifunctor: Catl x Cat2 --+ Cdt~ x Cat~, just assigning pairs to pairs of objects and morphisms. If for instance F I is covariant and F 2 contravariant, then F I x F 2 is co-contravariant. (9.1.9) The representation bifunctor

Mar(Catj -, -) == MarCat : Cat x Cat

--+

Set

defined below is contra-covariant:

Ob(Cat x Cat)

3

{A, B}

--+

Let ak E Mar(Cat;Ak,Bk),k

Mar(Cat;B I ,A2) 3 (3

--+

Mor(Cat; A, B)

= 1,2; then to the a2{3al

is assigned; one finds: {cA,cB}

--+

E

E

Ob(Set).

pair {al,a2} the map:

Mar(Cat;Al,B 2)

id(Mar(Cat;A,B)).

410

9. Categorial Viewpoint

(9.1.10) Consider two functors Fk,k = 1,2, from Cat into Cat', which are (i) both covariant, or (ii) both contravariant. A functorial morphism from F1 to F2 is a family of morphisms:

Ob(Cat):1 A-HPA E Mor(Cat';A~,A~), A ~

such that the following diagram commutes. Let

A

--+ Fie ob

Ak, B

then (i) OZtPA

--+ Fie ob

Bic,

= tPBoi, or

0

--+ Ok' Fic mar

(ii) OZtPB

--+ Fk ob

0 E

Ak,k = 1,2,

Mor(Catj A, B),

k = 1, 2j

= tPAoi,

respectively.

(9.1.10.1) The identical functorial morphism I from F to F is defined such that I: A --+ CA', for Fob : A --+ A'. (9.1.10.2) Let Fk,k = 1,2,3, be functors from Cat into Cat', and consider two functorial morphisms cP from F1 to F2, lJt from F2 to F3. They are composed to a functorial morphism lJt 0 cP from F1 to F3 , such that:

Ob(Cat):1 A-+'ljJAtPA E MOT(Cat';A~,A~), A lJto~

--+ Fk ob

Ak,k = 1,2,3.

(9.1.10.3) Denote by Ik, k = 1,2, the identical functorial morphisms from Fk to Fk. Then cP 011 = 12 0 cP = cP for every functorial morphism cP from F1 to F 2 . (9.1.10.4) A functorial morphism cP : Ob(Cat) :1 A --+ tPA E .Mor(Cat') is called functorial isomorphism, if and only if all the images tP A are isomorphisms; in this case the two involved functors F1 and F2 are called isomorphic, and one can define the inverse functorial morphism cP- 1 , such that cP- 1 0 cP = 11, cP 0 cP- 1 = 12 , inserting the involved identical functorial morphisms. (9.1.11) In many categories a product, or a coproduct, can be established. (9.1.11.1) Let P E Ob(Cat), and

11'k E Mor(CatjP,Ak)

11' := {11'kj

k E I} a family of morphisms

= Mor(P,Ak)'

The pair

{P, 11'}

= IT Ak kEf

is called product of the family {A k ;k E I} in the sense of Cat, if and only if

'v'bk

E

Mor(C,Ak);k

The morphisms 11'k, k

E

E

I}: 3 unique 'Y E Mor(C,P): 'v'k E I : 'Yk = 11'k'Y.

I, are then called projections of Ponto A k .

9.1 Categories and Functors

411

(9.1.11.2) Let Q E Ob(Cat), and 13 := {13k; k E I} a family of morphisms

13k

E

Mor(CatjAk,Q) == Mor(Ak,Q).

The pair

is called coproduct of the family {Akj k E I} in the sense of Cat, if and only if

'V{-Yk

E

Mor(Ak,C)jk

E

I}::3 unique 'Y E Mor(Q,C): 'Vk E I : 'Yk = 'Y13k.

The morphisms 13k, k E I, are then called embeddings of Ak into Q. (9.1.12) These two constructions are unique in the following sense. Let {P,1l"} and {P',1l"'} be products, {Q,13} and {Q',13'} coproducts ofthe family of objects {Ak; k E I} of a category Cat. Then there are isomorphisms (1 E Mor(P,P'),r E Mor(Q,Q'), such that 'Vk E I: 1l"k = 1l"k(1,13 = r13k. k Both product and coproduct are universal objects, the former universally attracting, the latter universally repelling, in the sense of appropriately chosen categories, due to the subsequent definition. (9.1.13) An object P E Ob(Cat) is called universally attracting or terminal, if and only if 'VA E Ob(Cat) : :3 unique a: E Mor(Catj A, P).

On the other hand, Q E Ob(Cat) is called universally repelling or initial, if and only if 'VA E Ob(Cat) : :3 unique a: E Mor(Cat; Q, A).

(9.1.13.1) If both P and P' are universally attracting, or both repelling, then:3 unique isomorphism (1 E Mor(Cat; P, P'). (9.1.13.2) For instance, in order to establish the coproduct

{Q,13} ==

II Ak kEI

as universally repelling object of some category Coprod, one defines the latter in the following way. The objects of Coprod are pairs ~:=

Let ljI

:=

{B,¢}, BE Ob(Cat), ¢:= {¢k;k

E

I}, ¢k

E

Mor(CatjAk,B).

{C, tP} be also an object of Coprodj then one defines the set

Mor(Coprod;~,ljI):= {J.L E

Mor(Cat;B,C)j'Vk

E

I: J.L¢k

= tPk}.

412

9. Categorial Viewpoint

(9.1.13.3) The product {P,1r} = IlkEIAk is universally attracting in the sense of an appropriately defined category Prod. Its objects and morphisms are defined as the following pairs and morphisms of Cat. ~:={B,4>},

BEOb(Cat), 4>:= {4>k;kE I}, 4>k EMar (Cat;B,Ak);

Mar(Prod;~,

1[1) := {J.L E Mar(Cat; B, C); 'Ilk E I : 1Pk 0 J.L = 4>d,

from

~

to an object 1[1:= {C,I[I} of Prod.

(9.1.14) Some examples of categories are presented below. (9.1.14.1) The category Set consists of sets as objects, with maps as morphisms. (9.1.14.2) The category Top of topological spaces, with continuous maps as morphisms, is some non-full subcategory of Set. (9.1.14.3) The category Abel of Abelian groups is some full subcategory of the category Group of groups, with respect to homomorphisms of groups. (9.1.14.4) The category of left (right) modules over a ring R, with the Rlinear mappings as morphisms, is denoted by RMod (ModR). (9.1.14.5) The category Ring consists of rings as objects, with ring homomorphisms as morphisms. (9.1.14.6) The category of algebras over a commutative ring R, with the algebra homomorphisms as morphisms, is denoted by AlgR . The categories of associative algebras, and of Lie algebras over R, are full subcategories of AlgR . The category of unital associative algebras over R is a non-full subcategory of Alg R , because its morphisms are expected to send unit to unit. (9.1.14.7) The category of associative (Lie) superalgebras over a commutative ring R, with the corresponding morphisms being understood to be even with respect to the involved Z2-grading, contains the full subcategory of associative (Lie) algebras over R, the latter just with trivial odd R-submodule. (9.1.14.8) Similarly one defines the categories of Z2-graded coalgebras, Z2graded bialgebras, and Z2-graded Hopf algebras over a commutative ring R, with the non-graded objects constituting corresponding full subcategories.

9.1 Categories and Functors

413

(9.1.15) The Cartesian product of a family of sets {Aki k E I}, with the projections:

II A j 3 {ajij E I} ~ak E Ak,k E I, jE[

de!

is some product in the categorial sense. (9.1.16) The direct product G of a family of groups {Gkj k E I} is some product in the categorial sense, with respect to the natural projections.

the last definition concerning units.

(9.1.17) There is some product in the sense of the category of rings, with respect to the natural projections. The sum and product of families of ring elements are constructed as the families of sums and products of the involved components. Thereby the Cartesian product of the sets of a family of rings {Rk j k E I}, with the units ek, is established as some ring R with the unit e:= {ekj k E I}, such that Vr, S E R: r

+ S := {rk + Ski k

E

I}, -r:= {-rki k

E

I}, rs:= {rkski k

E

I}.

Vk E I: R 3 r := {rii i E I} ~ rk E Rk. de!

(9.1.18) The tensor product of finitely many commutative unital associative algebras over a commutative ring R, for instance of finitely many commutative rings over the integers Z, is some coproduct in the category of such algebras. For instance in the case of two-fold tensor product, the embeddings are defined by:

with the units ek of Ak, k = 1,2. (9.1.19) Consider a family {Aki k E I} of left modules over a ring R. The direct product A of the corresponding Abelian groups becomes some R-left module, such that Va:= {akjk E I} E A,Vr E R: ra:= {rakjk E I}. With respect to the natural projections one thereby obtains some product in the sense of the category RMad. Moreover the direct sum

414

9. Categorial Viewpoint

E9Ak:= {{ak;k

E

I};card{k

E

I;ak:f.

ad E No}

kEI

of this family, which is some R-submodule of A, is some coproduct in the sense of RM00, with respect to the natural embeddings. Vk E I:

Ak

3

{3k

.

ak -----.{8kj ak;J de!

E

I}

E

E9 A j , 8kj:= {ae jEI

R R

for k = j ~ k...J.·· Lor r J

For any family of R-linear mappings {Ok : Ak -----. B; k E I}, one just uses the diagrams:

(9.1.20) The universal enveloping superalgebra E(L) of a Lie superalgebra L, over a commutative ring R, is some universally repelling object in the sense of the following category. Its objects are the homomorphisms of Lie superalgebras A : L -----. A L , into the super-commutator algebra A L of an associative unital superalgebra A over R; its morphisms a : A1 -----. A2 are the homomorphisms of unital associative superalgebras such that 00A1 = A2. (9.1.21) The category of Z2-graded differential algebras over a commutative ring R, with the corresponding homomorphisms as morphisms, allows for the following universal construction, due to A. Connes. The injection v of an associative superalgebra A into its non-unital universal differential envelope !1(A), over a commutative ring R of coefficients, is some repelling object in the sense of the following category. Its objects are the homomorphisms of associative superalgebras : A -----. D, into any Z2graded differential algebra Dover R, with the odd derivation 8 : D -----. D; its morphisms J1. : 1 -----. 2 are the homomorphisms of Z2-graded differential R-algebras J1. : D 1 -----. D2, 82 0 J1. = J1. 0 81, with an obvious notation of odd derivations, such that J1. 0 1 = 2, Here 0 denotes the composition of homomorphisms of associative superalgebras over R.

9.2 Abelian Categories (9.2.1) A category A is called additive, if and only if the axions (Ab 1), ... , (Ab 4) below are fulfilled.

(Ab 1) VA,B E Ob(A) : Mar(A;A,B) = Mar(A,B) is an Abelian group, the unit of which is denoted by a: A -----. B.

9.2 Abelian Categories

415

(Ab 2) The product of morphisms is Z-bilinear. V¢, ¢', '1/;, '1/;' E Mor(A): ('I/; + 'I/;')¢ = 'I/;¢ + 'I/;'¢, 'I/;(¢ + ¢') = 'I/;¢ + 'I/;¢', (-'I/;)¢

= 'I/;(-¢) = -('I/;¢)j

(Ab 3)

then ¢O

= O¢ = o.

3A o E Ob(A) : VA E Ob(A) :

card M or(Ao, A)

= card M or(A, A o) =

l.

This zero object Ao is also denoted by 0 sometimes. Then the only element of Mor (Ao,A) is monomorphic, and the only element of Mor(A,A o ) is epimorphic, for any object A. These two morphisms are also denoted by 0, for convenience. (Ab 4) For every finite family of objects, there exist both a product and a coproduct in the categorial sense. Obviously any two zero objects A o, A~ are isomorphic in the sense: Ao---+A~---+Ao, a'a=cAo, aa'=cA'. a'

Q

(9.2.2) Let I {¢k : M k

:=

---+

0

{l, 2, ... ,n} eN, or N, or Z. A sequence M k+ 1; k E I}

of homomorphisms of left (or right) modules over a ring R is called exact, if and only if Vk E I' : 1m ¢k = ker ¢k+lj here I' denotes {I, ... , n -I}, or N, or Z, respectively.

(9.2.2.1) Consider a homomorphism ¢ : M ---+ N of left (or right) modules over R. ¢ is injective, if and only if the sequence: {O} ---+ M ~ x ---+ ¢(x) E N is exact. ¢ is surjective, if and only if the sequence: M ~ x ---+ ¢(x) E N ---+ {O} is exact. (9.2.3) Consider a morphism ¢ E M or(A; A, B) = M or(A, B) of an additive category A. Take objects A', B", C, D of this category, and define ¢~, ¢*, ¢~, ¢* by means of the representation functors Mor(Aj C, -) and Mor(A; -,D), according to the following diagram. Here the mappings of morphisms ¢~, ¢~, ¢*, ¢*, which are induced by given ¢', ¢", and ¢, are homomorphisms of Abelian groups, due to (Ab 2).

416

9. Categorial Viewpoint

c

{o}--ftl~ A'

»

tP'

tP

A

»B

tP'

B"

»

~jy o

(OJ

Now fix A' and B". The morphism ¢/ is called kernel of ¢ and denoted by ker ¢, if and only if VC E Ob(A) : {O} ~ Mor(C, A') ~Mor(C,A)~Mor(C,B) ~:

~.

is an exact sequence. On the other hand, ¢" is called cokernel of ¢ and denoted by coker ¢, if and only if VD E Ob(A): {O} ~ Mor(B",D) ~Mor(B,D)~Mor(A,D) ~~

~.

is exact. Hence especially ¢¢' = 0, if ¢' is a kernel of ¢j ¢"¢ = 0, if ¢" is a cokernel of ¢. Obviously kernels are monomorphic, cokernels are epimorphic. Moreover any two kernels (cokernels) are isomorphic. For an easy proof, choose C and D in the above diagram appropriately. (9.2.4) For any kernel ¢' and cokernel ¢" of ¢ E Mor(A,B), the following conclusions hold.

¢'

= 0 ~ VC: ¢: = 0 ~ VC: Mor(C, A') = {O},¢* <==:}

¢"

injective

¢ monomorphic.

= 0 ~ VD : ¢~ = 0 ~ VD : Mor(B", D) = {O}, ¢* <==:}

injective

¢ epimorphic.

Here A, ... ,D denote objects of an additive category A. Hence, if both ¢' = 0 and ¢" = 0, then ¢ is bimorphic. Again denote the zero object of A by .40. ¢ is monomorphic, if and only if Ao ~ A is some kernel of ¢. ¢ is epimorphic, if and only if B ~ Ao is some cokernel of ¢. (9.2.5) An additive category A is called Abelian, if and only if the axioms (Ab 5), ... , (Ab 7) are also valid. (Ab 5) Every morphism ¢ of A admits kernels and cokernels.

9.2 Abelian Categories

417

(Ab 6) Let ¢/ be a kernel, and ¢" a cokernel of ¢. If ¢' = 0, then ¢ is some kernel of ¢". If ¢" = 0, then ¢ is some cokernel of ¢/. (Ab 7) If both a kernel ¢' phism.

= 0,

and a cokernel ¢"

= 0, then ¢

is an isomor-

(9.2.6) Let the categories A, B be additive. A covariant functor F : A is called additive, if and only if the map Fmor : Mor(Aj AI, A 2 )

--+

--+

B

Mor(Bj B l , B 2 )

is a homomorphism of Abelian groups, for any two objects Ak of A, and Fob: A k --+ Bk,k = 1,2.

(9.2.7) In an Abelian category A, the representation bifunctor: Ob(A) x Ob(A)

3

{A,B}

--+

Mor(AjA,B)

= Hom(A,B) E Ob(Abel),

into the category of Abelian groups, is denoted by Hom, and the morphisms are called homomorphisms.

(9.2.8) Let ¢ E Hom(A, B) be a homomorphism of an Abelian category, with kernel ¢' and cokernel ¢". Denote by im ¢ = ¢2 a kernel of ¢", and by coim ¢ = ¢l a cokernel of ¢/, called image and coimage of ¢. Then ¢ = ¢2¢1' (9.2.8.1) A sequence of homomorphisms 'l/J¢ : A --+ B --+ C is called exact, if and only if any image of ¢ is some kernel of'l/J. A finite or infinite sequence of homomorphisms is called exact, if and only if every occurring composition is exact in the above sense. (9.2.9) An additive functor F : A --+ B of Abelian categories is called left (right) exact, if and only if it preserves kernels (cokernels) in the sense, that ker(Fmor (¢» = Fmor(ker ¢)

(or coker (Fmor (¢» = Fmor(coker ¢»

for all ¢ E Hom(A). In this case exactness of sequences of the kind: --+ A --+ B --+ C (or A --+ B --+ C --+ 0) is preserved. F is called exact, if and only if it is both left and right exact; then it preserves exactness of any sequence.

°

(9.2.10) The covariant representation functor Hom(A, -) into the morphisms of an Abelian category is left exact.

418

9. Categorial Viewpoint

(9.2.10.1) In order to obtain the dual of an Abelian category, one must interchange kernels and cokernels. Hence one finds that the contravariant representation functor H om( -, B) is partially exact in the sense, that: coker

0:

---+ Hom(-,B)

ker (Hom( -, B)mor(O:))

for every homomorphism

0:.

(9.2.10.2) In the category VecK of vector spaces A over a field K, the representation functors H om(A, -) are exact. (9.2.11) The categories RMOO and MOOR of left and right modules over a ring R are Abelian. In both cases, H omR(A, B) is an Abelian group, for any objects A, B. Of course {O} is used as zero object. The inclusion: ker