Some properties of group-theoretical categories

SOME PR OPER TIES OF GR OUP-THEORETICAL CA TEGORIES SHLOMO GELAKI AND DEEP AK NAIDU Abstract. W e first show that ev ery group-theoretical catego ry is gr aded b y a certain double coset ring. As a conse quence, w e obtain a necessary and suffi- cien t condition for a group-theoretica l category to b e ni l potent. W e then giv e an explicit description of the simple ob jects in a group-theoretical catego ry (following [O2]) and of the group of in v ertible ob j ects of a group-the oretical category , in group-theoretica l terms. Finall y , under c ertain restrictive condi- tions, w e describ e the univ ersal grading group of a group-theoretical category . 1. Introduction Group-theore tica l categorie s w ere introduced and studied in [ENO] and [O1]. They co ns titute a fundamental class of fusion categories which are defined, as the name suggests, by a certain finite group data. F or example, fo r a finite group G its representation c ategory Rep( G ) is group-theor etical. As an indica tion of the cen trality of group-theor etical catego ries in the theory of fusion categories w e men tion the following obs e r v ation: all known co mplex semisimple Hopf algebras (as far as we kno w) ha ve g roup-theoretica l representation categor ies. In fact, it was asked in [ENO] whether it is true that an y complex se misimple Hopf alg e bra is group-theor etical. It is thus highly desirable to study group-theore tica l categor ies and understand as m uch as possible ab out them in the language of group theory . The notion of a nilpotent fusion category was introduced and studied in [GN]. F or e xample, it is not hard to show tha t if G is a finite group then Rep( G ) is nilpo ten t if and o nly if G is nilp otent. In [DGN O ] nilpotent modula r categories are studied, and in particular it is dis cussed when they are g roup-theor e tical. Ther efore a very natural question ar is es: what are necessary a nd sufficient conditions fo r a group-theor etical catego ry to b e nilp otent? The answer to this question is one of the main results of this pap e r (see Corollary 4.3). Other imp ortant inv ar ia nt s of a fusion categor y C are its p ointed sub categor y C pt (the subca tegory generated b y the gro up of inv ertible ob jects in C ), its adjoint sub c ategory C ad [ENO] a nd it s un iversal gra ding group U ( C ) [GN]. Descriptions of C pt for a general g r oup-theoretica l category C , and C ad , U ( C ) for a sp ecial c lass o f group-theor etical categor ies are other results of this pa per (see Theore m 5.2 and Prop osition 6.3). The organizatio n of the pap er is as follows. Section 2 contains necess ary prelimi- naries ab out fusion categories, mo dule categor ies, and gr oup-theoretical categories. W e als o r ecall some definitions from [GN] concerning nilp otent fusion c ategories and based rings. W e also recall some basic definitions and results from gro up theory . In Section 3 w e int ro duce the notion of a fusion categor y gr aded by a ba sed ring. Let H b e a subgroup of a finite group G . W e introduce a based ring which w e call Date : No vem ber 1, 2018. 1 2 SHLOMO GE LAKI AND DEEP AK NAIDU double c oset ring a rising from the set H \ G/H of double cosets o f H in G . W e give a necessary and sufficient condition for the double coset ring to b e nilp otent (see Prop osition 3.7). In Section 4 we first sho w that every g roup-theoretica l categor y is gr a ded by a certain double coset ring. As a consequence, we o btain a necessary a nd sufficien t condition for a group-theoretica l categ ory to be nilp otent. In Section 5 we give an explicit des cription o f the simple ob jects in a gro up- theoretical categor y (following Pr op osition 3 .2 in [O2]; see Theorem 5.1) and of the group o f in vertible o b jects of a gro up-theoretical ca tegory , in group-theoretical terms. In Section 6, we describ e the universal grading group of a gro up-theoretical category , under certain restrictive c onditions. Ac kno wledgment s. Part of this work was done while the first author was on Sabbatical in the departmen ts of mathematics a t the University of New Hampshire and MIT; he is grateful for their warm hospitality . The research of the first author was partially suppo rted b y the Isr ael Science F oundation (gr ant No. 125/ 05). The authors w ould like to tha nk P . Etingof and D. Niksh ych for useful discus s ions. 2. Prel iminaries 2.1. F usion categories and their mo dule categories. Throughout this pap er we w ork ov er an alg ebraically closed field k of c har- acteristic 0. All categories cons idered in this work are ass umed to be k -linea r and semisimple with finite dimensional Hom-spaces and finitely many isomor phis m classes of s imple ob jects. All functor s are ass umed to b e additiv e and k -linear . Unless otherwise stated a ll c o c y cles appearing in this work will have co efficients in the trivial mo dule k × . A fusion c ate go ry ov er k is a k -linear semisimple rigid tensor ca teg ory with finitely many isomorphism c la sses of s imple ob jects and finite dimensiona l Hom-spaces such that the neutral ob ject is simple [ENO]. A fusion category is said to be p ointe d if all its simple o b jects are inv ertible. A t ypical example of a p ointed catego ry is V ec ω G - the category of finite dimensional vector spaces over k graded by the finite group G . The morphisms in this catego ry are linear tr ansformations that respect the grading and the associa tivit y constraint is given by the normalized 3-co cycle ω o n G . Let C = ( C , ⊗ , 1 C , α, λ, ρ ) be a tensor catego ry , w he r e 1 C , α , λ , and ρ are the unit o b ject, the asso ciativity constraint, the left unit c onstraint, and the right unit constraint, r esp ectively . A right mo dule c ate gory ov er C (see [O1] a nd references therein) is a categor y M together with an exact bifunctor ⊗ : M × C → M and natural iso morphisms µ M , X, Y : M ⊗ ( X ⊗ Y ) → ( M ⊗ X ) ⊗ Y , τ M : M ⊗ 1 C → M , for all M ∈ M , X, Y ∈ C , suc h that the following t wo equations hold for all M ∈ M , X, Y , Z ∈ C : µ M ⊗ X, Y , Z ◦ µ M , X, Y ⊗ Z ◦ (id M ⊗ α X,Y ,Z ) = ( µ M , X, Y ⊗ id Z ) ◦ µ M , X ⊗ Y , Z , ( τ M ⊗ id Y ) ◦ µ M , 1 C , Y = id M ⊗ λ Y . Let ( M 1 , µ 1 , τ 1 ) and ( M 2 , µ 2 , τ 2 ) b e tw o right mo dule catego ries ov er C . A C - mo dule functor from M 1 to M 2 is a functor F : M 1 → M 2 together with natural isomorphisms γ M , X : F ( M ⊗ X ) → F ( M ) ⊗ X , for all M ∈ M 1 , X ∈ C , such that SOME PROPER TIES OF GROUP-THEORETICAL CA TEGORIES 3 the following tw o equations hold for all M ∈ M 1 , X, Y ∈ C : ( γ M , X ⊗ id Y ) ◦ γ M ⊗ X, Y ◦ F ( µ 1 M , X, Y ) = µ 2 F ( M ) , X , Y ◦ γ M ,X ⊗ Y , τ 1 F ( M ) ◦ γ M , 1 C = F ( τ 1 M ) . Two mo dule ca tegories M 1 and M 2 ov er C are e quivalent if there exists a mo dule functor fro m M 1 to M 2 which is a n equiv alence of categor ies. F or tw o mo dule categorie s M 1 and M 2 ov er a tensor category C their dir e ct sum is the catego ry M 1 ⊕ M 2 with the obvious module ca teg ory structure. A mo dule catego ry is inde c omp osable if it is not equiv alen t to a direct sum of t w o non-trivia l mo dule categorie s. Let M 1 and M 2 be tw o r ight mo dule categories ov er a tensor category C . Let ( F 1 , γ 1 ) and ( F 2 , γ 2 ) be mo dule functors fro m M 1 to M 2 . A natur al mo dule tr ansformation from ( F 1 , γ 1 ) to ( F 2 , γ 2 ) is a na tural transfor mation η : F 1 → F 2 such that the following equation holds for all M ∈ M 1 , X ∈ C : ( η M ⊗ id X ) ◦ γ 1 M , X = γ 2 M , X ◦ η M ⊗ X . Let C b e a tenso r category and let M b e a right module categ ory over C . The dual c ate gory of C with resp ect to M is the category C ∗ M := F un C ( M , M ) who se ob jects are C -mo dule functors from M to itself and mor phisms are natural mo dule transformatio ns. T he category C ∗ M is a tens o r catego ry with tensor pr o duct being comp osition of module f unctors. It is known that if C is a fusion c ategory and M is a se misimple k -linear indecomposable module categor y ov er C , then C ∗ M is a fusion category [ENO]. Two fusion categories C and D are said to b e we akly Morita e quivalent if there exists an indecomp osable (semisimple k -linear) right mo dule ca tegory M ov er C such that the categories C ∗ M and D are equiv alent as fusion categories . It was shown by M¨ uger [Mu] that this is indeed an equiv alence relation. Consider the fusion ca tegory V ec ω G , where G is a finite gr oup and ω is a nor - malized 3-co cycle on G . Let H b e a subgr oup of G such that ω | H × H × H is coho- mologically trivial. Let ψ be a 2-cochain in C 2 ( H, k × ) satisfying ω | H × H × H = dψ . The t wisted gro up a lgebra k ψ [ H ] is an asso ciative unital algebra in V ec ω G . Define C = C ( G, ω , H , ψ ) to b e the ca teg ory o f k ψ [ H ]-bimo dules in V ec ω G . Then C is a fusion category with tensor pro duct ⊗ k ψ [ H ] and unit ob ject k ψ [ H ]. Categorie s o f the form C ( G, ω , H , ψ ) a r e known as gr oup-the or etic al [ENO, Def- inition 8.40], [O 2]. It is known that a fusion c a tegory C is group- theo retical if and only if it is weakly Morita eq uiv alent to a p ointed categor y with re spe c t to some indecomp osable mo dule c ategory [ENO, Prop osition 8.42]. More prec isely , C ( G, ω, H, ψ ) is equiv alent to (V ec ω G ) ∗ ( H,ψ ) . 2.2. Nil p oten t based rings and nil p oten t fusion categories. Let Z + be the semi-ring of non-negative integers. Let R b e a ring with identit y which is a finite rank Z -mo dule. A Z + -b asis o f R is a basis B such that for all X , Y ∈ B , X Y = P Z ∈ B n Z X, Y Z, wher e n Z X, Y ∈ Z + . An element of B will be ca lled b asic . Define a non-degene r ate symmetric Z -v alued inner pro duct on R as follows. F o r all elemen ts X = P Z ∈ B a Z Z a nd Y = P Z ∈ B b Z Z o f R w e set (1) ( X, Y ) = X Z ∈ B a Z b Z . 4 SHLOMO GE LAKI AND DEEP AK NAIDU Definition 2 . 1 ([O1]) . A b ase d ring is a pa ir ( R, B ) c o nsisting of a ring R (with ident ity 1 ) with a Z + -basis B satisfying the following pr o pe r ties: (1) 1 ∈ B . (2) There is an inv olution X 7→ X ∗ of B such tha t t he induced map X = P W ∈ B a W W 7→ X ∗ = P W ∈ B a W W ∗ satisfies ( X Y , Z ) = ( X , Z Y ∗ ) = ( Y , X ∗ Z ) for all X , Y , Z ∈ R . By a b ase d subring of a based ring ( R , B ) w e will mean a bas ed ring ( S, C ) wher e C is a subset of B and S is a subring of R . Let us recall some definitions from [GN]. Let R = ( R, B ) b e a based ring and let C be a fusion c a tegory . Let R ad denote the based subring of R generated by all basic elements of R contained in X X ∗ , X ∈ B . Let R (0) := R , R (1) := R ad , and R ( i ) := ( R ( i − 1) ) ad , for every p ositive integer i . Similarly , let C ad denote the full fusion sub categor y of C generated by all simple sub o b jects of X ⊗ X ∗ , X a simple ob ject of C . Let C (0) := C , C (1) := C ad , and C ( i ) := ( C ( i − 1) ) ad , for ev ery p o sitive integer i . R is said to be nilp otent if R ( n ) = Z 1, for some n . The smallest n for whic h this happ ens is called the nilp otency class of R and is denoted by cl ( R ). C is said to be nilp otent if C ( n ) ∼ = V ec, for some n . The smallest n for whic h this happ ens is called the nilp otency class of C a nd is denoted b y cl ( C ). Note that a fusion categ o ry is nilp otent if and o nly if its Gr o thendieck ring is nilpo ten t. Also note that for any finite g roup G , the fusion ca tegory Rep( G ) of representations o f G is nilpotent if and only if the group G is nilpotent. Let C be a fusion category . W e ca n view C a s a C ad -bimo dule category . As such, it decomp ose s int o a direct s um of indecomp osable C ad -bimo dule categ ories: C = ⊕ a ∈ A C a , where A is the index set. It w a s s hown in [GN] that there is a canonical gro up s tructure o n the index set A. This group is called the u niversal gr ading gr oup of C a nd is denoted b y U ( C ). Every fusion categ ory is faithfully graded (in the sense of [ENO, Definition 5.9]) b y its univ ersal gr a ding group. 2.3. Some defini tions and results from group theo ry. The following definitions and results are con ta ined in [R]. Let H b e a subgroup of a group G . The subgro up H is said to b e subnormal in G if there exist subgroups H 1 , · · · H n − 1 of G such that H = H 0 ✂ H 1 ✂ · · · ✂ H n − 1 ✂ H n = G. F or any non- e mpty subsets X and Y of G , let X Y denote the subg roup generated by the set { y xy − 1 | x ∈ X , y ∈ Y } . Define a sequence o f s ubg roups H ( G, i ) , i = 0 , 1 , . . . , of G b y the r ules H ( G, 0) := G a nd H ( G, i +1) := H H ( G, i ) . So w e get the follo wing sequence G = H ( G, 0) ☎ H ( G, 1) ☎ H ( G, 2) ☎ · · · . Note that H ( G, 1) is the normal clos ure of H in G . The abov e sequence is called the series of suc c essive normal closur e of H in G . It is known that H is subnormal in G if and only if H ( G, n ) = H for some n ≥ 0. If H is subnorma l in G , the smallest n for whic h H ( G, n ) = H is called the defe ct of H in G . SOME PROPER TIES OF GROUP-THEORETICAL CA TEGORIES 5 Suppo se G is finit e. Then it is kno wn that G is nilp otent if and only if any subgroup o f G is subnormal in G . It is a lso known that if H is nilp otent and is subnormal in G , then the nor mal closure of H in G is nilp otent. Indeed, it can be shown that if H is nilp otent and is s ubnormal in G , then H is contained in the Fitting s ubgroup Fit( G ) of G (= the unique la rgest nor mal nilp otent subgro up of G ), and hence the normal closure of H in G m ust be nilp otent. 3. Fusion ca tegories graded by based rings and double coset rings In this section we define the notio n of a fusion c ategory graded by a based ring (generalizing the notio n of a fusion ca tegory gra ded by a finite g roup). W e then define th e double coset bas ed ring and give a necessary and sufficient condition for it to be nilp otent. 3.1. F usion categories graded b y based rings. Definition 3. 1. A fusio n catego ry C is said to b e gr ade d by a base d ring ( R, B ) if C decomp oses into a direct sum of full ab elian sub categ ories C = ⊕ X ∈ B C X such that ( C X ) ∗ = C X ∗ and C X ⊗ C Y ⊆ ⊕ Z ∈{ W ∈ B | W is contained in X Y } C Z , for all X, Y ∈ B . Remark 3.2. Note that the trivial compo nent C 1 is a fusion subca tegory of C . Let C b e a fusion catego ry whic h is gr a ded b y a based ring ( R, B ). Definition 3 . 3. F or a ny subcategor y D ⊆ C , define its su pp ort Supp( D ) := { X ∈ B | D ∩ C X 6 = { 0 }} . W e will say that C is fai thful ly gra ded b y ( R , B ) if C X 6 = { 0 } and Supp( C X ⊗ C Y ) = { W ∈ B | W is contained in X Y } , for all X, Y ∈ B . Remark 3.4. (i) Every fusion ca tegory is faithfully gra ded by its Grothendieck ring. (ii) Every fusio n categ o ry that is g r aded by a gro up G is gr aded by the ba sed ring ( Z G, G ). Recall that for any fusio n categ o ry C , C ad denotes the full fusion sub c ategory of C genera ted by all simple sub ob jects of X ⊗ X ∗ , X a simple ob ject o f C ; C (0) = C , C (1) = C ad , and C ( i ) = ( C ( i − 1) ) ad for every p ositive integer i . Also recall that for an y based r ing ( R, B ), R ad denotes the based subr ing of R g enerated by all basic elements o f R contained in X X ∗ , X ∈ B ; R (0) = R , R (1) = R ad , and R ( i ) = ( R ( i − 1) ) ad for every p ositive integer i . Prop ositio n 3.5. L et C b e a fusion c ate gory that is faithful ly gr ade d by a b ase d ring R = ( R , B ) . Then C is nilp otent if and only if R is nilp otent and the trivial c omp onent C 1 is nilp otent. If C is nilp otent, then its nilp otency class cl ( C ) satisfies the fol lowing ine qu ality: cl ( R ) ≤ cl ( C ) ≤ c l ( R ) + cl ( C 1 ) . Pr o of. Since the grading of C b y R is faithful, we ha ve Supp( C ( i ) ) = B ∩ R ( i ) for an y non-negative integer i . Indeed, note that even without faithfulness of the grading we hav e Supp( C ( i ) ) ⊆ B ∩ R ( i ) . F a ithfulness of the grading implies that B ∩ R ( i ) ⊆ Supp( C ( i ) ). Now supp ose that C is nilpo ten t of nilp otency class n . Then the trivial co mpone nt C 1 being a fusion sub category of C is nilpotent. Also, Supp( C ( n ) ) must b e eq ual to { 1 } . It follows that R must b e nilp otent. Conv ersely , suppo se that the trivia l comp onent C 1 is nilp otent and R is nilp otent of nilp otency class n . The n C ( n ) ⊆ C 1 and it fo llows that C m ust b e nilpo ten t. The statement ab out nilp otency cla s s should b e evident and the prop osition is prov ed. 6 SHLOMO GE LAKI AND DEEP AK NAIDU 3.2. The doubl e coset ring. Let H be a subgroup of a finite g roup G . Let R ( G, H ) denote the free Z -mo dule generated by the s et O of double cosets of H in G . F or any H xH, H y H ∈ O , the set H xH y H is a union of do uble co sets. Define the pro duct H xH · H y H by H xH · H y H := X H z H ∈O N H z H H xH , H y H H z H , where N H z H H xH , H y H = ( 1 if H z H ⊆ H xH y H , 0 otherwise . This multiplication rule on O extends, by linearity , to a m ultiplicatio n rule on R ( G, H ). The ident ity element of R ( G, H ) is g iven by the tr ivial double c o set H = H 1 G H . Ther e is a n inv olution ∗ on the set O defined as follows. F o r a ny H xH ∈ O , define ( H xH ) ∗ := H x − 1 H . It is stra ightf orward to c heck that R ( G, H ) is a based ring. Let S be a based subring of R ( G, H ). Define Γ S := [ X ∈S ∩O X . Note that Γ S is a subgr oup of G that contains H . Also note that Γ R ( G, H ) = G . Lemma 3.6 . The assignment S 7→ Γ S is a b ije ction b et we en the set of b ase d subrings of the double c oset ring R ( G, H ) and the set of sub gr oups of G c ontaining H . Pr o of. Let K b e a subgr oup of G that contains H . The do uble coset ring R ( K , H ) is a based subring of R ( G, H ). It is evident that the a ssignment K 7→ R ( K, H ) is inv erse to the a ssignment defined in the s ta temen t of the lemma. Prop ositio n 3.7 . The double c oset ring R ( G, H ) is nilp otent if and only if H is subnormal in G . If R ( G, H ) is nilp otent , then its nilp otency class is e qual to t he defe ct of H in G . Pr o of. Let R = R ( G, H ). Observe that Γ R ( i ) = H ( G, i ) , for all non-negative inte- gers i (see Subsection 2 .3 for the de finitio n of H ( G, i ) ). Note that R is nilp otent if and only if H ( G, n ) = H for some non-negative integer n . The latter co ndition is equiv alent to the condition that H is subnormal in G . Recall tha t if H is subnor mal in G , then the defect of H in G is defined to be the smalle s t no n- negative integer n such that H ( G, n ) = H . It follows that if R is nilp otent, then its nilp otency class is equa l to the defect of H in G . 4. Nil potency of a gr o up-theoretical ca tegor y In this section we give a necessa ry and sufficient condition for a gr o up-theoretical category to b e nilp otent. W e start with the following theorem. Theorem 4.1. L et C = C ( G, ω , H , ψ ) b e a gr oup-t he or etic al c ate gory. Then C is faithful ly gr ade d by the double c oset ring R ( G, H ) , with the trivial c omp onent b eing the re pr esentation c ate gory R ep ( H ) of H . SOME PROPER TIES OF GROUP-THEORETICAL CA TEGORIES 7 Pr o of. It follows from the res ults in [O 2] that the set of is o morphism class e s of simple ob jects in C are parametriz ed by pairs ( a, ρ ), wher e a ∈ G is a representative of a double coset X := H aH o f H in G (i.e., a basic elemen t X in R ( G, H )) and an irreducible pro jectiv e repre s en tation of H a := H ∩ aH a − 1 with a certain 2 − co cycle. Moreov er, the tensor pro duct of tw o simple o b jects X , Y , corresp onding to ( a, ρ ), ( b, τ ), re spec tiv ely , is supp orted on the union of the do uble cosets app earing in the decomp osition o f X Y . Therefore if we let C X , X := H aH , be the sub catego ry of C gener ated b y all simple ob jects whic h corresp ond to pairs ( a, ρ ), we get that C = ⊕ X C X , as required. It is clear that C H = Rep( H ). Remark 4.2. W e note that if N is the no rmal closur e of H in G then the group ring Z [ G/ N ] is a homomor phic image of R ( G, H ). Hence the gr oup-theoretica l category C = C ( G, ω , H , ψ ) is G/ N − graded. Corollary 4.3. L et C = C ( G, ω , H , ψ ) b e a gr oup-the or etic al c ate gory. Then C is nilp otent if and only if the normal closur e of H in G is nilp otent. If C is nilp otent, then its nilp otency class cl ( C ) satisfies the fol lowing ine quality: cl ( H ) ≤ cl ( C ) ≤ cl ( H ) + ( defe ct of H in G ) . Pr o of. By Theorem 4.1 and P rop osition 3 .5, it follows that C is nilpotent if a nd only if the double coset ring R ( G, H ) is nilpotent and H is nilp otent. By P r op osition 3.7, R ( G, H ) is nilp otent if and only if H is subnormal in G . Since G is a finite group, it follows from the r emarks in Subsectio n 2.3 that H is nilpo ten t and is subnormal in G if a nd o nly if the normal clo sure o f H in G is nilp otent . The statement ab out the nilp otency clas s of C follo ws immediately from Prop os itio n 3.5 and P r op osition 3.7. Example 4 .4. Let G b e a finite group and let ω b e a 3-co cycle o n G . It was shown in [O2] that the repre sent ation ca tegory Re p( D ω ( G )) of the twisted quantum double of G is equiv alent to C ( G × G, ˜ ω , ∆( G ) , 1), where ˜ ω is a certain 3- co cycle on G × G and ∆( G ) is the diago nal subgroup of G . It follows fr om Corollary 4.3 that Rep ( D ω ( G )) is nilp otent if and only if G is nilp otent. 5. The pointed s ubca tegor y of a gr oup-theoretical ca tegor y In this section w e des crib e the simple o b jects in a gro up-theoretical catego ry and then des crib e the gr oup of inv er tible ob jects in a g roup-theore tical categ ory . 5.1. Simp l e ob jects in a group-theoretical category. Let C = C ( G, ω , H , ψ ) b e a gro up-theoretical category . Let R = { u ( X ) | X ∈ H \ G/ H } be a set of representativ es of double cosets o f H in G . W e assume that u ( H 1 G H ) = 1 G . In [O2] it is explained how a simple ob ject in C giv es r is e to a pair ( g , ρ ), where g ∈ R and ρ is the isomorphis m class of an irreducible pro jective representation ρ o f H g with a cer tain 2- c o cycle ψ g . Let us recall this in details. F or each g ∈ G , let H g := H ∩ g H g − 1 . The group H g has a well-defined 2 -co cycle ψ g defined by ψ g ( h 1 , h 2 ) := ψ ( h 1 , h 2 ) ψ ( g − 1 h − 1 2 g , g − 1 h − 1 1 g ) ω ( h 1 , h 2 , g ) ω ( h 1 , h 2 g , g − 1 h − 1 2 g ) ω ( h 1 h 2 g , g − 1 h − 1 2 g , g − 1 h − 1 1 g ) . Let B = ⊕ g ∈ G B g be an ob ject in C . So B is equipp ed with is omorphisms l h, g : B g ∼ − → B hg and r g, h : B g ∼ − → B gh , g ∈ G, h ∈ H . T he s e isomorphisms satisfy 8 SHLOMO GE LAKI AND DEEP AK NAIDU the fo llowing identities: ω ( h 1 , h 2 , g ) ψ ( h 1 , h 2 ) l h 1 h 2 , g = l h 1 , h 2 g ◦ l h 2 , g , ψ ( h 1 , h 2 ) r g, h 1 h 2 = ω ( g , h 1 , h 2 ) r gh 1 , h 2 ◦ r g, h 1 and l h 1 , gh 2 ◦ r g, h 2 = ω ( h 1 , g , h 2 ) r h 1 g, h 2 ◦ l h 1 , g . The ab ov e three identities say tha t B is a left k ψ [ H ]-mo dule, B is a right k ψ [ H ]- mo dule, and that the left and rig ht mo dule structures o n B comm ute, resp ectively . It is clear that B is a direct sum of subbimodules supp o rted on individual double cosets of H in G . Suppo se B contains a s ubbimodule that is s uppor ted on a double coset r epresented by g . Then o ne get a pro jectiv e represe n tation ρ : H g → GL ( V ) with 2 -co cycle ψ g defined as fo llows. Let V := B g and (2) ρ ( h ) := r hg, g − 1 h − 1 g ◦ l h, g , h ∈ H g . The following theor em, sta ted in [O 2], asser ts that the a bove corresp ondence gives a bijection b etw een isomor phis m classes o f simple ob jects in C and iso mo r- phism classes of pairs ( g , ρ ). W e shall give an alterna tiv e proo f of the in verse corres p ondence by a direc t computation. Theorem 5. 1. The ab ove c orr esp ondenc e defines a bije ction b et we en isomorphism classes of simple obje cts in C and isomorphism classes of p airs ( g , ρ ) , whe r e g ∈ R and ρ is an irr e ducible pr oje ctive r epr esentation of H g with 2 − c o cycle ψ g . Pr o of. Given a pair ( g , ρ ), wher e g ∈ R and ρ : H g → GL ( V ) is an irr e ducible pro jective repr esentation with 2 -co cycle ψ g , we a s sign an ob ject B in C as follows. Let T b e a set of repre sent atives of H /H g . W e assume tha t 1 ∈ T . Let B := ⊕ t ∈ T ,k ∈ H B tgk , wher e ea c h comp onent is equal to V as a vector space. The right and left mo dule structur e s r and l , resp ectively , on B ar e defined as follows. (3) r tgk , h : B tgk ∼ − → B tgk h , v 7→ ψ ( k , h ) ω ( tg , k , h ) − 1 v . l h, tgk : B tgk ∼ − → B sg ( g − 1 pg ) h , v 7→ ψ ( h, t ) ψ ( s, p ) ψ ( g − 1 p − 1 g , g − 1 pg k ) × ω ( h, tg , k ) ω ( s, g, g − 1 pg ) ω ( h, t, g ) ω ( s, p, g ) × ω ( g , g − 1 pg , g − 1 p − 1 g ) ω ( g − 1 pg , g − 1 p − 1 g , g − 1 pg k ) ω ( sg , g − 1 pg , k ) ρ ( p )( v ) , (4) where s ∈ T and p ∈ H g are uniquely determined by the eq uation ht = sp . It is now straig h tforward to c heck that B is simple, and that the tw o corr esp ondences are inv erse to e ach other . 5.2. The group of inv erti ble ob jects in a group-theo retical category. F or any g ∈ N G ( H ) and f ∈ C n ( H, k × ), define g f ∈ C n ( H, k × ) by g f ( h 1 , · · · , h n ) := f ( g − 1 h 1 g , · · · , g − 1 h n g ) . SOME PROPER TIES OF GROUP-THEORETICAL CA TEGORIES 9 Pick any g 1 , g 2 ∈ N G ( H ) and let g 3 = g 1 g 2 k , k ∈ H . Define β ( g 1 , g 2 ) : H → k × , h 7→ ψ ( g − 1 2 g − 1 1 hg 1 g 2 k , g − 1 3 h − 1 g 3 ) ψ ( g − 1 1 h − 1 g 1 , g − 1 1 hg 1 ) ψ ( g − 1 2 g − 1 1 h − 1 g 1 g 2 , g − 1 2 g − 1 1 hg 1 g 2 k ) × ω ( g − 1 1 hg 1 , g − 1 1 h − 1 g 1 , g − 1 1 hg 1 ) ω ( g 1 , g − 1 1 hg 1 , g − 1 1 h − 1 g 1 ) ω ( g − 1 1 hg 1 , g 2 , k ) ω ( g 2 , g − 1 2 g − 1 1 hg 1 g 2 , k ) × ω ( g − 1 2 g − 1 1 hg 1 g 2 , g − 1 2 g − 1 1 h − 1 g 1 g 2 , g − 1 2 g − 1 1 hg 1 g 2 k ) ω ( g 2 , g − 1 2 g − 1 1 hg 1 g 2 , g − 1 2 g − 1 1 h − 1 g 1 g 2 ) ω ( g 2 , g − 1 2 g − 1 1 hg 1 g 2 k , g − 1 3 h − 1 g 3 ) . (5) It is straightf orward (but tedious) to verify that (6) ψ g 3 = d ( β ( g 1 , g 2 )) ψ g 1 ( g 1 ( ψ g 2 )) . Let K := { g ∈ R | g ∈ N G ( H ) and ψ g is co ho mologically trivia l } . F or any g 1 , g 2 ∈ K , define g 1 · g 2 := u ( g 1 g 2 ). It follows from (6) that with this pro duct rule K is a gro up that is isomor phic to a s ubgroup of N G ( H ) /H . F or each g ∈ K , fix η g : H → k × such that dη g = ψ g . W e take η 1 := β (1 , 1 ) − 1 . F or any g 1 , g 2 ∈ K , define (7) ν ( g 1 , g 2 ) := η g 1 ( g 1 η g 2 ) η g 1 · g 2 β ( g 1 , g 2 ) . Let b H := Hom( H , k × ) and define a gro up K ⋉ ν b H as follows. As a set K ⋉ ν b H = K × b H and for any ( g 1 , ρ 1 ) , ( g 2 , ρ 2 ) ∈ K ⋉ ν b H , define ( g 1 , ρ 1 ) · ( g 2 , ρ 2 ) = ( g 1 · g 2 , ν ( g 1 , g 2 ) ρ 1 ( g 1 ρ 2 )) . Theorem 5.2. The gr oup G ( C ) of isomorphism classes of invertible obje cts of C is isomorphi c to the gr oup K ⋉ ν b H c onstructe d ab ove. Pr o of. By Theorem 5.1, G ( C ) is in bijection with the s e t L = { ( g , ρ ) | g ∈ K, ρ : H → k × such that dρ = ψ g } . The set L b ecomes a group with pro duct ( g 1 , ρ 1 ) · ( g 2 , ρ 2 ) = ( g 1 · g 2 , β ( g 1 , g 2 ) ρ 1 ( g 1 ρ 2 )) . The identit y element o f L is (1 , β (1 , 1) − 1 ). Le t B , B ′ be ob jects in C cor resp onding to ( g 1 , ρ 1 ) , ( g 2 , ρ 2 ) ∈ L , r esp ectively . So B = ⊕ h ∈ H k g 1 h and B ′ = ⊕ h ∈ H k g 2 h , where each comp onent is equal to the ground field k . The rig ht and left mo dule structures on B , B ′ are defined via (3 ) a nd (4). Let A := k ψ [ H ]. W e ha ve B ⊗ A B ′ = ( k g 1 A ) ⊗ A ( ⊕ h ∈ H k g 2 h ) = k g 1 ⊗ ( ⊕ h ∈ H k g 2 h ). T aking in to account (3) and (4) we ca lculate that the pro jective r epresentation (defined in (2)) ρ : H → k × with 2-co cy cle ψ g 3 , corres p onding to B ⊗ A B ′ , where g 3 = g 1 · g 2 , is given by β ( g 1 , g 2 ) ρ 1 ( g 1 ρ 2 ). So G ( C ) is iso morphic to the gro up L . The map L → K ⋉ ν b H : ( g , ρ ) 7→ ( g , η − 1 g ρ ) establishes the desir ed isomorphism a nd the theorem is proved. 6. The universal grading group of cer t ain group-theoretical ca tegories Recall that ev ery fusion category C is faithfully gra ded by its univ er sal gra ding group U ( C ): C = ⊕ x ∈ U ( C ) C x . In this se c tion we describe U ( C ) for certain group- theoretical ca tegories. 10 SHLOMO GE LAKI AND DEEP AK NAIDU Lemma 6. 1. L et D b e a fusion c ate gory and let E b e a fusion su b c ate gory of D . The map U ( E ) → U ( D ) define d by the rule x 7→ y if and only if E x ⊆ D y ∩ E is a homomorph ism. This homomorp hism is inje ctive if and only if D ad ∩ E = E ad . Pr o of. W e hav e univ ersa l gr adings: D = ⊕ y ∈ U ( D ) D y and E = ⊕ x ∈ U ( E ) E x . F rom the former grading we obtain E = D ∩ E = ⊕ y ∈ U ( D ) ( D y ∩ E ). Note that this grading need not b e faithful. Since E ad ⊆ D ad ∩ E , each comp onent D y ∩ E is a E ad -submo dule category of E . So , for ev er y x ∈ U ( E ) there is a unique y ∈ U ( D ) such that E x ⊆ D y . This gives rise to a homomor phism U ( E ) → U ( D ). It is evident that this homomo rphism is injective if and only if D ad ∩ E = E ad . Lemma 6. 2. The universal gr ading gr oup U ( Rep ( K )) of the r epr esentation c ate- gory of a fin ite gr oup K is isomorphic to the c en t er Z ( K ) of K . Pr o of. This is a sp ecial case o f Theo rem 3.8 in [GN] ( H b eing the gr oup algebr a of K ). Prop ositio n 6. 3. L et C = C ( G, 1 , H , 1) . Su pp ose H is normal in G . Then t her e is a split exact se quenc e 1 → Z ( H ) → U ( C ) → G/H → 1 . Ther efor e, U ( C ) is isomorphi c to the semi-dir e ct pr o duct G/H ⋉ Z ( H ) . Pr o of. By Theorem 4.1, we have a grading of C by the g roup G/H : C = ⊕ x ∈ G/H C x , where C x is the full a belia n sub catego ry o f C cons is ting of ob jects supp orted on the coset x . Let E := C 1 . W e will firs t show that C ad = E ad . Let R be a s et representatives o f cosets of H in G . Reca ll that simple ob jects o f C cor resp ond to pairs ( a, ρ ), where a ∈ R and ρ is a n irr educible represe ntation of H . Let B be the ob ject in C corresp onding to ( a, ρ ) defined via (3) a nd (4). The dual ob ject B ∗ corres p onds to the pair ( b, ( b ρ ) ∗ ), where b ∈ R is the representative of the coset a − 1 H . The repr e sent ation (defined in (2 )) corr esp onding to B ⊗ k [ H ] B ∗ is given by ρ ⊗ a (( b ρ ) ∗ ) ∼ = ρ ⊗ ρ ∗ . This establishes the equality C ad = E ad . By Theorem 4.1, E ∼ = Rep( H ) a nd Lemma 6.2 implies that U ( E ) ∼ = Z ( H ). By Lemma 6.1, we g e t an injectiv e homomorphism i : Z ( H ) → U ( C ). F ro m [GN, Corollar y 3 .7 ] w e get a surjective homomorphism p : U ( C ) → G/H which is defined as follows. Note that E co ntains C ad . Therefo r e, e ach C x is a C ad -submo dule category of C . So, for every y ∈ U ( C ) there is a unique p ( y ) ∈ G/H such that the c ompo nent C y of the universal grading C = ⊕ z ∈ U ( C ) C z is co ntained in C p ( y ) . W e claim that the sequence 1 → Z ( H ) i − → U ( C ) p − → G/H → 1 is exact. W e have C ad = E ad ∼ = Rep( H ) ad ∼ = Rep( H / Z ( H )). By [E NO, Prop ositio n 8.20], it follows that | U ( C ) | = | Z ( H ) | | G | | H | and ther efore | Ker p | = | Z ( H ) | . So, it suffices to show that Ker p ⊆ Im i . W e have Ker p = { y ∈ U ( C ) | C y ⊆ E } . P ick an y y ∈ Ke r p and let K := { y ∈ U ( C ) | C y ∩ E 6 = { 0 }} . Then E = ⊕ k ∈ K ( C k ∩ E ) is a faithful grading of E . No te that y ∈ K . By [GN, C o rollary 3.7], ther e exists z ∈ U ( E ) such that E z ⊆ C y , i.e., y ∈ Im i . This establishes the exactness of the a forementioned sequence. Finally , we show that the aforementioned sequence splits. Let D b e the full fusio n sub c ategory of C generated by simple ob jects in C co rresp onding to pairs ( a, ρ 0 ), where a ∈ R and ρ 0 is the trivial repres e n tation of H . Note tha t D ∼ = V ec G/H and U ( D ) ∼ = G/H . Also note tha t C ad ∩ D = D ad ∼ = V ec. So, by Lemma 6 .1 w e obtain an injection j : G/H → U ( C ). W e claim that p ◦ j = id G/H . Pic k any x ∈ G/ H and let j ( x ) = y , i.e., D x ⊆ C y . W e hav e C y ⊆ C p ( y ) which implies that D x ⊆ C p ( y ) . It fo llows that p ( y ) = x and the pr op osition is proved. SOME PROPER TIES OF GROUP-THEORETICAL CA TEGORIES 11 References [DGNO] V. Drinf eld, S. Gelaki, D. Nikshyc h and V. Ostri k, Gr oup-the or etical pr op erties of nilp o- tent mo dular c ate gories , arXi v:0704.0195. [ENO] P . Etingof, D. Nikshyc h, and V. Ostrik, On fusion c ate g ories , Ann. of Math. 162 (2005), 581-642. [GN] S. Gelaki, D. Ni ksh yc h, Nilp otent fusion c ate gories , Adv ances in Mathematics, to appear, arXiv:math/0610726v2. [Mu] M. M ¨ uger, F ro m subfactors to c ategories and top olo gy I. F r ob enius algebr as in and Morita e quivalenc e of tensor cate gories , J. Pure Appl. Algebra 180 (2003), 81-157. [O1] V. O strik, Mo dule ca te g ories, we ak Hopf algeb r as and mo dular invariants , T ransform . Groups 8 (2003) , no.2, 177-206. [O2] V. Os tr i k, Mo dule c ate gories over t he Drinfeld double of a finite gr oup , Int . M ath. Res. Not., 2003, no. 27, 1507-1520. [R] D. Robinson, A c ourse in the the ory of gr oups , Graduate texts in mathematics 80 , Springer-V er lag, New Y ork, 1982. Dep ar tment of Ma thema tics, Technion - Israel Institute of Technology, Ha if a 3200 0, Israel. E-mail addr ess : gelaki@math .technion.ac.il Dep ar tment of Ma thema tics and St a tistics, University of New Hampshire, Durham , NH 038 24, USA. E-mail addr ess : dnaidu@unh. edu