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A generalization of the quasi-Hopf algebra D ω(G)Daniel Bulacu a & Florin Panaite ba Faculty of Mathematics , University of Bucharest , Bucharest 1, RO-70109, RomaniaE-mail: Str. Academiei 14b Institute of Mathematics of the Romanian Academy , Bucharest, RO-70700, RomaniaE-mail: P.O.Box 1-764Published online: 27 Jun 2007.

To cite this article: Daniel Bulacu & Florin Panaite (1998) A generalization of the quasi-Hopf algebra D ω(G),Communications in Algebra, 26:12, 4125-4141, DOI: 10.1080/00927879808826398

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COMMUNICATIONS IN ALGEBRA, 26(12), 4125-4141 (1998)

A generalization of the quasi-Hopf algcbra Dw(G)

Daniel Bnlacu University of Bucharest, Faculty of Mathematics

Str. Acaderuiei 14, RO-70109 Bucharest 1. Roniania. e-mail: dbulacu~Qal.math.u~lib~~c.ro

Florin Panaite Institute of Mathematics of the Romanian Acadcniy

P. 0. Box 1-764, RO-70700 Bucharest, Ro~riania e-mail: fpanaiteiQstoilow.iinar.ro

1 Introduction

Quasi-bialgebras and quasi-IIopf algebras were introduced hy Llri~lfcl'd ([4], [ 5 ] ) and used in his treatment of the Iinizhnik-Zamolodchikov equations. From a purely algebraic point of view, they appear naturally as follows: if ('4, A.5) is a tialgebra and F' E A ~3 A is an invertible element such that ( E @ 1)(1.') = ( I 3 E ) ( F ) = 1 (such an element was called by Drinfel'tl "gauge tmnsformation") and we define another comultiplication on A by AF(a) = FA(u)FT1, then ( A , A F , ~ ) is no longer a bial- gebra, but it is a quasi-bialgeLra. 1x1 t,his way, we can get a lot of quasi-bia1gehra.s (and quasi-Hopf algebras).

In 131. Dijkgraaf, Pasquier and Hoche introduced a very important example of a quasi-Hopf algebra, denokd by nu((;) (where G is a finite group and w is a nor- ~nalized S-cocycle on G). which does not appear by using tliis procedure. It was introduced by physical reasons, in relation to the work of Dijkgraaf arid Witten on topological field theories, and it is similar to the Drinfel'd doublr of a finite dimen- sional IIopf algebra (it is also quasitriangular). In [ I ] , Altschultrr and Coste proved that Dw(G) is a ribbon quasi-Hopf algebra, and then used 1Iw(C; ) i r ~ order to cow struct invariants of knots and links.

111 this paper, we shall generalize this construction, as follows. Let H be a finite dimensional cocornrnutative Hopf algebra and w : H 13 H ;; (I -+ k a ~iorrnalized 3-cocycle (in the cohomology of commutative algebras over cocornmutative Hopf

Copyright O 1998 by Marcel Dekker, Inc www.dekker.com

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4126 BULACU AND PANAITE

algebras introdured by Sweedler in [14] ) . We shall construct a quasi-Hopf alge- bra structure on 11- c3 11, which will be denot.ed by 1 I w ( H ) . Its algebra structure will be a crossed poduct (as in [14], [ lo]) between 11- and H , wilt1 a 2-cocycle a : H @ 11 -+ H* obtained from w. The comultiplication will be a kind of a "crossed coproduct" (as in [2] or [ 9 ] ) between H* and H. via a map v : H -+ H* @ H* obtained from w (but which is not, in general, a 2-cocycle, so the cornultiplication is not, in general, coassociative). We shall prove then that D d ( H ) is a quasitriangular quasi-Hopf algebra. ITnlilic the case of D w ( G ) , it is not true, in general, that D w ( H ) has a ribbon crlerr~t.r~t (this is similar to the case of Drir~fel'd doubles, see (81). We shall give a necessary and sufficient condition such that t,he natural element which generalizes the ribl~on element of D w ( G ) is a ribhon element for U w ( H ) . We shall see then how D U ( H ) depends on the cohomology class of d.

In the final section, we shall study the integrals of D w ( H ) . We shall prove some results similar to the ones obtained by Radford in [12] for a Drinfel'd double : D w ( H ) is always unimodular. and it is semisimple if and only if H and H* are semisimple.

There is another possibility to see the quasi-Hopf algebra D w ( H ) , suggested by the referee. Namely. it can be viewed as a particular case of a "bicrossproduct" between two Hopf algebras H and Ii' provided ( I f , lt') is all abelian matched pair of Hopf algebras (see Remark 3.2 below).

Finally, let us note that, if the 3-cocycle w is trivial, then D w ( l l ) has the multipli- cation of the Drinfrl'd double D(H), but its comultiplicat,ion is t,he one of If' 9 H . instead of the one of f f""~ @ H , a s in the case of D ( H ) .

2 Preliminaries

In this section, we shall recall some definitions and results arid we shall fix the notation that will be used in the rest of the paper. Throughout, k will be a fixed field and all algebras, h e a r spaces etc. will be over k ; unadorned @ means : k

For coalgebras and Hopf algebras, we shall use the framework of [Is]; in particular. for coalgebras, we shall use Y-notation: A(h) = C hl 8 h z , ( I G A ) ( A ( h ) ) = ( A @ I ) ( A ( h ) ) = C hl ,: h2 @ h3, etc.

Definition 2.1 Let 4 be a k-algebra, A : A -+ A A, E : A + k two alyt brrr homomorphisms. A ts called a quasz-bzalgebra zf thcrt t n s t s an znvertzblt d r m t nt @ E A 8 A 9 A such that, for all elements a E A, we hnrv

(2.1) ( 1 @ A)(A([i)) = @((A €4 I ) ( A ( a ) ) @ - ' ,

(2 .2) ( E @ I ) ( A ( a ) ) = a and ( I @ E ) ( A ( u ) ) = u ,

(2.3) ( I @ I ~3 A)(@)(A @ I @ I ) ( @ ) = ( 1 @ @ ) ( I E 1 I ) ( @ ) ( @ E I ) , (2.4) ( I @ ~ ~ l ) ( @ ) = l @ l ,

where I = ~ d * . The m a p A zs called the coproduct or thr c07nult1plzcat~on and ; t h f countt.

A zs callcd a quasr-Hopf ulgrbra ~ f , mol tov tr , fh trc fri.4 art ant)-aufonzorphr~rn s

of the algebra .4 and fltrr~ents a and /3 of A suth ! h n f , l o / all n E A ~ u r harr

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A QUASI-HOPF ALGEBRA 4127

I f R sat isfies these three conditiot~s, it is called an H-rnat~,is. PYOIII t 1 1 c w rclations, olle car1 deduce the quasi-Yang-Raxter equation:

(2.15) l~12@312R13@&R23~ = @ , ~ ~ 1 ~ ~ . ~ @ ~ ~ l H ~ : ~ @ 2 1 : ~ f Z 1 2

Also. it is easy to see that:

(2.19) (: $2 [)(HI = (I (3 €)(HI = 1

Let A be a quasi-Hopf algebra and H E A (:: A an H-~t~a t r i s . I f \vc denote R = K t i?, RZ and @ - I = 1 .rl I-: .r2 i:: .r3, and i f w e tlcfir~c.:

(2.20) u = C s ( . r 2 ~ . s ( x ~ ) s ( t t ' ) t r l t 1 . ~ ~ 1 .

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4128 BULACU AND PANAITE

then it can b~ proved (see [ I ] ) t l ~ a t u is an i r~wrt ih le element o f A, ~ ( u ) = 1 and

( 2 . 2 1 ) a 2 ( a ) = ncrrr-' for all n E A.

From th is relation it follows i~nrnediately that s2(71) = 11 and t.he e l~rnent u s ( l r ) = s ( u ) u is central.

Definition 2.3 (scr [ I ] ) Lrl '4 bt (1 qunsrtrcclngular quoat-Hopf algcbrn MTc say thaf r7 E A 7s a quasr-rzbbon r l en~en t for 1 cj zl w l ~ c f i e s l h f tond~lrons:

( R l ) 1 9 ' = u s ( u )

(EL) ~ ( 1 , ) = u

(NR) ~ ( 7 1 ) = 1

( R 4 ) A(u71- I ) = f - ' ( ( 8 & ~ j ( . f ~ ~ ) ) ( u ~ l - ~ : lto-I j ,

where f and u a i t the ones definrd 111 (2.11) and (2.20) . If the ~ l c incn l 1 , i s ,

moreover, central, ~t 7s called a rzbbon rlentrnf . I j A has a rzbbon r l e n l ~ n t . tt 15

called a rzbbon quasz-Kopj algebra.

It was proved in [ I ] that ( R 4 ) is equivalent t o the relation:

(2.22) A(u) = ( 0 (3 ~ ) ) ( R r l Rj-' .

provided cr i s invertible.

Definition 2.4 (scr [7], L'hapler "i l') ii nor ph1sn1 of quasz-btalgebras 1s n m a p T : ( A ' , A',E'. @ I ) + ( ' 4 , A,:, @) ulhfch is a rnorphtcm of algrbras b r tuwu t h f 1111-

derlyzng algrbrns, such that

( 2 23) (T c",)A' - A T ( 7 c h T $3 l ' ) ( a l ) = 0, ET = C'

It 1s an ~sornorph~s~rt ofquuar-btalgcbra~ if, rn addr t~or~ , ct la tr~cerftblt If 4' and 1 are quasttr-langular quasr-bralgebras rmth 11-rrzatrrces K t and H , re spec t~z~ ly , t h t n T zs ca l l~d n m o r p h ~ s m of qunsz lr~ang~~lar . quasz-bzalgcbins rf ct rs a ir~orphesrrz of the underlytnq qilasr-bralgebrns such thaf ( T ~9 T ) ( R ' ) = R

Definition 2.5 ( s f r 171) Let A = ( , ~ , A , E , @ ) be a quasi-bralgebm. A gocrgc trprla- jormatzon on A 1s an tnncr.tible rlement F E A ,3 A such / ha t .

(2.24) ( E @ I ) ( F ) = ( I @ E ) ( F ) = 1.

Using a gauge t~ra~isforrnation F o n A, ire can build a new quasi-bialgebra A!. as follows. Define an algebra morphistn AF : A + A :;) '4 by :

(2.25) AF(o) = F A ( n ) F - ' for all (1 E A:

and an element OF E '4 0 .1 13 A , by:

(2.26) aF = F2,(I O A)(F)@(A 8 l)(F ' - ' )F; , ' .

Proposition 2.6 (arc (71) Thr cdgebro Ab = ( . I , AF. E . ) I S n qrror~-brt~lgrl),rr. M O I P O I I ~ I ' , tf A I S q r ~ a ~ ~ t r t a n q d w u11tl1 K- rnntr rr I<, thrrr ,IF = ( A , IF . 5. Q k . I I , ) with RF = R F 1 1s n qr~asrtriar~gulnr qunsr-61nlg~l)ra.

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A QUASI-HOPF ALGEBRA 4129

We recall some facts concerning crossed products and cohomology (see [14]). Let H be a cocomrnutative Hopl'algebra a.nd A a cornrnut~ative algebra.; A is called a left H-module algebra if it has a left It-module structure (denoted by H @ A + A, h c.1 n H it . a ) such that:

(2.27) h . ( u b ) = C ( h l . c l ) (h2 . 6 ) for all h E H and ( I , 6 E A,

(2.28) h . 1 = ~ ( h ) l for all h E H .

For instance, k itself is a left ti-nloclule algebra, with module structure given by h . X = c(h)X for all h E H and X E k.

Let H be a cocommutative Ilopf algebra, A a commutatiw left N-module algebra and o : H @ H -+ A a k-l~near map, which is normalized (that is, ( ~ ( 1 , h ) = o ( h , 1) = ~ ( h ) l for all h E H) and convolution invertible. Supposc that, moreover, (T satisfies the 2-cocycle condition:

this multiplication is associative and 1 6 1 is a unit, hence As H becomes an algebra, which will be denoted by A#,N and will he called the crossed product of A and H . Note that crossed products call be defined in some more general conditions, see

[lo]. Suppose again that If is a cocommut~ative Hopf algebra and A is a conirnutative

left !I-module algebra, and denote by Q : H @ A + A the H-module structure of A. We shall denote by Rcy:(ll. A ) the set of k-linear maps g : H@4 -+ '4 which are normalized (that is, g(hl @... 811,) = c ( h l ) ... ~ ( h , ) l whenever at least one of the h,'s equals 1 ) and convolutio~i invertible. lVe shall denote bj' Zq(lI. A ) , B9(H, A ) and Hq(N. A ) the q-cocycles, q-coboundaries and q-cohomology group of the complex determined by Rcg!(H, A) and the maps Dq : Regl (H, A ) 4 R E ~ ; + ' ( ( H , A ) , given by:

(2.31) Dq(u,) = [$(I 0 u ) ] * (u-l(n) I D ... C I ) ] * [u ( I '2 111 .3 1 ::. ... S I)]*

... * [ u f l ( I . ... 0 m ) ] * [ uT1 E ] ,

where we denoted by rn the multiplication of I3 and by u-' the convolution inverse of u.

3 The construction of D w ( H )

Let H be a finite dimensional cocommutative Hopf algebra with antipode S and w : If @ H @ H + k a normalized 3-cocycle, that is, w is k-linear, convolution invertible and satisfies the conditions:

( 3 . 1 ) ( ~ ) C ~ ( ~ I , ~ I , Z I ~ I ) ~ J ( X ~ Y ~ . : L . ~ Z ) = ~ W ( Y I , ~ I . ~ I ) ~ ( . ~ I , Y Z ; ~ . ~ L ) W ( . ~ L . y7,:>)

for all ,r, y, 2 . 1 E H ,

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4130 BULACU AND PANAITE

(h ) d (1 , z . y ) = Y ( X , 1, y ) = w(.r,y, 1 ) = s ( x ) E ( ~ ) for a11 x , y E H.

We int.roduce now the structures of Dw(If) . We shall not ~ e r f o r m all the compu- tations, but we shall point out t.hc relations that are used at every step. During all the cornp~it~ations, the cocommutativity of H will he essential.

We introduce t.he following notation: g a .r = C S(.r1)yx2 for all g , s f H. Define 0 : If 8::: H H 3 k, by the relation:

It is easy to see that 0 is a150 normalized and convolution invertible. Using several times the cocycle relation of w , one can get the following relation:

Since H is cocommutative, H " becomes a commutative left H-module algebra, with the action : H H* -+ H * , h 8 p ++ h p , where h p = C hl - p - S ( h 2 ) , where we denoted by - and - the left and the right actions of H on H* given by: (h - p)(a) = p(ah) and ( p - h)(a) = p(ha) for all h, a E H and p E H'. Hence, (h p ) ( a ) = p(a a h) for all h, a E H and p E H'.

Define u : H 3 H 4 Ii' by u(x , y)(g) = B(g; x, y). Since 0 is normalized and cori- volution invertible, o i3 also normalized and convolutio~~ invertible, then the relat~on (3.3) is equivalent to the fact that u is a 2-corycle, that is:

Hence, we can considet the crossed product H*#, ti, which will be denoted h j D w ( H ) , and which is an assoc~ative algebra with unit : @ 1. Its multiplication is:

( 3 . 5 ) (p f2 h)(p' r h h ' ) = C p ( h 1 - - S'(h2))a(h3, hl) :3 h4h;

for all p , p' E H* and h . h' f H.

Let us define the map 7 . H @ H @ H + k by:

(3.6) ~ ( g , h; R . ) = C 4 g 1 , h ~ , . r l )w(x~,g2 Q R.S. h j Q ~. .+)d-~(g3 ,xg , h3 Q 6 6 )

for all x ,g . h E H

for all 1 2 , hi . .c. y E H

Define u : H + ( H 3 If)'. by o ( h ) ( x $Qi y ) = ~ ( x . y: h) ldentifying ( H - H) ' with H*@ H*, we shall write. for any h E H, u(h) = C ul(h) 9 u2(h) f H' 3 H'. antl thi\ relation is equ~\a l rn t to u(h)(x F3 y) = C ul (h) (x)v j (h) (y) for all x, y E 11 Sin<f' ic,

is normalized a n t l convolution invertible. so are 7 ~tild V .

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A QUASI-HOPF ALGEBRA 4131

for all p E f l * , h E H . where l ( p ) = C p l c> pz in the coalgebra structure of H * . Iising the relation (:3.7) it follows that ~ ( ( p ~ : . ? h ) ( ~ ' t 3 h ' ) ) = A ( p h ) _ l ( p ' @ h l ) for all p.p' E I1 and h, h' E H and by ~ ( 1 ) = E:.?E it follows t,hat A(E . I ) = (5,: I ) @ ( E Q ~ ) . 111 conclusion, A is an algebra Iv.m~olr~orphisrn.

I f we define c : D w ( H ) -, k . by ~ ( p $3 h ) = p ( l ) ~ ( h ) for all p E H' . h E H , it is easy to see that E is also an algebra homomorphism.

Identifying (H @? I I @ H)' with fi* c;. ti* 5 ti*, we shall write &. = 3 4 Nu3 E H' :: If* ~;j H* and we' = C p , :$ p2 3 }L3 E H* @ H* H * . \\'it11 this notation, define @ E D w ( H ) @ D w ( t I ) ~ ; j D w ( f l ) , by:

lisir~g again the 3-cocycle relation of d, one can check the following identity:

for all 11, .r. y , : E H, and using it we can get:

(:3.1 1 ) ( f c<' A ) ( A ( p @ h ) ) = @ ( ( A :;' l ) ( A ( p @ h)))@-I

for all p E H'. h E H. Since 1 and 6' are normalized. we get imniediately that :

(3.12) ( :: I ) ( A ( p 6 h ) ) = p . h = ( I S E ) ( A ( ~ C3 h ) )

for all j ) E H* and h E H , and

(:j.l:i) ( I ::, E I ) ( @ ) = ( 5 :- 1 ) : ( f ,: 1 )

Finally, thv 3-cocycle condition for w is even equivalent to t 11e rt-lation:

(3.14) ( I 5 I @ A ) ( @ ) ( A p I : I ) ( @ ) = ( ( E L, 1 ) 3 @ ) ( I :: A I ) ( @ ) ( @ 2 ( E G 1 ) )

In conclusion, ( D w ( H ) . A. s. Q ) is a quasi-bialgebra.

Now, define a , 13 E D w ( H ) , by:

(3.15) ck = E $2 1, f j = do 2 1.

\vhere E I i * is defined by J o ( h ) = CG,(hl . S ( h z ) . h3) for all 11 E ti. \Ve d&ne a map s : D w ( H ) -+ D w ( H ) , by:

(3 .16) s ( p t3 h ) = C[E @ ,S(h,)][n- '(hz. .~(h3))S*(Pu;1(I~q))~;'(h4) -. I ]

for all p E H*, h E H , where wc denoted by v-' the convolutior~ in\.erse of u , with the notation v - ' ( h ) = ~ v ; ' ( h ) v ; ' ( h ) E H' @I H', and by Y the antipode of H ' , that is S * ( p ) = p o S for all p E H*. U'e shall prove that ., is an antipode for D w ( I l ) . Using the relation ( B . R ) , is follows that:

(3 .17) C O(S; S ( h l ) , h 2 ) = C B(hl.rS(h2 ); h3. S ( h 4 ) )

for all .r. h E H . and using t.l~i. ; relation wt- obtain:

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4132 BULACU AND PANAITE

(9.18) C s ( ( p ( I : / ~ ) ~ ) ( p ' ~ h ) ~ = e ( p O h ) ( i ! : ~ , l )

for all p E If*, h E H. where we denoted. as usual. A(p : h ) = C ( p K h)l 43 ( p @ h)2. Then, using the re la ti or^ (3.10), it follows t h t :

(3.19) C y ( s l , 5'(.73); hl)$o(.r:3 a h2) = C $S(.r l ) , .r2: 1 1 ) h(.r :~) ,

and by this relat.ion we grt.:

(3.20) C ( p 0 h) l$s ( (pO h)n) = €(PC> h)P

for all p E H*, h E H. ['sing the notation of (3.9). we can get immediately:

(3.21) C(pI @ 1).3s(p2 ;:> 1)(p3 0 1) = E 0 1.

Then, it is obvious that @ - I is given by the formula:

(9.22) (I-' = C(u1 (i) 1 ) @ (w2 @ 1) @ (($3 @ 1)

The 3-cocycle relation for LJ yields:

(3.23) Cw(x1, S(rz) , ~3)w(S(xr ) , 1 5 , S(x6)) = E ( X )

for all x E H, and then it follows that:

(3.24) C s(wI @ l)(w, @ l)/?s(ws @ 1) = E @ 1

Now, applying the relations (3.7) and (3.3), we obtain:

(3.25) s((p @ h)({ '2 h ' ) ) = s(p' 8 h ' ) ~ ( ~ @ h)

for all p,; E H* and h , k' E If. Since obviously S ( E 2 I ) = i tg I, it follows that s is an algebra antihoniolnorphism.

Ol>viously, is an invertible element in DU(l l ) . its irivc?rsc k i n g 13-' = 1.7 1, where PC' is the cor~volution inverse of Do. By the relation (3.23), it follows also that p-' = s ( B ) . Finally, using the relations (3.7). (3.17) and (3.19), it follows that:

(3.26) s2(p 8 h ) = 3 - ' ( p 5 h)$

for all p E H*. h E H. In particular, it follows that s is I,ijective. In conclusion, ( D w ( H ) , A, E , @. s , o , d ) is a quasi-IIopf algebra.

We prove that D"( H ) is a quasitriangular quasi-Hopf algebra, with R-matrix defined by R = ~ ~ = , ( t . : 3 1) 9 ( E 2 e,) E D d ( H ) D d ( H ) , where {el . ..., t,,} is a basis of H and { c ; ..... c,} the dual basis of H' (of course, R does not depend on the choice of the hasis of H).

By expressing 6' and 7. with w. one can check the relation:

(3.27) C y ( x ~ , !/I 4 x 2 : hl)@(yz; 23. h ~ ) = CO(y1: ~ L I . . X I Q h2)y(y~ . x2: h3)

for all x, y , h E H, and using this relation it follows that:

(3.28) ACoP(p E:\ h) R = RA(p @ h)

for all p E H*, h E H. Then, one can prove the following two relations:

(3.29) (a 0 I ) ( R ) = @ , I , R ~ @ ; & R ~ @ ,

(3.30) (1 8 A)( H ) = @ ~ ~ l R ~ ~ @ 2 ~ 3 R i 2 @ - 1 .

by showing that they are rquivalent, respectively. to the clefinitions (3.2) and (3.6) of 6' and 7.

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A QUASI-HOPF ALGEBRA 4133

Finally, I)y a direct computation. it call Ijo proved that IZ is invertible. its inverse being the rlenlent:

(3.31) R-' = C:=,(e: I ) o ( a - ' ( S ( ( c , ) l ) . ( t , ) 2 ) ,.) .S((F,)~))

(for KH-' = (~(3 1) 61 ( E @ 1). one call use the relation (3.17)). As a final conclusion, we ha\.? obhined the following

Theorem 3.1 If H 1s a fiwztt dirr~tn.slorrul rocomtrtutnt~c~c Ilopf algebra und zs a norr~~alrted 9-cocyclr on H , then L)"(ll) is n qu(~.~rtr.~ar~gular quasz-llopJ crlgebm

Remark 3.2 As the referee kindly pointed out, there is some construction on quasi- Ilopf algebras which contains D w ( H ) and Hofstetter's Hopf algebra li#(,,,)H ([6], 11. 273) a.s special cases. Let (H, t i ) be a.n aljelian matched pair of Hopf algebras (for the drfinition see [6] or 1161). We enlarge the double complex on [GI, p. 267, by adding a first column, so we consider the double complex (CP.9, DP,q,dP,q), for p 2 0, q > 1, where CP.9 = Reg(HP, liq), DP'q : Reg(HP, I<q) -+ Reg(Hp+'. Kq) and dPs' : Reg(HP, Ii'q) + Reg(HP, Iiq+'), where Ha = I;. Th? 2-cocycles Z 2 ( H , I<) of the associated total complex consist of triples (w, x, u ) with w E R e g ( k , I i3) , x E Rey(H. IiZ), a E Reg(H2, I<). satisf~ing the relations: d0.3(w) = z . /)".'(a) = E

(that is. w and a are respectively a 3-cocycle and a 2-cocycle). D 1 , 2 ( ~ ) = d i ) ~ l ( u ) and DO."(d) = dl.'(*). Then, starting with a 2-cocycle (w, K , a ) . we call construct a quasi-tIopf algebra Ii'#(,,,)Ii by generalizing Hofstetter's co~~stru:tion. The clenlent @ arises from w naturally.

Now. if If is a cocommutative Hopf algcbra and w is a normalized 3-cocycle on it, and we define Ii = H*, then (H. I { ) is an abelian matched pair of Hopf algebras with the adjoint action of H on li and with trivial coaction E i -r N .. Ii. If we define u and I/ as above, from t,hr relations (3.7) arid (3.10) it follows that the triple (w. u. a ) is a 2-cocycle. Then, the quasi-Hopf algebra D w ( H ) is precisely El'#(,,,)li.

\ t e define the element u E D d ( H ) by the formula:

(3.32) 11 = C .s(x2/3s(x3)).s(R2) R'xl ,

where we denoted a-' = C x l @ x2 a .I' and R = C R1 @ RL. By computing and using the relation (3.23), one can check that:

We shall see under what conditions the element:

is a ribbon element for Dw(H) . Sinre s2(u) = u a ~ ' for all a E D w ( l i ) (sre the Preliminaries) and s2(u) = ,3-'ai3 for all a E 1Iu(H) (see 3.26). it follows that tj

is a ccntral element and also u i j = $11. Then. using the notation introduced in Prellrninaries, we have:

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4134 BULACU AND PANAITE

Proposition 3.3 The element u is a ribbon element for D U ( H ) af and only zf p2u = S ( U ) (or, equivalently, using (3.3.3), if and only tf u = C:=l 3 ~ ~ e : S ( e , ) ) .

Let A be a quasitriangular quasi-Hopf algebra. Denote by G ( A ) = { I E A 1 1 zs invert ib le ,s( l ) = I-' ccnd A(1) = ( 1 @ l ) ( s 8 s)( f i ; ' ) f ) . For I E G ( A ) one obtains i ( 1 ) = 1, as in the case of ordinary Hopf algebras, by applying E @ 5 to the both sides of the relation A ( / ) = (1 @ l f ( s @ s ) ( f i l ) f . We prove that there exists a one- one correspondence between quasi-ribbon (respectively ribbon) elements of A and certain elements of C(A). This is similar to 181, Th.1.

Lemma 3.4 Let A be a quas~trzangular quasz-Hopf algebra wzth R-matrix R and let u be the element defined by (2.20). Suppose that u I S a quasz-rzbbon eienxnt of A ; zf we set h = u - ' ~ ( u ) and 1 = u- ' v , then : ( a ) l2 = h and

( b ) 1 E G ( A ) .

Proof : (a) By (H2) is follows that s 2 ( v ) = v ; hence, applying (2.21), we obtain that P L and P J comniutr. Bv ( R l ) we have v2 = Z L S ( U ) = u2h Thus l2 = u - ~ I , ' = h since u and v conln~ute.

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A QUASI-HOPF ALGEBRA 4135

Theorem 3.5 Lcl A br a quu,~r/r~~c~r:grilnr qr~aac-Hopfcllgrbrn u,lfh H-rt1a1rr.r H urtd kt u and It he as 111 L~rrtrrla 3.4. Tlrrn:

( o ) 1 H ti1 define9 a onr-onr corrc.spondencc

( b ) Suppose that 1 E G ( A ) satrsJcs 1" h . Then ir = ul I S a rtbbon clcincnt o f A fund only zf s 2 ( a ) = I-'a1 for all a E A.

Proof : Let 1 E G ( A ) with 1' = /I. By Lenlrna 3.4, t,o prove part ( a ) wc. r~eed o ~ i l y t o show that v = 111 is a quasi-ribbon rlert~ent of A.

By s(1) = 1-I, we get s2(1) = 1. h ( > n c ~ , I)y (2.21), it follows that 11 ar~d 1 commutc . Thns I!' = 1 1 2 1 2 = 112h = 1 1 2 ~ - 1 d ( 1 1 ) = 11~(U) , 6 0 ( R l ) l l0 ld~ for Zl.

To show that ( K 2 ) l~olds for 1 1 , we first note that h is invertible with h-' = s(1r)-'tr. Now I - ' = k ' l , which follows from tllc equation l2 = h . Tl~ereforr s ( u ) = ~ ( 1 1 1 ) = s(11)*5(1) = s(t1)l-' = ~ ( u ) h - ~ l = , s ( l r ) s ( l l ) - l l~ l = ,111 = 11.

Note that ( R 3 ) is immediate sinct, ~ ( r i ) = ~ ( 1 ) = 1 . Since 1 E G ( A ) and 11 antl i t co~u~nr l t e (lwcause n a ~ ~ d 1 c o n i ~ n r ~ t r ) is follows that

A(cl11-I) = A(/- ' ) = f - ' ( s t";) ~ ) ( t ~ ~ ) ( l - ' t , - ' 1 - I ) = J-'( .s t:! s ) ( ~ ~ ~ ) ( u ~ ~ - ~ 8:: 1171-1), tllereforc ( R 4 ) holds.

Since 1) = 111 is central if and only i f /-lo1 = ucr.u-' for all (I E 4 . part ( h ) Collows by part ( a ) and (2.21).

Remark 3.6 ( 1 ) G ( A ) I S a group # ( s i? s ) ( f z l ) = f . (2) I!\tng (9.11,, zt zs easy to s w that, zJt1 r c a quas~- i~~hbon rlcrrtf~tt /hrlr ( 1 3 ' 1 1 ) j = f ( r c > 1 x 1

Remark 3.7 M tth the abow i r ~ t o f t o n rt Jollouw that ( R l ) hold, for I . = lu lf and o i l ( 1 ) = ( 1 1 ) s f ) f Because for D w ( l l ) . (Rd) hold. for tht tlrnirnt I J = Ru. uv obtnzil that /3 E C : ( I I w ( H ) ) . By (3.26) and T h t o r f m 3 j rt follows that u = ,du zs (L rzbbon elemcnt for D w ( l f ) tf and only 7f B'II = s ( u ) 7'1ir> l i anolht r proof of f'r oposdton 9.7

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4136 BULACU AND PANAITE

for all a , b, c E f l , where f - I is the convolution inverse of t . We shall tlenote by 0, o', 0,q0 and u, rr'. a d , , ) the rnaps associatwl to w. w', d(t),

respwt ivfaly. Jlefine now the map r = T~ : tl + H*, by T ( T ) ( ~ ) = 1 t(.rt,yl 4 x2)l- l (yz,x3) for

all r,g E H. Obviously ~ ( 1 ) = E and T i5 convolution invertible. Since w' = wd( t ) , is follow.; that O' = OOd(t), herlcc .

A straightforward computation yields:

Now, for the cohomology H n ( H , N*), since r E Reg:(H, H-). we can consider D1(r ) E BZ(H, H*), and it is easy to see that :

hence Od(t)(g; .r, y ) = D1(r)(x, y)(g) . that is ud(t) = D1(r). Identifying (N @ H)' = H* @ ti-, we shall write t = C t l B i z E If' ZJ H*. Define

now F E D w ( H ) @ Dw(H), by F = C(tl I.: 1 ) (3 ( I 2 @ 1). Since t is normalized and convolution invertible (let t - ' = C 11 0.) 11 be t,he convolution inverse of t ) . it follows immediatc4y that F is invertible with inverse C.'-I = C ( l l (c3 1)@(12::: 1 ) . and satisfies the relation ( E 9 I ) ( F ) = ( I O z ) ( I . ' ) = 1. tha.t is F is a gauge transformation on nw(li).

Define now T : D"'(H) -r U W ( I l ) by T ( p fl h ) = Xpr(h1) hz. Hy a direct computa.tion, using the facts that o' = U O ~ ( ~ ) and uqt) = D1(r ) , i t f o l l o ~ s that T is an algebra isomorphism.

Consider now DW(H)F-I, that is D w ( H ) with its algebra.structure, hut with co~nul- tiplication given by AF-l(a) = F - ' A ( a ) F , for all a E D w ( H ) . Using the fact that

= + y ~ d ( ~ ) . it follows that (T 8 T) o A' = AF-1 o T. Using the relati011 d' = wd(t), it follows that (?'@TOT)(@') = OF-, . whcre QF-I = F ~ l ( I O A ) ( ~ - ' ) Q ( ~ : : I ) (F)FIz . Obviously, E o T = E ; hence T is an isoniorpliism of quasi-bialgehras. Fron~ the def- inition of T , is f01Iu~s that (T 8 T ) ( R ' ) = l i p * , where RF-I = Fs'HE'.

In conclusion, T is an ison~orphism of quasitriangular qu&+hialgel)ras between D"'(H) and Dw(H)F-I. Using [ 7 ] , Th.XV.3.5 and Th.SV.B.9, we o1)tain also the following :

Proposition 3.8 Ifw and LJ' are ac uboue, then there e ~ i s t s a brardrd trnsor epvzv-

aler~ce betureen the brazded tensor cal~yorrcs Dd(H) - mod and Uw ( H ) - mod.

Remark 3.9 If w and w' are as ahow. wc can show that, using thca ~ ~ o t a t i o n of Re- mark 3 . 2 , tlrc 2-cocycles (w, v , o) a d (w'. v' , n ') are cohomologous. lr~tlcctl, we have already proved that w' = wd(f) and IT' = u D 1 ( r ) . Now, from = -,-,,i(,, i t follows

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A QUASI-HOPF ALGEBRA 4137

4 Integrals for D d ( H )

We start this section by recalling some facts ahout integrals for ordinary Hopf algebras (see 1121, 1131). I f A is a finitc dirne~~sional IIopf algebra and t E '4 is a I ~ I I -

zero left. integral for A . there exists a unique o E ,41yk(A, k ) such that tu = cr(a)t for all a E A; N does not depend on the choice of t and it is called the distinguished grouplike ele~nrnt of .4'.

If 71 E Algc(A. k ) , set L,, = {s E A I as. = rl(a).r for all a E A ) and R, = {y E A I ya = q(a)y for all a E '4).

It was proved in 1131 that:

(4.1) L, = R,, = R,,, for all 17 E Algh(A, k).

We shall define integrals for quasi-Hopf algebras exactely as for finite dimensional Hopf algebras.

Definition 4.1 L F ~ A bc ajlnztr dzinenszonal quclst-1lol)falgebra. An clcinfnf z E 1 zs called a left ( r . f sp tc / i~~ly rtghl) zntcgml for. A , z j (I: = E(U)Z ( i e s p t c t ~ ~ d y :u = ~ ( a ) z ) for all a E .I Il r shall dcrtote by Jp ( i ~ ~ p c i ~ ~ t l y I:) the set of lrft (1~5prc- tzvely rzght) rnftgml~ of A

It is a very well-known fact the existence anti uniqut~riess of iiltegrals for a h i t c dimensional Hopf algcbra. We do not know yet if a sirnilar result holds for quasi-Hopf algebras ; nevertheless, we can prove such a result for O"(I i ) .

Proposition 4.2 LC/ A E /I* be a non-zero lcft (H rlqht, srnce H* rs cornrnrcfaliz~t) zntegral of II* and A E I1 a non-zero rzght ~ n t e ~ r a l of / I . Then X @ A t s a rion-xro left and rzghf trrt~yrnljor the qunsz-Hopf algebra U " ( f 1 )

Proof: Let p E I/', h E H: we have : (A @ A)(p @ h) = C X(.tl -- p - S(A1))u(A3, 1 1 , ) (3 .Ilhl = C(A1 - p - .5(.\2))(l)~(A3, hl ) ( l )A @ A 4 h 2 (since X is an iutegral for H ' ) = p(1)A @ Ah = p ( l ) ~ ( h ) A @ A (since A is a right integral for H ) = ~ ( p @ h)(X f< A ) . Hence A (3 A is a right integral for UU(H) . Wc. shall prove that it is also a left integral. Wc ha\.c,:

(p C3 h ) ( X \,: , \) = C p ( h I - A - S ( h 2 ) ) ~ ( h : 3 . . \ I ) , h 4 A 2

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4138 BULACU AND PANAITE

U) [ I?] . pag. 303, we have the following relation:

(1'1) CXzi:XJa-'S'(XI)=X i,

w h r ~ e rr E H* 1s the distinguished grouplikeelement ot H' arid (1-' 15 ~ t s convolution In\Prse. By evaluating on an elen1t.111 1. ,; y E H c;3 H , we get.

(4 .5) C a ( h l ) h Z A = ~ ( h ) ! \ for all h E H

Since h is a right integral for If, S(.2) is a left integral for Ii : hence, by the definition of a we get:

(4.6) S ( A ) n = a(a )S(A) for all a E If.

Then a ( a ) A = S(S(A)a) = S(a)A (since H is cocornmutative. so SZ = t d ) . Hence, we get : C n ( h l ) h 2 A = C hZS(hl)A = € ( h ) . A , qed.

Proposition 4.3 dimk ~ y ~ ( ~ ' = 1.

Proof: Consider again X and '1 as in the previous proposition, and /L a non-zero left integral on U w ( H ) . We shall w ~ i t r 11 = C:=, p, @ h,, with { h , } linearly independent. Let p E H* ; since ,u is a left integral, we have (p@l)g = p(1)p. that is C:=, ppl@h, = C:=l p(l)p, @ h,, so we get pp, = p(l )p, for z = 1, ..., n, hence all the p, 's are integrals fol H ' . By the uniqueness of integrals of N', there exist a , E k , 1 = 1 , ..., 1 1 , such that p, =a,X for z = 1 ,..., n.

I n conclusion, we have obtained that p = X 8 r , with r E H. r # O To end the ploof, it will be enough to prove that T is right integral of 11 1,ct h E If ; since p is a lrft ~ n t e g ~ a l , we have ( ~ i ' h ) p = c ( h ) p . hence (:(- h)(X I ) = : ( h ) h > . I , that i \

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A QUASI-HOPF ALGEBRA

( 4 . ) C [ h l - X - . C ( h 2 ) ] a ( h , , , . ~ , ) ; / I ~ . I . ~ = f ( h ) X . I .

By waluat i~lg un an cxlemrl~t y (3 p E lf ::' f f^ and using again ( 4 . 4 ) . we ohtain finally:

(1.8) C n ( h l ) h 2 . r = ~ ( h ) x for all h E I f .

This relat.ion is equivalent lo:

(4 .Y) c r - ' ( h ) s = h x for all h E f f ,

and this inearls that J E L,-1. B u t , from (1.1). we obtairl that L,,-I = R,-1, = R,, so x E I?,, that is:

(4.10) s h = ; (h ) . r for all h E f f ,

so x is a right integral of f f , qed.

Lemma 4.4 Lrt .4 bc n f i n ~ t e dzmens~onal quasi-Hopj anlgtbra. Supposr l h t r f t.rlsts z E A a non-zero left and rzght zntegral of A and d2112~ JI4 = 1 . Then J,' = h A .

Proof: Let {i E A be a non-zero right integral of A . It is easy t.o see that .$(/L) is a left integral, where s is the antipode of A. Since dinth hA = 1, therca exists 6 E k such that . s (p ) = 6:. Since s ( z ) is also a left and right integral. thcrr exists c E X:

.A such that z = cs(:). hence, since s is bijective, we get / I = hcz, that is / I E 1, . qed.

Definition 4 . 5 A jinztt dzintnszonal quasz-Hopf algebra A u d l bc collrd unirnodular tf Jp = J; and rl~nri: J," 1.

From wl~a t we I1aL.e proved before, we obtain the following:

Theorem 4.6 U d ( H ) 1s a unimodular quasz-Hopi algcbra.

In [ l l ] it was proved the following version of Maschke's theoren1 for quasi-Hopf algebras : i f ,4 is a finite dimensional quasi-Hopf algebra, then A is semisiri~ple if and only if z ( J t ) # 0. lrsing this and the above results, we obtain thc following:

Proposition 4.7 I ) " ( H ) zs semtstnlple IJ arrd orlly cf H and H' arc scrntstnlplr.

Remark 4.8 Thc " t f" part of the proposzt~on can be obtatned also from a genrml result on crossrd products (see [IO]. Th.7.4.2) : let A#,H be a c r o w d product wzlh H a fintte dzrnensional semzstmple fiopf algebra ; ~f 4 1s semtszmplr A r t l n ~ a n , then so zs A#,H.

ACKNOWLEDGMENTS

The authors would like to thank the referee for his helpful comments, which inproved a first version of this paper. In particular. the Kcmark 3.2 and t 11c suggestion for t.he Remark 3.9 are due to the refwee.

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4140 BULACU AND PANAITE

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[%I S. Discilescu, 3. Haianu, 1'. %hang, Finite Iiopf-Calois ("oextensions, Crossed Coproducts and Duality, J. Algebra 178(1995), 400413.

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[4] V. G. Drinfel'd, Quasi-Hopf algebras and the Iir~izhrlik-Zarnolodchikov equations. In A. Belavin et al. (eds), "Problems of modern quantum field theory", Springer-Verlag. 1990.

[5] V. G. Drinfel'd, Quasi-Hopf algebras. Leningrad Math. J. 1(1990), 1419- 1457.

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[8] L. H. Kauffman and D. E. Radford, A Necessary and Sufficient Condition for a Finite-Dimensional Drinfel'd Double to Be a Ribbon Hopf Algebra, J . Algebra 159(1993), 98-1 14.

[9] S. Majid and Y. S. Soibelman, Bicrossproduct Structure of the Quantum Weyl Group, .J. Algebra 163(1991), 68-87.

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A QUASI-HOPF ALGEBRA 4141

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Received: August 1997

Revised: February 1998

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