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Advances in Mathematics 185 (2004) 136–158

Products in Hochschild cohomology andGrothendieck rings of group crossed products$

S.J. Witherspoon

Department of Mathematics and Computer Science, Amherst College, Amherst, MA 01002, USA

Received 17 October 2002; accepted 20 May 2003

Communicated by David J. Benson

Abstract

We give a general construction of rings graded by the conjugacy classes of a finite group.

Some examples of our construction are the Hochschild cohomology ring of a finite group

algebra, the Grothendieck ring of the Drinfel’d double of a group, and the orbifold

cohomology ring for a global quotient. We generalize the first two examples by deriving

product formulas for the Hochschild cohomology ring of a group crossed product and for the

Grothendieck ring of an abelian extension of Hopf algebras. Our results account for

similarities in the product structures among these examples.

r 2003 Elsevier Inc. All rights reserved.

MSC: primary 16E40; 16E20

Keywords: Hochschild cohomology; Hopf algebras; Crossed products

1. Introduction

Cibils and Solotar first noticed that the cup product in the Hochschild cohomologyof a finite group algebra is similar to the tensor product in the category of modulesfor the Drinfel’d (or quantum) double of the group (or equivalently of Hopfbimodules for the group algebra) [12,13]. This observation led them to conjecture aparticular formula for the product in the Hochschild cohomology ring, which wasproven by Siegel and this author [31]. However, a question remained: Why are theseproduct structures so similar? Recent work of Bouc provides an answer to this

ARTICLE IN PRESS

$Research supported by National Security Agency Grant MDA904-01-1-0067.

E-mail address: [email protected].

0001-8708/03/$ - see front matter r 2003 Elsevier Inc. All rights reserved.

doi:10.1016/S0001-8708(03)00168-3

question; they both give examples of rings constructed in a prescribed way fromGreen functors (the cohomology ring of the group, and the Grothendieck ring ofmodules for the group algebra, respectively) [8].

However, there are more examples of cohomology and Grothendieck rings havingsimilar product structures that do not arise from Bouc’s construction. Theseexamples have components that are indexed by conjugacy classes of the group(or more generally by orbits under another group action), but that do not come froman underlying Green functor as the components are not themselves algebras. Theproduct satisfies a particular formula with respect to this additive structure,analogous to product formulas for Hochschild cohomology of the group algebra andfor modules of the Drinfel’d double of the group. In this paper, we follow the spiritof Bouc’s work, recording (in Section 2) precisely the conditions necessary to recoverthese examples. We generalize Bouc’s ring construction in Theorem 2.2, where wegive two potential product formulas, (i) and (ii): The second (ii) is a directgeneralization of Bouc’s formula. The first (i) differs from (ii) by integer scalarfactors, and applies in case the conjugacy class-graded ring is a subring of invariantsof a larger ring (which may not always be the case). Thus formula (ii) applies to moreexamples and retains more information in positive characteristic.

In the remainder of Section 2, we describe several examples of rings graded byconjugacy classes in which the product structure is known and coincides with ourconstruction. In Sections 3 and 4, which are largely self-contained, we prove productformulas for two further classes of examples. These examples are all of independentinterest, and justify our construction. While most of them do not arise from Bouc’sconstruction for Green functors, they retain many properties of Green functorsleading to their distinctive product structure.

Example 2.10 is orbifold cohomology. Given an orbifold with (almost) complexstructure, Chen and Ruan defined an orbifold cohomology ring [11]. In case theorbifold is a global quotient, that is the quotient of a manifold by an action of a finitegroup, Fantechi and Gottsche [22] and Uribe [34] gave an equivalent definition of itsorbifold cohomology ring. As a consequence of their results, the cup product in theorbifold cohomology of a global quotient coincides with that given by our Theorem2.2(i). If orbifold cohomology were defined more generally for a quotient of atopological space by a finite group action, we expect it would also satisfy theproperties given in Section 2, and hence our Theorem 2.2 would endow it with a ringstructure. The difference between the two possible products defined in Theorem 2.2would be meaningful, as the coefficients may be a ring of positive characteristic. Weoffer the classifying space of a finite group as an example to illustrate this difference.Its orbifold cohomology should be the Hochschild cohomology of the group algebra.

In Section 3, we prove a product formula for the Hochschild cohomology ring of acrossed product S#sG of an algebra S and a finite group G; thus generalizing theproduct formula for Hochschild cohomology of a group algebra [31, Theorem 5.1].As a consequence, such a cohomology ring satisfies the properties given in Section 2,and its product coincides with the product given in our Theorem 2.2(ii). Such crossedproducts and their (co)homology are featured in many papers: Lorenz and Cornickgave an additive decomposition of the Hochschild homology of group-graded rings

ARTICLE IN PRESSS.J. Witherspoon / Advances in Mathematics 185 (2004) 136–158 137

(of which crossed products are a special case) into components indexed by conjugacyclasses of the group [15,26], generalizing the well-known case of a group algebra.Several authors studied the (co)homology of specific types of crossed products. Alevet al. [1] and Alvarez [2] studied the cohomology of a crossed product of a Weylalgebra with a finite group, and their calculations were used by Etingof andGinzburg [20] to understand deformations of these crossed products. One class ofsuch examples arises from an orbifold that is a global quotient. In this case, G actson a corresponding commutative algebra S of functions, and the (noncommutative)

crossed product ring S#sG (as well as the commutative ring of invariants SG) is ofinterest in connection with the orbifold. See for example Connes’ book [14]. Whensuch a crossed product has a nontrivial twisting cocycle s (see the definitions inSection 3), the orbifold is said to have discrete torsion, and this case has been ofparticular interest. Caldararu et al. [9] used Hochschild cohomology to finddeformations of such a crossed product arising from an orbifold with discrete torsionthat is related to an example of Vafa and Witten [35].

A comparison of Example 2.10 (orbifold cohomology) and Theorem 3.16(Hochschild cohomology of a crossed product) shows that the product structuresof these cohomology rings associated to a given orbifold are very similar. Ourconstruction in Section 2 of rings graded by conjugacy classes, of which each of theserings is an example, accounts for this similarity.

In Section 4, we consider representations of a finite abelian extension of Hopfalgebras, that is a Hopf algebra H that is a crossed product arising from an action ofa finite group L on another group G: Such a Hopf algebra is perhaps the simplestpossible extension, and is a fundamental building block for other finite-dimensionalHopf algebras. (For example all semisimple Hopf algebras of dimension 16 (over C)have this form [24].) In case H is semisimple, all simple H-modules were determinedby Kashina et al. [25]; they are partitioned into classes indexed by the L-orbits of G:Here we first use Clifford theory to extend their result to the nonsemisimple case. Wethen prove a formula for the tensor product of any two such H-modules, and showthat it coincides with the product given by our Theorem 2.2(ii). This generalizesa formula for the tensor product of modules for the Drinfel’d double of G

(or equivalently of Hopf bimodules of the group algebra [12]). Our methods workequally well when H is only a quasi-Hopf algebra, and thus apply to the twisted

Drinfel’d double of G as well. Our Theorem 2.2 gives one explanation of why thisformula for the tensor product of H-modules is similar to the product formula forthe Hochschild cohomology ring HH�ðkL; kGÞ of Example 2.7.

2. Hochschild constructions

Let G and L be finite groups with a left action of L on G by automorphisms.(Often we will let L ¼ G act on G by conjugation.) Write the action of xAL on gAG

as xg; and let Lg denote the stabilizer of g in L; that is

Lg :¼ fxAL j xg ¼ gg:

ARTICLE IN PRESSS.J. Witherspoon / Advances in Mathematics 185 (2004) 136–158138

Let fAðgÞ j gAGg be a set of free modules over a commutative ring k: Assumethere are k-linear and k-bilinear maps

cg;x : AðgÞ-AðxgÞ and mg;h : AðgÞ � AðhÞ-AðghÞ ð2:1Þ

and that cg;x is an isomorphism of k-modules, for all g; hAG; xAL: Let

A :¼MgAG

AðgÞ

and let cx : A-A be the linear function which is cg;x on the component AðgÞ: If aAA;

write a ¼P

gAG ag with agAAðgÞ: We will also denote by ag any element of AðgÞCA:

Assume the following properties hold:

(H1) (group action) c1 ¼ id and cx3cy ¼ cxy for all x; yAL;

(H2) (compatibility) cx3mg;h ¼ mxg;xh3ðcx � cxÞ for all g; hAG; xAL; and

(H3) (unity) there is an element 1AAð1Þ with cxð1Þ ¼ 1 for all xAL and m1;gð1; agÞ ¼ag ¼ mg;1ðag; 1Þ for all gAG; agAAðgÞ:

Notice that property (H1) implies there is an action of L on the k-module A: Further,A will be an associative algebra with an action of L as automorphisms if thefollowing additional property is satisfied:

(H4) (associativity)

mde; f ðmd;eðad ; beÞ; gf Þ ¼ md;ef ðad ;me; f ðbe; gf ÞÞ

for all d; e; fAG; adAAðdÞ; beAAðeÞ; gf AAð f Þ:

In this case the product on A is defined componentwise by the maps mg;h; and A is a

group-graded algebra, graded by the group G: The action of L by automorphisms iscompatible with the grading. Let

AL :¼ faAA j cxðaÞ ¼ a for all xALg;

the k-submodule of L-invariants of A; also an associative algebra in case properties(H1)–(H4) are satisfied.

We would like to weaken property (H4) and still obtain an associative algebra

structure on the invariants AL: The example of Hochschild cohomology of thegroup algebra kL; with product formula given in [31, Theorem 5.1] and generalizedin [8, Theorem 6.1], suggests how to do this. Consider the following propertyinstead:

(H40) (associativity)Xðd;e; f ÞATg

mde; f ðmd;eðad ; beÞ; gf Þ ¼X

ðd;e; f ÞATg

md;ef ðad ;me; f ðbe; gf ÞÞ;

for all gAG; and a; b; gAAL; where Tg is the set described below.


The set Tg is any set of representatives of equivalence classes in

fðd; e; f ÞAG � G � G j def ¼ gg

given by orbits under the following group actions. The stabilizer Lg of g acts on pairs

ðde; f Þ; that is on the product of the first two factors, and the last factor. For each f ;the stabilizer Lgf 1 ð¼ LdeÞ acts on pairs ðd; eÞ; that is on the first two factors.

We may equally well consider Tg to be a set of representatives of equivalence

classes given by orbits under the action of Lg on pairs ðd; ef Þ and for each d; the action

of Ld1g (¼ Lef ) on pairs ðe; f Þ: The bijection Tg-Tg given by ðd; e; f Þ/ðgfg1; d; eÞyields a correspondence of orbit representatives. By property (H2), the choice of

representatives does not matter as we are working in the ring of invariants AL:Clearly property (H4) implies (H40), but not conversely. We have already observed

part (i) of the following theorem, and part (ii) describes a product on AL in case theabove weaker property (H40) holds.

Theorem 2.2. Let fAðgÞ j gAGg be a set of free k-modules, with maps cg;x and mg;h as

in (2.1).

ðiÞ If A satisfies (H1)–(H4), then A is an associative ring with multiplication m given

componentwise by mg;h for all g; hAG; and AL is a subring under the inherited

multiplication.ðiiÞ If A satisfies (H1)–(H3) and (H40), then AL is an associative ring with product

defined componentwise by

ða � bÞg ¼X

ðh;kÞALg\G�Ghk¼g

mh;kðah; bkÞ

for all a; bAAL and gAG:

We remark that if properties (H1)–(H4) are satisfied, then the L-invariants

AL have two distinct products given by Theorem 2.2(i) and (ii). The product

on AL inherited from A as in (i) is given by

ða � bÞg ¼X

ðh;kÞAG�Ghk¼g

mh;kðah; bkÞ

¼X

ðh;kÞALg\G�Ghk¼g

j Lg : Lh-Lk j mh;kðah; bkÞ; ð2:3Þ

where jLg : Lh-Lkj is the index of the subgroup Lh-Lk in Lg: This differs from

product (ii) by the integer scalar factors jLg : Lh-Lkj: Thus the distinction is

meaningful in positive characteristic, where product (ii) potentially contains moreinformation.


Proof of Theorem 2.2. The proof of (i) is straightforward. We will prove (ii). First

note that by property (H2), if a; bAAL; then a � bAAL as well, as the product formula

in (ii) yields ða � bÞxg ¼ xðða � bÞgÞ: By property (H3), AL has a multiplicative identity.

We will show that (H40) is equivalent to the associativity of the product defined in

(ii). Let a; b; gAAL and gAG: Then

ðða � bÞ � gÞg ¼X

ðh;kÞALg\ðG�GÞ; hk¼g

ðd;eÞALh\ðG�GÞ; de¼h

mh;kðmd;eðad ;beÞ; gkÞ;

which is the same as the left-hand side of property (H40), whereas

ða � ðb � gÞÞg ¼X

ðh;kÞALg\ðG�GÞ; hk¼g

ðe; f ÞALk\ðG�GÞ; ef¼k

mh;kðah;me; f ðbe; gf ÞÞ;

the right-hand side of property (H40). &

Let g1;y; gt be a set of representatives of orbits of L on G: There is another wayto write the product of Theorem 2.2(ii), in terms of the additive decomposition

AL ¼MgAG

AðgÞ !L

DMt

i¼1

AðgiÞLgi : ð2:4Þ

Write Li :¼ Lgiand ai :¼ agi

for aAA:

Corollary 2.5. Let aiAAðgiÞLi and bjAAðgjÞLj in the additive isomorphism (2.4). Then

under the product of Theorem 2.2(ii),

ai � bj ¼XxAD

mygi ;yxgjðyai;

yxbjÞ;

where D is a set of representatives of double cosets Li\L=Lj; and k ¼ kðxÞ and y ¼ yðxÞare chosen so that ygi

yxgj ¼ gk:

Proof. The elements ai and bj correspond in AL toP

yAL=Li

yai andP

zAL=Lj

zbj ;

respectively. By Theorem 2.2(ii),

XyAL=Li

yai

0@ 1A �X

zAL=Lj

zbj

0@ 1A0@ 1Agk

¼X

ðygi ;zgjÞALk\ðG�GÞ

ygizgj¼gk

mygi ;zgjðyai;

zbjÞ:


Letting x ¼ y1z so that z ¼ yx; this looks like the sum in the corollary, other thanthe set over which the sum is taken. Noting that the choice of y ¼ yðxÞ in thecorollary is unique only up to multiplication by an element of Lk; and similarly in theabove sum, y; z are unique only up to multiplication by elements of Li;Lj;

respectively, we see that the two sums are the same. &

We give several examples in the remainder of this section. The first example is justthe group algebra.

Example 2.6 (Group algebra). Let AðgÞ :¼ kg for each gAG; that is AðgÞ is the one-dimensional vector space with basis g: Define linear maps mg;h : kg � kh-kgh by

mg;hðg; hÞ ¼ gh: Then A satisfies properties (H1)–(H4), and A ¼ kG under the

product of Theorem 2.2(i). The ring of invariants AL is ðkGÞL: In particular, if

L ¼ G; then AL is the center of kG:

The invariant subring ðkGÞL of the above example is isomorphic to HH0ðkL; kGÞ;the degree 0 component of the Hochschild cohomology ring of kL with coefficientsin kG: The next example is thus a generalization. The definition of Hochschildcohomology is given in Section 3; see also [5,21] for group cohomology.

Example 2.7 (Hochschild cohomology). For each gAG; let AðgÞ :¼ H�ðLgÞ ¼Ext�kLg

ðk; kÞ where Lg acts trivially on the coefficient ring k: The conjugation maps

are given by conjugation of group cohomology rings, and

mg;hðag; bhÞ :¼ corLgh

Lg-LhðresLg

Lg-Lhag ^ resLh

Lg-LhbhÞ; ð2:8Þ

where res and cor denote the standard restriction and corestriction maps of groupcohomology. Then (H1)–(H3) are well-known properties of group cohomology. Acomparison of [31, Lemma 4.2 and Theorem 5.1] with Corollary 2.5 above shows

that (H40) holds, and that under the product of Theorem 2.2(ii), ALDHH�ðkL; kGÞas rings.

In the above example, the difference between products (i) and (ii) of Theorem 2.2is important: If there were a definition of mg;h satisfying (H4) instead of (H40), the

subring of invariants AL of A would in general not be isomorphic to HH�ðkL; kGÞ:For example, if k is a field and n40; then HHnðkL; kGÞa0 only if the characteristicof k divides jLj: In this case, the integer scalar factor jLgh : Lg-Lhj of (2.3) may often

be 0 even when the corresponding product in Hochschild cohomology is nonzero.The next example, due to Bouc [8], is a generalization of Example 2.7 arising from

any Green functor. This was the motivating example for our present work.

Example 2.9 (Green functor). Let L ¼ G and let AðÞ be any Green functor for G

over k: That is, to each subgroup HoG; there is assigned a k-algebra AðHÞ; togetherwith conjugation maps cx : AðHÞ-AðxHÞ (for all xAG and HoG), restriction maps


rHK : AðHÞ-AðKÞ and transfer maps tH

K : AðKÞ-AðHÞ (for all KoHoG). These

maps are required to satisfy certain properties (see for example [33]). Let AðgÞ :¼AðCðgÞÞ where CðgÞ ¼ Gg is the centralizer of g in G; and

mg;hðag; bhÞ :¼ tCðghÞCðgÞ-CðhÞðr

CðgÞCðgÞ-CðhÞag � r

CðhÞCðgÞ-CðhÞbhÞ:

Then (H1)–(H3) are standard properties of Green functors. Property (H40) is

equivalent to the associativity of the product on AG proven by Bouc [8, Theorem

6.1]. The ring AG of our Theorem 2.2(ii) is precisely Bouc’s ring AGðGÞ: Examplesinclude, among others, the crossed Burnside ring (obtained from the Burnside ringfunctor), the Hochschild cohomology ring of kG (obtained from the groupcohomology functor), and the Grothendieck ring of the Drinfel’d double of G

(obtained from the Grothendieck ring functor for a group algebra). The last two ofthese examples are generalized in Sections 3 and 4 to rings that do not arise fromBouc’s construction, and their definitions appear there.

The final example in this section is orbifold cohomology of a global quotient.

Example 2.10 (Orbifold cohomology). Let L ¼ G and let Y be a complex mani-fold with a left action of a finite group G: Let AðgÞ :¼ H�ðY gÞ; the singularcohomology of the submanifold Y g of elements invariant under g: The map

cg;x : H�ðY gÞ-H�ðY xgÞ is induced by the action of x on Y : Let

mg;hðag; bhÞ ¼ i�ðagjY/g;hS � bhjY/g;hS � cðg; hÞÞ;

where i : Y/g;hS-Y gh is the natural inclusion, the pushforward i� is defined via

Poincare duality, and cðg; hÞAH�ðY/g;hSÞ are particular classes (see [22] or [34] forthe details). By Fantechi and Gottsche [22, Theorem 1.18] (see also [34]), properties(H1)–(H4) hold, and A of our Theorem 2.2(i) is the associative algebra H�ðY ;GÞ of[22] (see also [34]). The subring AG is the orbifold cohomology H�

oð½Y=G Þ of the

orbifold ½Y=G :

The product on orbifold cohomology in the above example differs from the

product on AG given in our Theorem 2.2(ii) by scalar factors (see (2.3)). Thus

we have two different products on AG ¼ H�oð½Y=G Þ: If orbifold cohomology

were defined more generally for a quotient of a topological space by a finite groupaction, it may be that there would be no obvious ring A from which to take the

subring AG of invariants as the orbifold cohomology. In addition, if coefficients aretaken in a ring k of positive characteristic, the product of Theorem 2.2(i) may often

be 0 on the invariants AG; due to the (integer) scalar factors that appear in (2.3).Under these conditions, Theorem 2.2(ii) may yield the desired cohomologyring directly.

To illustrate these issues, we consider the classifying space BG ¼ EG=G of thefinite group G (see [6] for the definition). Given a; b; cAG with ab ¼ c; there is an


arrow in EG from b to c labeled by a: A fourth element gAG acts on the

corresponding cell in EG by sending it to an arrow from gb to gc labeled by gag1:This action commutes with the action of G on EG by right multiplication by

elements of G; and so yields an action of G on BG: The invariant subspace ðBGÞg ofgAG may be identified with BCðgÞ; the classifying space of the centralizer of g in G:

Letting g1;y; gt be a set of representatives of the conjugacy classes of G; theorbifold cohomology of BG should thus be (additively)

H�oð½BG=G Þ ¼

MgAG

H�ðBCðgÞÞ !G

DMt

i¼1

H�ðBCðgiÞÞ;

as the group CðgÞ acts trivially on H�ðBCðgÞÞDH�ðCðgÞÞ: If coefficients are taken ina fixed field k; we may rewrite this as

H�oð½BG=G ÞD

Mt

i¼1

H�ðCðgiÞÞDHH�ðkGÞ:

That is, H�oð½BG=G Þ is additively isomorphic to the Hochschild cohomology of the

group algebra kG: We may now use the definition of mg;h from Example 2.7 (with

L ¼ G), making H�oð½BG=G Þ into a ring isomorphic to HH�ðkGÞ:

3. Hochschild cohomology of group crossed products

In this section we prove a product formula (Theorem 3.16) for the Hochschildcohomology ring of a group crossed product, generalizing the formula [31, Theorem5.1] for group algebras. As a consequence, such a Hochschild cohomology ringsatisfies the properties of Section 2, and its product formula coincides with that ofTheorem 2.2(ii).

Let S be an algebra over a commutative ring k; and assume S is projective as a k-module. Let G be a finite group with an action by automorphisms on S; denoted

ðg; sÞ/gs for gAG; sAS: Let s : G � G-ZðSÞ� (the units in the center of S) be atwo-cocycle, that is

gsðh; kÞsðg; hkÞ ¼ sðg; hÞsðgh; kÞ ð3:1Þ

for all g; h; kAG:1 Assume that s is normalized, that is sðg; 1Þ ¼ 1 ¼ sð1; gÞ for allgAG:

The crossed product algebra R :¼ S#sG (or S#skG) has underlying vector spaceS#kG (where # ¼ #k) and multiplication given by

ðs#gÞðt#hÞ ¼ sðgtÞsðg; hÞ#gh

ARTICLE IN PRESS

1We require the image of s to be central in S to be consistent with having an action of G on S: More

generally, the action of G could be twisted by s with noncentral image. See for example [28, Chapter 7].

S.J. Witherspoon / Advances in Mathematics 185 (2004) 136–158144

for all s; tAS and g; hAG:We will write s %g :¼ s#g to shorten notation. Note that S isa subalgebra of R; but kG is not. For each gAG; the element %gAR is invertible with

ð %gÞ1 ¼ s1ðg; g1Þg1:

The action of G on S becomes an inner action on R; as

gs ¼ %gsð %gÞ1 ð3:2Þ

for all gAG; sAS:A number of authors have studied Hochschild (co)homology of such group

crossed products or more general algebras. Sanada [30] and -Stefan [32] gave spectralsequences relating Hochschild cohomology to that of the components S and G:(-Stefan treated the more general case of a Hopf Galois extension.) Cornick [15] andLorenz [26] gave spectral sequences to describe components of Hochschild homologyof group-graded algebras that are indexed by the conjugacy classes of G: Here wewill describe the overall product structure of the Hochschild cohomology ringHH�ðRÞ in terms of components indexed by conjugacy classes.

We will first give a decomposition of the Hochschild cohomology ring HH�ðRÞinto such components. We recall the definition of Hochschild cohomology: As R isprojective over k; its Hochschild cohomology may be defined as

HH�ðRÞ :¼ Ext�ReðR;RÞ;

where Re :¼ R#Rop acts on the left on R by left and right multiplication. Moregenerally, the Hochschild cohomology of R with coefficients in an R-bimodule M isHH�ðR;MÞ :¼ Ext�ReðR;MÞ: (So HH�ðRÞ ¼ HH�ðR;RÞ:) This may be expressed in

terms of the (acyclic) Hochschild complex:

?!d3 R#4 !d2 R#3 !d1 Re !m R-0 ð3:3Þ

is an Re-free resolution of R; where m is the multiplication map and

dnðr0#r1#?#rnþ1Þ ¼Xn

i¼0

ð1Þir0#?#ririþ1#?#rnþ1:

Dropping the term R from complex (3.3) above, and applying HomReð;MÞ; wehave the Hochschild (cochain) complex

HomReðRe;MÞ!d�1HomReðR#3;MÞ!

d�2HomReðR#4;MÞ!

d�3 ? ð3:4Þ

Thus HHnðR;MÞ ¼ Kerðd�nþ1Þ=Imðd�nÞ:If H is a subgroup of G; let

DðHÞ :¼MhAH

S %h#Sð %hÞ1;


a subalgebra of Re; and let D :¼ DðGÞ: Let g1;y; gt be a set of representatives of theconjugacy classes of G:

Lemma 3.5. Let R ¼ S#sG: There is an isomorphism of vector spaces

HH�ðRÞDMt

i¼1

Ext�DðCðgiÞÞðS;SgiÞ:

Proof. Note that RDRe#DS as an Re-module (see [7, Lemma 3.3], valid for anycommutative ring k). Thus by the Eckmann–Shapiro Lemma [5, Corollary 2.8.4] wehave

HH�ðRÞDExt�ReðRe#DS;RÞDExt�DðS;RÞ:

Now, as a D-module, RD"ti¼1 RCi

; where Ci is the conjugacy class of gi and

RCi¼ "hACi

S %h: Therefore HH�ðRÞD"ti¼1 Ext

�DðS;RCi

Þ:As a D-module, RCi

is isomorphic to the coinduced module HomDðCðgiÞÞðD;SgiÞ;where the action of D is given by ððr0#r1Þ � f Þðs0#s1Þ ¼ f ðs0r0#r1s1Þ forfAHomDðCðgiÞÞðD;SgiÞ: The map RCi

-HomDðCðgiÞÞðD;SgiÞ defined by

ð %hÞrð %gÞð %hÞ1/ððð %hÞ1# %hÞ/r %gÞ ð3:6Þ

is an isomorphism of D-modules, where h ranges over a set of coset representativesfor CðgiÞ in G: The lemma now follows from another application of the Eckmann–Shapiro Lemma,

Ext�DðS;RCiÞDExt�DðCðgiÞÞðS;SgiÞ: &

Remark 3.7. Another way to view the isomorphism given by (3.6) in the lemma is tonotice that RCi

DD#DðCðgiÞÞSgi; and this induced module is isomorphic to the

coinduced module HomDðCðgiÞÞðD;SgiÞ: Passman has provided us with a proof that

the induced and coinduced modules for finite group crossed products areisomorphic.

We will show that HH�ðRÞ satisfies the properties of Section 2, and in the processwill determine how the Hochschild cup product behaves with respect to the additivedecomposition of Lemma 3.5. We will need to develop some general theoryregarding such Hochschild cohomology rings first.

Let M be a DðHÞ-module for a subgroup HpG; and K a subgroup of H: Let

?-P2-P1-P0-S-0 ð3:8Þ

be a projective resolution of S as a DðHÞ-module. Since DðHÞ is free as a DðKÞ-module, this restricts to a projective resolution of S as a DðKÞ-module, and so there


is a restriction map

resHK : Ext�DðHÞðS;MÞ-Ext�DðKÞðS;MÞ;

defined at the chain level by the inclusion HomDðHÞðPn;MÞ-HomDðKÞðPn;MÞ: IfgAG; conjugation by ð %gÞ1 is an algebra isomorphism from DðgHÞ to DðHÞ: Denoteby gM the DðgHÞ-module corresponding to M under this algebra isomorphism. Thenwe have conjugation maps

g� : Ext�DðHÞðS;MÞ-Ext�DðgHÞðS; gMÞ

induced by the algebra isomorphism DðgHÞ B-DðHÞ: They may be described at the

chain level as follows. Let

?-gP2-gP1-

gP0-S-0

be the projective resolution of S; as a DðgHÞ-module, corresponding to (3.8). Denoteby gp an element of gPn thus corresponding to pAPn:We let g�ð f ÞðgpÞ ¼ gf ðpÞ:Notethat hAH acts trivially by conjugation on Ext�DðHÞðS;MÞ: The corestriction map

corHK : Ext�DðKÞðS;MÞ-Ext�DðHÞðS;MÞ

is defined at the chain level by corHK ð f ÞðpÞ ¼

PgAH=K gf ðg1pÞ: Strictly speaking,

this sum is over a set of coset representatives of H=K ; but since K acts trivially onHomDðKÞðPn;MÞ; the choice of representatives does not matter. This induces a map

on cohomology as it commutes with the cochain maps (see for example [5, Section3.6] for the case of group cohomology).

We recall the Hochschild cup product on HH�ðRÞ; defined at the chain level on the

Hochschild complex (3.4) when M is itself a ring (see [23]): If fAHomReðR#m;MÞand f 0AHomReðR#n;MÞ; then f#f 0AHomReðR#ðmþnÞ;MÞ is defined by

ð f#f 0Þðr1#?#rmþnÞ

¼ f ðr1#?#rmÞf 0ðrmþ1#?#rmþnÞ:

In order to translate this into a product on Ext�DðS;RÞ; we will give a D-projectiveresolution of S that induces to the Hochschild complex (3.3) for R: For each nX0; let

Dn be the D-submodule of R#ðnþ2Þ consisting of sums of all s0g0#?#snþ1gnþ1

(siAS; giAG) such that g0?gnþ1 ¼ 1 in G: Thus D0 ¼ D; and each Dn is a projectiveD-module. Then

?!d3 D2 !d2 D1 !

d1 D0 !m

S-0 ð3:9Þ

is a projective resolution of the D-module S; where the maps are restrictions of themaps from the Hochschild complex (3.3). The above complex (3.9) is exact, as there


is a chain contraction sn : Dn1-Dn given by

snðr0#?#rnÞ ¼ r0#?#rn#1:

The Hochschild complex (3.3) is just complex (3.9) induced from D to Re: Thus theisomorphism ExtnReðR;RÞ B

-ExtnDðS;RÞ of the Eckmann–Shapiro Lemma is given at

the chain level simply by restricting maps from HomReðR#ðnþ2Þ;RÞ to HomDðDn;RÞ:The Hochschild cup product on HH�ðRÞ now yields a cup product on Ext�DðS;RÞ asfollows. If fAHomDðDm;RÞ and f 0AHomDðDn;RÞ then f#f 0AHomDðDmþn;RÞ isdefined by

ð f#f 0Þðr1#?#rmþnÞ ¼ %gf ðr10#r2#?#rmÞ

� f 0ðrmþ1#?#rmþn1#rmþn0Þð %gÞ1; ð3:10Þ

where r1 ¼ %gr10 and rmþn ¼ r0mþnð %gÞ1 are chosen so that r1

0#r2#?#rm andrmþ1#?#rmþn1#rmþn

0 are in Dm and Dn; respectively. Similarly, we have aHochschild cup product on Ext�DðHÞðS;RÞ for any subgroup H of G:

Lemma 3.10. Let K and H be subgroups of G: The following relations hold for all

aAExt�DðHÞðS;RÞ and bAExt�DðKÞðS;RÞ:

(i) (Frobenius property) If KoH then

corHK ðresH

K ðaÞ ^ bÞ ¼ a ^ corHK ðbÞ and corH

K ðb ^ resHK ðaÞÞ ¼ corH

K ðbÞ ^ a:

(ii) (Mackey property)

resGKðcorG

HðaÞÞ ¼XxAD

corKK-xHðresxH

K-xHðx�aÞÞ;

where D is a set of double coset representatives for K\G=H:

Proof. Property (i) holds at the chain level by the definitions, and (ii) follows from aparticular ordering of G=H (see [21, Theorem 4.2.6]). &

In fact, the assignment of Ext�DðHÞðS;RÞ to each subgroup H of G is a Green

functor (see [33] for the definition). The additional properties that we will need arestraightforward:

g�3resH

K ¼ resgHgK 3g� and g�

3corHK ¼ cor

gHgK 3g� ðgAG and KoHoGÞ ð3:11Þ

and

resKL 3res

HK ¼ resH

L and corHK 3corK

L ¼ corHL whenever LoKoHoG: ð3:12Þ


We will need explicit maps giving the isomorphism of Lemma 3.5. For each gAG;the inclusion map yg : S %g-R and projection map pg : R-S %g are both DðCðgÞÞ-homomorphisms. Thus they induce the following maps on cohomology:

y�g : : Ext�DðCðgÞÞðS;S %gÞ-Ext�DðCðgÞÞðS;RÞand

p�g : Ext�DðCðgÞÞðS;RÞ-Ext�DðCðgÞÞðS;S %gÞ:

Let g1;y; gt be a set of representatives of conjugacy classes of G; and write yi :¼ ygi

and pi :¼ pgi: The isomorphism Ext�DðS;RÞ B

-"ti¼1 Ext

�DðCðgiÞÞðS;SgiÞ is given explicitly

by z/ðp�i resGCðgiÞðzÞÞi; and its inverse sends aAExt�DðCðgiÞÞðS;SgiÞ to corG

CðgiÞ3y�i ðaÞ:

(See [31, Lemmas 4.1 and 4.2] for more details in the case of a group algebra.)Let g; hAG: The following further properties are straightforward:

If WpCðgÞ; then h�3y�g ¼ y�hg3h

� and h�3p�g ¼ p�hg3h

�

as maps from Ext�DðW ÞðS;S %gÞ to Ext�DðhW ÞðS;RÞ: ð3:13Þ

If W 0pWpCðgÞ; then y�gand p�g commute with resWW 0 and corW

W 0 : ð3:14Þ

If WpCðgÞ-CðhÞ; then p�g3y�h ¼ dg;hid

as maps from Ext�DðW ÞðS;S %hÞ to Ext�DðW ÞðS;S %gÞ: ð3:15Þ

We now have the following product formula. For aAExt�DðCðgiÞÞðS;SgiÞ; write

giðaÞ ¼ corGCðgiÞ3y

�i ðaÞ; which is the image of a in Ext�DðS;RÞDHH�ðRÞ under the

isomorphism of Lemma 3.5.

Theorem 3.16. Let aAExt�DðCðgiÞÞðS;SgiÞ; bAExt�DðCðgjÞÞðS;SgjÞ; and giðaÞ and gjðbÞtheir images in Ext�DðS;RÞDHH�ðRÞ under the isomorphism of Lemma 3.5. Then the

cup product of giðaÞ and gjðbÞ in HH�ðRÞ is given byXxAD

gkðcorCðgkÞyCðgiÞ-yxCðgjÞp

�kðy

�ygires

yCðgiÞyCðgiÞ-yxCðgjÞy

�a ^ y�yxgjres

yxCðgjÞyCðgiÞ-yxCðgjÞðyxÞ�bÞÞ;

where D is a set of representatives of double cosets CðgiÞ\G=CðgjÞ; and k ¼ kðxÞ and

y ¼ yðxÞ are chosen so that gk ¼ ygiyxgj:

Proof. By the Frobenius andMackey properties (Lemma 3.10), giðaÞ ^ gjðbÞ is equal to

corGCðgiÞðy

�i aÞ ^ corG

CðgjÞðy�j bÞ

¼ corGCðgiÞðy

�i a ^ resG

CðgiÞcorGCðgjÞy

�j bÞ


¼XxAD

corGCðgiÞðy

�i a ^ cor

CðgiÞCðgiÞ-xCðgjÞres

xCðgjÞCðgiÞ-xCðgjÞx

�y�j bÞ

¼XxAD

corGCðgiÞ-xCðgjÞðres

CðgiÞCðgiÞ-xCðgjÞy

�i a ^ res


�y�j bÞ

Inserting the identity map id ¼Pt

k¼1 gkp�kres

GCðgkÞ; and applying the Mackey property

and properties (3.11)–(3.15), giðaÞ ^ gjðbÞ equalsXt

k¼1

XxAD

gkp�kres

GCðgkÞcor

GCðgiÞ-xCðgjÞðres

CðgiÞCðgiÞ-xCðgjÞy

�i a ^ res


�y�j bÞ

¼Xt

k¼1

Xx;y

gkðp�kcorCðgkÞW res

yCðgiÞ-yxCðgjÞW y�ðresCðgiÞ

CðgiÞ-xCðgjÞy�i a ^ res


�y�j bÞÞ

¼Xt

k¼1

Xx;y

gkðcorCðgkÞW p�kðy�ygi

resyCðgiÞW y�a ^ y�yxgj

resyxCðgjÞW ðyxÞ�bÞÞ;

where y runs over a set of double coset representatives for CðgkÞ\G=CðgiÞ-xCðgjÞ; andW ¼ CðgkÞ-yCðgiÞ-yxCðgjÞ: Now y�ygi

resyCðgiÞW y�a ^ y�yxgj

resyxCðgjÞW ðyxÞ�b is in the

image of the map y�ygiyxgj

from Ext�DðW ÞðS;SygiyxgjÞ to Ext�DðW ÞðS;RÞ; and so if we

apply p�gk; this can only be nonzero when gk ¼ ygi

yxgj: But each x determines

a unique k and double coset CðgkÞyðCðgiÞ-xCðgjÞÞ for which this holds. So

we may take k ¼ kðxÞ; y ¼ yðxÞ; and then yCðgiÞ-yxCðgjÞpCðgkÞ; so W ¼yCðgiÞ-yxCðgjÞ: &

Finally, we show how HH�ðRÞ fits into the construction of Section 2. For eachgAG; let AðgÞ :¼ Ext�DðCðgÞÞðS;S %gÞ: The conjugation maps ch ¼ h� yield isomorph-

isms of k-modules AðgÞ B-AðhgÞ: Define mg;h :;AðgÞ � AðhÞ-AðghÞ by

mg;hðag; bhÞ ¼ corCðghÞCðgÞ-CðhÞðp

�ghðy

�gres

CðgÞCðgÞ-CðhÞag ^ y�hres

CðhÞCðgÞ-CðhÞbhÞÞ:

Then properties (H1)–(H3) of Section 2 are straightforward. A comparison ofCorollary 2.5 and Theorem 3.16 shows that property (H40) holds, so that Theorem

2.2(ii) yields a product on AG: By (2.4), Corollary 2.5, Lemma 3.5 and Theorem 3.16,there is an algebra isomorphism

AGDMt

i¼1

ExtDðCðgiÞÞðS;SgiÞDHH�ðRÞ:

4. Representations of abelian extensions of Hopf algebras

In this section we prove a product formula (Theorem 4.8) for the Grothendieckring of modules for an abelian extension of Hopf algebras. As a consequence, this


Grothendieck ring satisfies the properties of Section 2, and its product formulacoincides with that of Theorem 2.2(ii). This generalizes a known formula formodules of the Drinfel’d double of a finite group (or equivalently of Hopf bimodulesof the group algebra [12]). Related results are in [12,36].

We assume in this section that k is an algebraically closed field as we will useSchur’s Lemma, but we do not place any restrictions on the characteristic of k: Allour modules will be finite dimensional.

Let L be a finite group acting on another finite group G: This induces an action of

L on the linear dual ðkGÞ� of the group algebra kG: For each gAG; let pgAðkGÞ�denote the function dual to g in the basis G of kG; that is

pgðhÞ :¼ dg;h ðhAGÞ:

We have xpg ¼ pxg for all xAL; gAG: Let s : L � L-ððkGÞ�Þ� and t : L-

ððkGÞ�Þ� � ððkGÞ�Þ� be maps giving rise to a (quasi) Hopf algebra H :¼ðkGÞ�#t

sL (or ðkGÞ�#tskL) as follows. As an algebra, H is the crossed

product ðkGÞ�#sL defined in Section 3. Again we write pg %x :¼ pg#x to shorten

the notation; ðkGÞ� will be a Hopf subalgebra of H; whereas kL will not. For eachx; yAL; let

sðx; yÞ ¼XgAG

sgðx; yÞpg; and tðxÞ ¼X

g;hAG

tg;hðxÞpg#ph; ð4:1Þ

for scalars sgðx; yÞ and tg;hðxÞ: Then the product in H may be written

ðpg %xÞ � ðph %yÞ ¼ dg;xhsgðx; yÞpgxy:

Define the coproduct by2

Dðpg %xÞ :¼X

h;kAGhk¼g

th;kðxÞph %x#pk %x:

In order that H ¼ ðkGÞ�#tsL be a (quasi) Hopf algebra, t (as well as s) must satisfy

certain properties. See [3] for the general case of a Hopf algebra, [25] for (cocentral)abelian extensions in particular, [10] for general facts about quasi Hopf algebras, andthe references given at the end of the following example for a special case.

Example 4.2. Let G act on itself by conjugation. Let o : G � G � G-k� be anythree-cocycle, that is

oða; b; cÞoða; bc; dÞoðb; c; dÞ ¼ oðab; c; dÞoða; b; cdÞ

for all a; b; c; dAG: Assume that o is normalized so that oða; b; cÞ is equal to 1

whenever one of a; b; or c is 1. There is an associated two-cocycle s : G � G-ðkGÞ�

ARTICLE IN PRESS

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S.J. Witherspoon / Advances in Mathematics 185 (2004) 136–158 151

given by (4.1) where

sgðx; yÞ ¼ oðg; x; yÞoðx; y; ðxyÞ1gxyÞ

oðx; x1gx; yÞ ;

and an associated function t : G-ðkGÞ� � ðkGÞ� given by (4.1) where

tg;hðxÞ ¼oðg; h; xÞoðx; x1gx; x1hxÞ

oðg; x; x1hxÞ :

The twisted Drinfel’d (or quantum) double DoðGÞ :¼ ðkGÞ�#tsG is a quasi Hopf

algebra. In case o is a coboundary, this is isomorphic to the Drinfel’d double DðGÞ :¼ ðkGÞ�#G; a Hopf algebra. These (quasi) Hopf algebras and their representationsappear in [4,19,27,36,37].

We will first apply Clifford theory to the crossed products H ¼ ðkGÞ�#tsL to

obtain a description of all simple H-modules. Such a description was first found in[25] by more direct methods in case H is semisimple. Note that we do not need thecoalgebra structure of H to describe H-modules. Clifford theory for group-gradedrings generally is developed in [17,18], and more specifically for crossed products thetheory is in [16, Section 11C]. However we will use the terminology and notation of[29], recalling the needed results as we go.

Up to isomorphism, the simple ðkGÞ�-modules are the ideals kpg ðgAGÞ: Thestabilizer of the ðkGÞ�-module kpg in L is the subgroup of all xAL such thatxðkpgÞDkpg: This is then the subgroup

Lg ¼ fxAL j xg ¼ gg:

By (3.1), the function sg : G � G-k� defined by (4.1) restricts to a two-cocycle

Lg � Lg-k�: Therefore we may form the twisted group algebra

ksgLg :¼ k#sg

Lg;

that is the algebra with basis f %x j xALgg and multiplication %x � %y ¼ sgðx; yÞxy: This is

in fact isomorphic to the subalgebra pgLg of H generated by all pg %x (xALg), so no

confusion should arise from the choice of notation.For each gAG; we define the subalgebra of H;

Hg :¼ ðkGÞ�#sLg:

Let Hg#ðkGÞ�kpg be the Hg-module induced from the ðkGÞ�-module kpg: Let

E :¼ EndHgðHg#ðkGÞ�kpgÞop;


the endomorphism algebra of the module Hg#ðkGÞ�kpg; taken with multiplication

opposite that of composition of functions.3

Lemma 4.3. There is an algebra isomorphism EDksgLg:

Proof. There are isomorphisms of vector spaces

EndHgðHg#ðkGÞ�kpgÞDHomðkGÞ� ðkpg;Hg#ðkGÞ�kpgÞ

DM

xALg

HomðkGÞ� ðkpg; %x#kpgÞ:

Each such endomorphism is thus determined by the image of the element pg ¼1#pg; and this must be a linear combination of the %x#pg ðxALgÞ: The product of

the functions fx;fy; where fxðpgÞ ¼ %x#pg and fyðpgÞ ¼ %y#pg; is given by the

opposite of composition:

ðfy3fxÞðpgÞ ¼ fyð %x#pgÞ ¼ %xfyð1#pgÞ ¼ %xð %y#pgÞ ¼ sðx; yÞxy#pg:

As sðx; yÞ ¼P

hAG shðx; yÞph; and the tensor product is over ðkGÞ�; this is in fact

equal to sgðx; yÞxy#pg: Therefore ðfy3fxÞðpgÞ ¼ sgðx; yÞfxyðpgÞ: &

The endomorphism algebra EDksgLg plays a crucial role in the Clifford

correspondence. (See for example [38,39].) In order to use the results of [29], wherethe algebra E is not explicitly mentioned, we point out that

sgðx; yÞ ¼ s1g ðy1; x1Þs1

g ðx1; xÞs1g ðy1; yÞsgðy1x1; xyÞ;

as follows from (3.1). Letting aðx; yÞ ¼ sgðy1; x1Þ; this shows that sg is

cohomologous to a1: It may be checked that a is the cocycle of [29, Proposition

1.2] that was used for the Clifford correspondence there. As sg and a1 are

cohomologous, there is an algebra isomorphism ksgLg

B-ka1Lg; given in this case by

%x/s1g ðx1; xÞ %x:

Notice that any H-module must contain at least one of the simple ðkGÞ�-modules

kpg; on restriction to ðkGÞ�:Given such an H-module, its H-submodule generated by

pg is a direct sum of copies of kpxg (xAL) on restriction to ðkGÞ�; as phð %xpgÞ ¼dh;xgph %x: Therefore any simple H-module is a direct sum of copies of conjugates of

some kpg; on restriction to ðkGÞ�: For the Clifford correspondence, we will also

consider Hg-modules: Note that any simple Hg-module containing kpg on restriction

to ðkGÞ� is in fact a direct sum of copies of kpg; as Lg is the stabilizer of kpg:

The Clifford correspondence comes in two steps. In the first step, there is abijection between the set of (isomorphism classes of) simple ksg

Lg-modules

(that is E-modules), and the simple Hg-modules whose restriction to ðkGÞ� contains

ARTICLE IN PRESS

3We must take the opposite multiplication as we write our functions on the left.

S.J. Witherspoon / Advances in Mathematics 185 (2004) 136–158 153

kpg [29, Proposition 1.2]. The second step of the Clifford correspondence is

a bijection between the set of simple Hg-modules whose restriction to ðkGÞ�contains kpg; and the simple H-modules whose restriction to ðkGÞ� contains kpg

[29, Proposition 1.1]. This second step is simply given by tensor induction ofmodules. An explicit description of such H-modules is given in [29, Theorem 1.3]. Inour situation, it provides a new proof of the following proposition which appears as[25, Theorem 3.3] in case H is semisimple. The action described in the propositionbelow looks simpler than that of [29, Theorem 1.3] due to application of the algebraisomorphism ksg

LgDka1Lg described above.

Let g1;y; gt be a set of representatives of orbits of L on G: Write pi :¼ pgi;

si :¼ sgi; Li :¼ Lgi

; and Hi :¼ Hgi: By the above analysis, each simple H-module

contains one of kp1;y; kpt on restriction to ðkGÞ�: Thus each arises from a simpleksi

Li-module as stated.

Proposition 4.4 (Kashina–Mason–Montgomery). The simple H-modules are pre-

cisely the modules bVV :¼ H#Hiðpi#VÞ induced from Hi; where V is a simple ksi

Li-

module, and i ranges over f1;y; tg: The action of Hi on V 0 :¼ pi#V is given by

ðph %xÞðpi#vÞ ¼ dgi ;hpi# %x � v:

More generally, for any gAG and ksgLg-module U ; we let U :¼ H#Hg

ðpg#UÞ;where U 0 :¼ pg#U is the Hg-module given by ðph %xÞðpg#uÞ ¼ dg;hpg# %x � u: As the

idempotentP

yAL=Lgpyg acts as the identity on such a module, and as 0 on H-

modules corresponding to other orbits of L on G; each H-module is a sum of

indecomposable modules U corresponding to L-orbits on G: Each U is then inducedfrom an Hg-module of the form U 0 ¼ pg#U where U is a ksg

Lg-module. Thus we

obtain a decomposition of the category of H-modules into a product of thecategories of ksi

Li-modules (i ¼ 1;y; t). This yields an additive decomposition ofGrothendieck groups [25, Corollary 3.4]:

K0ðHÞDMt

i¼1

K0ðksiLiÞ: ð4:5Þ

As H is a Hopf algebra, the tensor product of modules induces a ring structure onK0ðHÞ; but the Grothendieck groups K0ðksi

LiÞ are not necessarily rings themselves.We will see next how the product on K0ðHÞ behaves with respect to the additivedecomposition (4.5), that is we will give a formula for the tensor product of twoH-modules corresponding to ksi

Li- and ksjLj-modules.

If H is not semisimple, we may alternatively replace K0ðHÞ by the representationring of H; that is the ring generated by isomorphism classes of H-modules withdirect sum for addition and tensor product for multiplication. The image of eachmodule in this ring is the sum of images of indecomposable modules, eachcorresponding to an L-orbit on G: Theorem 4.8 below governs the tensor product oftwo such indecomposable modules.


First suppose that g; hAG and x; yALg-Lh: As D is an algebra map, the following

relation holds:

sgðx; yÞshðx; yÞ ¼ sghðx; yÞt1g;hðxÞt1

g;hðyÞtg;hðxyÞ ð4:6Þ

(see [25, (4.8)]). That is, sgh is cohomologous to sg � sh on Lg-Lh: Therefore there is

an isomorphism

c : ksghðLg-LhÞ B

-ksg�shðLg-LhÞ ð4:7Þ

given by cð %xÞ ¼ tg;hðxÞ %x: This will be important in taking tensor products of

modules, as the tensor product of a ksgðLg-LhÞ-module with a ksh

ðLg-LhÞ-module

is naturally a ksg�shðLg-LhÞ-module. We will want to induce such a module to a

ksghLgh-module. This involves first applying isomorphism (4.7), and then applying

induction from the subalgebra ksghðLg-LhÞ to the algebra ksgh

Lgh:

Note that xLj ¼ Lxgj: We will use an arrow down ðkÞ to denote restriction of

modules to the indicated subalgebra of ksgLg; and an arrow up ðmÞ to denote tensor

induction of modules from a subalgebra to ksgLg: The following theorem does not

give a formula for decomposing the tensor product of two simple (respectively,indecomposable) modules fully into a direct sum of simple (respectively, indecompo-sable) components, although it is a first step towards such a decomposition.

Theorem 4.8. Let V be a ksiLi-module, W a ksj

Lj-module, and bVV ; bWW the

corresponding H-modules as described in Proposition 4.4. Then as H-modules,

bVV# bWWDXxAD

dUðxÞUðxÞ;

where D is a set of representatives of double cosets Li\L=Lj; and UðxÞ is the

ksgi ðxgj ÞLgiðxgjÞ-module

UðxÞ ¼ ðVkLi

Li-xLj#xWk

xLj

Li-xLjÞm

Lgi ðxgj ÞLi-xLj

:

We obtain a formula for the product in K0ðHÞ in terms of the additive

decomposition (4.5) by identifying dUðxÞUðxÞ with the isomorphic H-module dyUðxÞyUðxÞ wherek ¼ kðxÞ and y ¼ yðxÞ are chosen so that gk ¼ ygi

yxgj: Then the formula in the

theorem closely resembles the formulas in Corollary 2.5 and Theorem 3.16.

Proof of Theorem 4.8. First, we will check that the underlying ðkGÞ�-modules are

isomorphic. The Grothendieck ring of ðkGÞ�-modules is just the group algebra ZG:

The underlying ðkGÞ�-module of bVV (respectively, of bWW ) is dimV (respectively,dimW ) copies of the sum OðgiÞ of the elements in the orbit of gi (respectively, OðgjÞof gj). The underlying ðkGÞ�-module of dUðxÞUðxÞ is jLgiðxgjÞ : Li-xLjjðdimVÞðdimWÞcopies of the sum OðgiðxgjÞÞ of elements in the orbit of giðxgjÞ: In ZG; the product of


sums of orbit elements is

OðgiÞ � OðgjÞ ¼XxAD

jLgiðxgjÞ : Li-xLjjOðgiðxgjÞÞ:

(This follows from standard properties of the trace map for the L-algebra ZG:See for example [33].) Multiplying both sides by ðdimVÞðdimWÞ; we see that

the underlying ðkGÞ�-modules in the statement of the theorem are indeedisomorphic.

We will next identify the appropriate H-submodules of bVV# bWW ; keeping in mind

its underlying ðkGÞ�-module structure. For each xAD; we will describe the HgiðxgjÞ-

module generated by pgibVV#pxgj

bWW : Consider the map

ðVkLi

Li-xLj#xWk

xLj

Li-xLjÞm

Lgi ðxgj ÞLi-xLj

-HgiðxgjÞðpgibVV#pxgj

bWWÞ ð4:9Þ

given by %yðv#wÞ/ %yððpgi#vÞ#ðpxgj

#wÞÞ for y ranging over a set of coset

representatives of Li-xLj in LgiðxgjÞ: Note that distinct coset representatives y will

generate disjoint subspaces %yðpgibVV#pxgj

bWWÞ; and each has dimension

ðdimVÞðdimWÞ; so the above map is bijective. It may be checked that the actionof ksgi ðxgj Þ

LgiðxgjÞ on the left side of (4.9) corresponds to the action of HgiðxgjÞ on the

right-hand side of (4.9), using isomorphism (4.7). The induced H-module then hasdimension jL : Li-xLjjðdimVÞðdimWÞ; as desired. &

For each gAG; let AðgÞ :¼ K0ðksgLgÞ; the Grothendieck group of ksg

Lg-modules.

If H is not semisimple, we may alternatively take AðgÞ to be the additive groupgenerated by ksg

Lg-modules, with direct sum for addition. Additively, the

representation ring of H decomposes into a direct sum of such groups, analogousto (4.5). The following statements apply equally well to this ring.

We obtain an isomorphism pgLgB- pxgLxg of subalgebras of H for each xAL; via

the action of L (see (3.2)). This induces an isomorphism ksgLg

B-ksxg

Lxg: Thus we

have conjugation maps cg;x : AðgÞ B-AðxgÞ: Define mg;h : AðgÞ � AðhÞ-AðghÞ by the

following for a ksgLg-module Vg and a ksh

Lh-module Wh:

mg;hðVg;WhÞ ¼ ðVgkLg

Lg-Lh#Whk

Lh

Lg-LhÞmLgh

Lg-Lh:

Then properties (H1)–(H3) are straightforward, and (H40) is equivalent to theproduct formula of Theorem 4.8 (compare with Corollary 2.5). As cg;x ¼ id when

xACðgÞ; by (2.4) we have additive isomorphisms

ALDMt

i¼1

K0ðksiLiÞDK0ðHÞ;

and in fact the product on AL given by Theorem 2.2(ii) coincides with that on K0ðHÞ:


Acknowledgments

The author thanks Alejandro Adem, Don Passman, Arun Ram, and Frank Sottilefor helpful discussions.

References

[1] J. Alev, M.A. Farinati, T. Lambre, A.L. Solotar, Homologie des invariants d’une algebre de Weyl

sous l’action d’un groupe fini, J. Algebra 232 (2000) 564–577.

[2] M.S. Alvarez, Algebra structure on the Hochschild cohomology of the ring of invariants of a Weyl

algebra under a finite group, J. Algebra 248 (2002) 291–306.

[3] N. Andruskiewitsch, Notes on extensions of Hopf algebras, Canad. J. Math. 48 (1996) 3–42.

[4] P. Bantay, Orbifolds, Hopf algebras, and the moonshine, Lett. Math. Phys. 22 (1991) 187–194.

[5] D.J. Benson, Representations and Cohomology I: Basic Representation Theory of Finite Groups and

Associative Algebras, Cambridge University Press, Cambridge, 1991.

[6] D.J. Benson, Representations and Cohomology II: Cohomology of Groups and Modules, Cambridge

University Press, Cambridge, 1991.

[7] P.R. Boisen, The representation theory of fully group-graded algebras, J. Algebra 151 (1992)

160–179.

[8] S. Bouc, Hochschild constructions for Green functors, Comm. Algebra 31 (2003) 403–436.

[9] A. Caldararu, A. Giaquinto, S. Witherspoon, Algebraic deformations arising from orbifolds with

discrete torsion, math. KT/0210027, to appear in J. Pure Appl. Algebra.

[10] V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994.

[11] W. Chen, Y. Ruan, A new cohomology theory for orbifold, math.AG/0004129.

[12] C. Cibils, Tensor product of Hopf bimodules over a group algebra, Proc. Amer. Math. Soc. 125

(1997) 1315–1321.

[13] C. Cibils, A. Solotar, Hochschild cohomology algebra of abelian groups, Arch. Math. 68 (1997)

17–21.

[14] A. Connes, Noncommutative Geometry, Academic Press, New York, 1994.

[15] J. Cornick, On the homology of group graded algebras, J. Algebra 174 (1995) 999–1023.

[16] C.W. Curtis, I. Reiner, Methods of Representation Theory, with Applications to Finite Groups and

Orders, Vol. I, Wiley, New York, 1981.

[17] E.C. Dade, Compounding Clifford’s theory, Ann. Math. 91 (2) (1970) 236–290.

[18] E.C. Dade, Clifford theory for group-graded rings, J. Reine Angew. Math. 369 (1986) 40–86.

[19] R. Dijkgraaf, V. Pasquier, P. Roche, Quasi Hopf algebras, group cohomology and orbifold models,

Nucl. Phys. B (Proc. Suppl.) 18B (1990) 60–72.

[20] P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed

Harish-Chandra homomorphism, Invent. Math. 147 (2) (2002) 243–348.

[21] L. Evens, The Cohomology of Groups, Oxford University Press, Oxford, 1991.

[22] B. Fantechi, L. Gottsche, Orbifold cohomology for global quotients, math.AG/0104207.

[23] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math. 78 (1963) 267–288.

[24] Y. Kashina, Classification of semisimple Hopf algebras of dimension 16, J. Algebra 232 (2000)

617–663.

[25] Y. Kashina, G. Mason, S. Montgomery, Computing the Frobenius-Schur indicator for abelian

extensions of Hopf algebras, preprint.

[26] M. Lorenz, On the homology of graded algebras, Comm. Algebra 20 (2) (1992) 489–507.

[27] G. Mason, The quantum double of a finite group and its role in conformal field theory, in Groups

’93, London Math. Soc. Lecture Notes, Vol. 212, Cambridge University Press, Cambridge 1995,

pp. 409–417.

[28] S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series in

Mathematics, Vol. 82, Amer. Math. Soc., Providence, RI, 1993.


[29] S. Montgomery, S.J. Witherspoon, Irreducible representations of crossed products, J. Pure Appl.

Algebra 129 (1998) 315–326.

[30] K. Sanada, On the Hochschild cohomology of crossed products, Comm. Algebra 21 (8) (1993)

2727–2748.

[31] S.F. Siegel, S.J. Witherspoon, The Hochschild cohomology ring of a group algebra, Proc. London

Math. Soc. 79 (3) (1999) 131–157.

[32] D. -Stefan, Hochschild cohomology on Hopf Galois extensions, J. Pure Appl. Algebra 103 (1995)

221–233.

[33] J. Thevenaz, G-Algebras and Modular Representation Theory, Oxford University Press, Oxford,

1995.

[34] B. Uribe, Orbifold cohomology of the symmetric product, math.AT/0109125.

[35] C. Vafa, E. Witten, On orbifolds with discrete torsion, J. Geom. Phys. 15 (1995) 189–214.

[36] S.J. Witherspoon, The representation ring of the quantum double of a finite group, J. Algebra 179

(1996) 305–329.

[37] S.J. Witherspoon, The representation ring of the twisted quantum double of a finite group, Canad.

J. Math. 48 (1996) 1324–1338.

[38] S.J. Witherspoon, Clifford correspondence for finite dimensional Hopf algebras, J. Algebra 218

(1999) 608–620.

[39] S.J. Witherspoon, Clifford correspondence for algebras, J. Algebra 256 (2002) 518–530.


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