On the classification of certain fusion categories

On the classification of certain fusion categories Da vid Jordan, Eric Larson Abstract W e adv a nce the cla ssification of fusion catego ries in t w o directions. Firstly , w e completely classify integ ral fusion categories — and consequen tly , semi- simple Hopf algebras — of dimension pq 2 , where p and q are distinct primes. This case is esp ecially in teresting b ecause it is the simplest class of dimensions where not all in tegral f usion categories are group -theoretical. Secondly , we classify a certai n family of Z / 3 Z -graded fusion categories, whic h are general- izations of the Z / 2 Z -graded T ambara-Y amag ami categories. Our pro ofs are based o n the recen tly dev elop ed theory of extensions of fusion categories. 1 In tro du ction and r e sults Recall that a fusion category is a semi-simple r ig id monoidal category with finitely man y simple ob j ects X i , with simple unit X 0 = 1 , suc h that ∀ i , End( X i ) = C . The goal of this pap er is to obtain t w o classification results for f usion categories, and t o apply o ne of them t o the classification of semi-simple Hopf algebras. W e b egin b y explaining how the main results of the pap er fit into t he exis ting literature. Classification of fusion catego ries is a n imp ortant and difficult problem. More sp ecifically , by the Ocnean u rig idity theorem (see [ENO1]) , there are finitely man y fusion categories of a giv en dimens ion a nd, in particular, finitely many fusion cate- gories with a giv en fusion ring R (called c ate gorific ations of R ). This leads to the natural pro blems o f classifying all fusion categories of a giv en dimension and cate- gorifications of a giv en f usion ring. In full generalit y these problems ar e v ery hard; for example, the first problem includes the classification of finite groups and Lie groups. Ho w eve r, for certain dimensions and certain fusion ring s these problems are some- times tractable and lead to interesting results. F or example, there exist classifications of fusion categories of dimension p , p 2 (see [ENO1]), pq ([EGO ]) or pq r ([ENO2]), where p, q , r are distinct primes. In [TY] categorifications o f T am ba r a-Y amagami 1 rings are classified. In addition, there is a simple description of gr oup-the or etic al cat- egories, i.e. categories that are Morita equiv a len t to a po in ted catego ry , (see [ENO1]), and all categories whose dimension is a prime p o w er are of this t yp e (see [D GNO]). In this pap er we extend these classification results in t w o directions. O ur first result is the classific ation of integral fusion categories of dimension pq 2 , where p and q are distinct primes. This case is intere sting b ecause it is the first class of dimensions for which an in tegral fusion category need not b e group-theoretical, and significan tly new metho ds are needed to get a classific ation. 1 The second result is the classification of categorifications of certain fusion rings R p,G , asso ciated with a finite group G . These rings are generalizations of the T ambara-Y amagami rings (whic h corresp ond to p = 2). Namely , w e obtain a complete classification of categorifications of such rings for p = 3, when the order of the group G is not divisible b y 3. Our main results are Theorems 1.1 and 1.4 b elo w. Theorem 1.1. L et p and q b e primes, and C b e an i n te gr al 2 fusion c ate gory of F r ob enius-Perr on dimension pq 2 . The n, ex actly one of the fol lowing is true: • C is a gr oup-the or etic al c ate gory. • p = 2 , and C is a T amb ar a - Y amagami c ate gory [TY] c orr esp ondin g to an anisotr opic quadr atic form 3 on ( Z /q Z ) 2 ; ther e ar e two e quivalenc e classes of such c ate gories. • The prime p is o dd and divides q + 1 , and C is one of the c ate gories C ( p, q , { ζ 1 , ζ 2 } , ξ ) we shal l explici tly c onstruct. Her e, ζ 1 6 = ζ 2 ∈ F q 2 ar e such that ζ p 1 = ζ p 2 = 1 , but ζ 1 ζ 2 6 = 1 , and ξ ∈ H 3 ( Z /p Z , C ∗ ) ∼ = Z /p Z . Ther e ar e ( p 2 − p ) / 2 e quivalenc e classes of such c ate gories. Corollary 1.2. Al l semi-simple Hopf alge b r as of dim e n sion pq 2 ar e gr oup-the or etic a l. R emark. Another pro of of Corollary 1.2, based on differen t metho ds, is giv en in [ENO2]. Our second theorem concerns categorifications of a certain f usion ring R p,G at- tac hed to a finite gro up G and a prime p . 1 All in tegral fusio n ca tegories of dimension pq r classified in a recent prepr int [ENO2] are g roup- theoretical a nd th us the techniques of [ENO2] do not work in our s ituation. 2 If p and q a r e o dd, then the as s umption that C is in tegral is redundant. 3 i.e, of the form x 2 − ay 2 , where a is a quadratic non-r esidue. 2 Definition 1.3. Let G b e a finite group whose order is a square 4 and let p ∈ N . Then, the f usio n ring R p,G is the ring generated by the group ring Z [ G ] and X 1 , . . . , X p − 1 , with relations g ⊗ X i = X i ⊗ g = X i , X ∗ i = X p − i X i ⊗ X j = ( p | G | X i + j if i + j 6 = p, P g ∈ G g if i + j = p. Theorem 1.4. L et A b e a finite gr oup of or de r not divisible by 3 . Then the fusion ring R 3 ,A admits c ate gorific ations if, and only if, A is ab elian, of the form A ∼ = L N i =1 ( Z /p n i i Z ) a i , wher e p i ar e prime s , p airs ( p i , n i ) a r e distinct, and a i ar e even inte gers. In this c ase, ther e ar e 3 Q i  a i 2 + 1  c ate gorific a tions. The pro ofs of these theorems a re based on the solv ability of fusion categories of dimension p a q b pro v ed in [ENO2], the new theory of extensions of fusion categor ies dev elop ed in [ENO3], and some intricate linear algebra ov er finite fields. The organization of this pap er is as follows. Section 2 contains a review of standard definitions, and also results fro m recen t literature which w e will need. In Section 3, w e exhibit non-trivial gradings o n the fusion categories of study , and analyze these g radings with the metho ds from [ENO3]. Sections 4 and 5 presen t the pro ofs of Theorem 1.1 and Corollary 1.2, resp ectiv ely . In Section 6, w e fo cus on the case of Z / 3 Z -gra ded extensions of V ec A . In Section 7 we prov e Theorem 1.4. R emark. It is not difficult to extend Theorem 1.1 to classify non-in t egr a l categor ies of dimension pq 2 . This is b ecause the dimensions of all ob jects in suc h a category are neces sarily square ro o t s of p ositiv e in tegers. This forces a Z / 2 Z - grading on the category , whic h means either p or q is 2. Then a case b y case analysis yie lds a complete list. W e hav e not included these computations a s they a r en’t of part icular in terest. R emark. It is also p ossible to extend L emmas 6 .1 and 6.2, and thu s Theorem 1.4 to the situation where A con ta ins no elemen ts of order 9 (i.e. the 3-comp onen t of A is a v ector space ov er F 3 ). Ho w ev er, it seems that our metho ds break down if A has a more complicated 3- comp onen t. R emark. So me p ossible directions of future rese arc h w ould b e a generalization of Theorem 1.1 to categories of dimension pq n , n ≥ 3 , and of Theorem 1.4 to p > 3. These problems reduce to describing orbits of actions of certain reductiv e subgroups of O (2 n, F q ) on the Lagrangian Grassmanian. While in general these pro blems ma y 4 This a s sumption is unnecessary when p = 1 o r p = 2. 3 b e intractable, we think t ha t under reasonable simplifying assumptions o ne can get manageable and interes ting classific ations. Ac kno wledgemen ts The author s w ould like to w armly thank P a v el Etingof and Victor Ostrik for p o sing the problem, and for many helpful con v ersations as t he w ork progressed. W e are grateful to Victor Ostrik for explaining to us how Corollar y 1.2 could b e easily deriv ed from our results. The w ork of b oth authors w as suppor ted b y the Researc h Science Institute, and conducted in the Departmen t of Mathematics a t MIT. 2 Preliminaries In this se ction, w e recall sev eral basic notions ab out fusion cat ego ries. F or more details, see [ENO1, ENO2, ENO3, O1]. F or the remainder of the pap er, C and D are fusion categories, G is a n y finite group, and A is a finite a b elian group. Definition 2.1. A fusion (o r b ase d ) ring R is an asso ciativ e ring whic h is free of finite rank as a Z -mo dule, with fixed Z -basis B = { X i } containing X 0 = 1 , and a n in v olutio n ∗ : B → B extending to a n an ti-inv olution R → R , suc h that: (i) for all i, j, X i X j = P N ( i, j, k ) X k , where N ( i, j, k ) are non-negativ e integers, (ii) N ( i, j ∗ , 0) = δ ij . Definition 2.2. The fusion ring of C , denoted K ( C ), has as its basis the isomorphism classes of simple ob jects of C , with N ( i, j, k ) equal to the m ultiplicit y of X k in X i ⊗ X j , and ∗ defined by the duality in C . A c ate gorific ation of a fusion ring R is a fusion category C suc h t hat K ( C ) = R . A fusion ring can hav e mor e than one categorification, or none at all. F or example, consider the group ring Z [ G ] of a finite gro up G (with basis { g ∈ G } ). Categorifi- cations of these rings are kno wn as p ointe d c ate gories . One suc h categorificatio n is the catego ry of G -graded ve ctor spaces, V = ⊕ g ∈ G V g , with the trivial asso ciativity isomorphism. W e can construct other categorifications by letting the asso ciativity isomorphism α b e defined on the graded comp onents b y ( U g ⊗ V h ) ⊗ W k ξ ( g ,h,k ) − − − − → U g ⊗ ( V h ⊗ W k ) , for some 3-co cycle ξ ∈ Z 3 ( G, C ∗ ). W e denote the resulting category V ec G,ξ (or just V ec G if ξ is trivial). It is w ell kno wn that V ec G,ξ dep ends only on the cohomology class 4 of ξ , and these categories are the only p oin ted categories. Th us, categorifications of Z [ G ] are parameterized b y the set H 3 ( G, C ∗ ) / Aut( G ). On the other hand, the t wo- dimensional fusion ring with the basis { 1 , X } and the fusion rules X 2 = 1 + nX has t w o categorifications when n = 0 , 1, and no categorifications for n > 1 (see [O3]). Definition 2.3. The F rob enius-P erron dimension FPdim X i of X i is the largest p ositiv e eigenv alue of the matrix N i with en tr ies N ( i, j, k ) ( such an eigenv alue exists b y the F rob enius-P erron theorem). F or categories of group represen tations (or more generally , represen ta tions of a semi-simple quasi-Hopf algebra), this is the ve ctor space dimension; how ev er, in general the dimension need not b e an integer — it is o nly a n algebraic inte ger. If a ll FPdim X i are in tegers, w e call t he category inte gr al . Definition 2.4. The dimension | C | of C is the sum of the squares of dimensions of all simple ob jects of C . F or the catego ry o f r epresen tations of a semi-simple quasi-Hopf algebra H , |C | is the v ector space dimension of H , b y Masc hke’s theorem. Definition 2.5. W e sa y that C is gr ade d by G if C decomp oses as a direct sum C = L g ∈ G C g , suc h t hat C g ⊗ C h ⊆ C g h . Let us denote the trivial comp o nent of the grading C e . When G is abelian, w e refer to the trivial comp onen t as C 0 . By an extension of D b y G , w e mean a G -graded fusion catego ry C with C e = D . Lemma 2.6. [O 2] L et C b e a G -gr ade d fusion c ate gory whose trivial c om p onent C e is p ointe d, with some c omp onent C g c ontaining a unique si m ple obje ct. Then C e ∼ = V ec A , with A a b elian. Pr o of. The category C g defines a fib er functor on C e , whic h implies that C e = Rep H for some comm utativ e Ho pf algebra H . Th us, ξ = 0 , H = F un( G ). No w, Rep H ∗ is the dual category to C e with resp ect to C g , whic h is t he same as C e . Thu s H ∗ is comm uta tiv e and G is a b elian. Definition 2.7. The Pic ar d gr oup of C , denoted Pic( C ), is the set of all equiv alence classes of in v ertible C -bimo dule catego ries under the op eration of the tensor pro duct. Th us, Pic( C ) is t he group of equiv alence classes of Morita auto equiv a lences of C . Definition 2.8. Tw o fusion categories C and C ′ are e quivalent , if t here is an in v ert- ible 5 tensor f unctor: C → C ′ . If we ha v e t w o catego ries C and C ′ graded by the same group G , then w e sa y tha t they are gr ading-e quivalent if there is some inv ertible tensor functor: C → C ′ whic h restricts to a functor C g → C ′ g for eac h g ∈ G . 5 by an inv er tible functor, w e mean a functor with a quasi-inv erse. 5 Theorem 2.9. [ENO3] Pic(V ec A ) is the split ortho gonal gr oup O ( A ⊕ A ∗ ) . F or completenes s, let us sk etc h a pro of. The ke y p oint is that Morita equiv alences b et w een fusion categories C and D ar e in bijection with braided equiv a lences b et w een their Drinfeld cen ters Z ( C ) and Z ( D ), a nd this equiv alence maps t ensor product of bimo dule categories to composition of functors. (See [ENO2 ], The orem 3.1.) In particular, the group of Morita auto -equiv alences of C is naturally isomorphic t o the group of braided auto equiv alences o f Z ( C ). In the case C = V ec A , Z ( C ) is V ec A ⊕ A ∗ , with braiding given b y the standard (split) quadratic form. Th us, the gro up of braided auto equiv alences of Z ( C ) is isomorphic to O ( A ⊕ A ∗ ), and the r esult follows. Theorem 2.10. [ENO3] Fix a fusion c ate gory C e . Then c ate gories C , gr ade d by G , with trivia l c omp onent C e ar e cl a s sifie d, up to gr ading-e quivalenc e, by triples ( ρ, h, k ) , wher e ρ : G → Pic( C e ) is a homomorphism, h ∈ H 2 ( G, In v ( Z ( C e ))) , and k ∈ H 3 ( G, C ∗ ) . 6 Ther e ar e obstructions φ 1 ( ρ ) ∈ H 3 ( G, In v ( Z ( C e ))) , and φ 2 ( ρ, h ) ∈ H 4 ( G, C ∗ ) whic h must vanis h for C ( ρ, h, k ) to exi s t. Her e, w e c o n sider this data up to the ac tion of the gr oup of tensor auto e quivalenc es o f C e . 3 Cyclic e xtensio n s of V ec A Let us fix primes p and q , and a g enerator ǫ of Z /p Z . Prop osition 3.1. L et | A | b e c oprime to p . T h en c ate gorific ations of R p,A ar e p a- r ameterize d by the d ata ( ρ, ξ ) , whe r e ξ ∈ H 3 ( Z /p Z , C ∗ ) ≃ Z /p Z , and ρ : Z /p Z → O ( A ⊕ A ∗ ) is a homomo rphism such that if we write ρ ( i ) =  α i β i γ i δ i  , wher e α i : A → A, β i : A ∗ → A, γ i : A → A ∗ , δ i : A ∗ → A ∗ , then β i is an isom o rp hism for al l i 6 = 0 . Two such c a te gorific ations ar e e quivalent if, and only if, they ar e r elate d by the natur al action of Aut( Z /p Z ) a nd the sub gr oup of the ortho gonal gr oup of elements of the form ν =  ψ 0 ϕ ψ − 1 ∗  , wher e ψ ∗ ϕ is skew - s ymmetric. 6 Actually the data h and k b elong to torsor s ov er the groups H 2 ( G, Inv( Z ( C e ))), and H 3 ( G, C ∗ ) resp ectively r a ther than to the gr o ups themselves. This is a tec hnical p oint whic h is not going to matter for o ur consider ations. Here In v( D ) denotes the group o f inv er tible ob jects of D . 6 Pr o of. Clearly , an y categorification C of the fusion ring R p,A is Z /p Z -gra ded. F rom Lemma 2.6, C 0 = V ec A . W e m ust ha ve that β i is an isomorphism for all i 6 = 0 since C i = h X i i , and FPdim( X i ) 2 = | A | = | Im β i | . Now, | Inv( Z ( C 0 )) | divides | Z ( C 0 ) | , since the dimension of an y sub category divides the dimension of the catego r y . But, | Z ( C 0 ) | = | C 0 | 2 . Because w e are a ssuming that p is coprime to | A | , p is coprime to | In v ( Z ( C 0 )) | , whe reb y H ∗ ( Z /p Z , In v ( Z ( C 0 ))) = 0. (In pa r t icular this implies that the third cohomology group is trivial.) W e also hav e no choice fo r the second piece of data, since the second cohomology group is a lso trivial. Finally , the second ob- struction v a nishes b ecause H 4 ( Z /p Z , C ∗ ) = 0 . Therefore, by [ENO3], suc h catego r ies are determined up to g rading-equiv alence b y t he data ( ρ, ξ ). It is clear tha t if C ′ , C ′′ are tw o categorifications of R p,A , then an y equiv alence b et w een C ′ and C ′′ , will preserv e the grading, up to t he action of Aut( Z /p Z ). Thu s, since the subgroup of the orthog onal group whic h acts on the da t a ( ρ, ξ ) is the group of a ut o equiv alences of C 0 = V ec A , w e conclude t he statemen t of this pr o p osition. No w we consider what happ ens when instead o f requiring that eac h graded com- p onen t C g ( g 6 = 0) con ta ins a unique simple ob j ect, w e only require that the graded comp onen t C ǫ con tains a unique simple ob ject. Since this condition is not inv ar ian t under t he action of Aut( Z /p Z ), w e classify these categorifications up to grading- equiv alence. Theorem 3.2. L et p ∈ N b e r elatively prime to | A | . T hen, Z /p Z -gr ade d c a te gories C with trivia l c om p onent V ec A such that C ǫ c ontains a unique simple obje ct, ar e p ar ameterize d up to gr ading-e quiva lenc e by an element of H 3 ( Z /p Z , C ∗ ) ≃ Z /p Z , to gether with a map α : A → A , and an isomorph i sm γ : A → A ∗ , such that γ ∗ α is skew-symmetric, and  α Id ( γ − 1 γ ∗ ) − 1 0  p = Id . (1) Pr o of. As in Prop osition 3.1, our categories are determined b y the data ( ρ, ξ ), except that w e only require β 1 to b e an isomorphism. T o sp ecify the homomor phism ρ , it suffices to g iv e the image of the generator ρ ( ǫ ), sa y: M =  α β γ δ  ∈ O ( A ⊕ A ∗ ) (2) suc h that β is in v ertible and M p = Id. How ev er, we m ust consider suc h matr ices M up to conjugation b y elemen ts of the f orm:  Id 0 ϕ Id  , (3) 7 where ϕ is sk ew-symme tric, since conjugation by elemen ts of the form  ψ 0 0 ψ ∗ − 1  amoun ts to the change o f basis ( α, γ ) 7→ ( ψ − 1 αψ , ψ ∗ γ ψ ) . Claim 3.3. F or a giv en matrix in the for m (2), there is exactly one matrix of the form (3) whic h conjugates it into a matrix where δ = 0. Pr o of. Observ e that when w e conjugate (2) by (3) w e obtain  Id 0 ϕ Id   α β γ δ   Id 0 − ϕ Id  =  α − β ϕ β ⋆ ϕβ + δ  So, a matrix in the for m of (3) conjugat es M into a matrix where δ = 0 if, a nd only if, ϕ = − δ β − 1 = − β − 1 ∗ β ∗ δ β − 1 , whic h is sk ew-symmetric because M ∈ O ( A ⊕ A ∗ ). Th us, w e hav e reduced the problem to classify ing the set o f all matrices M in the form:  α β γ 0  ∈ O ( A ⊕ A ∗ ) , whose p th p o w er is the identit y matrix. The condition that M ∈ O ( A ⊕ A ∗ ) can b e expresse d as γ ∗ α b eing sk ew-symmetric, and β = γ ∗ − 1 . Therefore, w e w an t to find linear maps α, γ , with γ in v ertible, such that γ ∗ α is sk ew-symmetric and  α γ ∗ − 1 γ 0  p = Id . (4) No w, conjugat ing M by an ything in the general linear gro up do es not ch ange the prop ert y that M p = Id, so w e can r eplace M with  Id 0 0 γ ∗− 1   α γ ∗ − 1 γ 0   Id 0 0 γ ∗  =  α Id ( γ − 1 γ ∗ ) − 1 0  Therefore (4) may b e replaced with the condition (1). R emark. One ma y easily deduce Theorem 3 .2 of [TY], as a corollar y t o Theorem 3 .2 ab ov e. Lemma 3.4. L e t C b e a cyclic q -gr oup. Then, up to e quivalenc e, ther e is exactly one non-de gener ate skew-symmetric map γ : C ⊕ C → ( C ⊕ C ) ∗ . 8 Pr o of. Iden tical to the pro o f of the cor r espo nding theorem f o r v ector spaces. Lemma 3.5. L et a ∈ Z , such that q 6 | a 2 − 4 , and let C b e a cyclic q -gr oup. Then, up to e quivalenc e, ther e is exactly one isom o rphism γ : C ⊕ C → ( C ⊕ C ) ∗ such that ( γ − 1 γ ∗ ) 2 = aγ − 1 γ ∗ − Id . Pr o of. W rite C = Z /q n Z . First, assume to the con trary that q n − 1 γ − 1 γ ∗ is multipli- cation b y some constan t q n − 1 λ . Then, q n − 1 γ = ( q n − 1 γ ∗ ) ∗ = ( q n − 1 λγ ) ∗ = q n − 1 λ 2 γ . Th us, mo dulo q , λ 2 = 1, so q n − 1 Id = q n − 1 ( λ Id) 2 = q n − 1 ( aλ Id − Id) = q n − 1 ( aλ − 1) Id. Th us, 1 = λ 2 = (2 /a ) 2 mo dulo q , which contradicts o ur a ssumption that q do es not divide a 2 − 4. Th us, the characteristic p olynomial of x = γ − 1 γ ∗ is t 2 − at + 1, since that is t he minimal p olynomial and of t he correct degree. Now, w e claim that there is exactly one equiv alence class for x , namely the class of x =  a 1 − 1 0  . (5) Since the ch aracteristic p olynomial of x is t 2 − at + 1 , x has the form x =  a − d b c d  , where d ( a − d ) − bc = 1, and a 6 = 2 d mo dulo q if b = c = 0 mo dulo q , b ecause we already pro v ed that q n − 1 x w as not m ultiplication by a scalar. If d ( a − d ) − bc = 1, it is straight-forw ar d to chec k  a − d b c d   ( a − d ) y + b y cy + d 1  =  ( a − d ) y + b y cy + d 1   a 1 − 1 0  . Th us, w e are done if there is y ∈ F q so that − cy 2 + ( a − 2 d ) y + b = det  ( a − d ) y + b y cy + d 1  6 = 0 , as w e can take an arbitr a ry lift of y in to Z /q n Z to finish. Since w e cannot ha v e b = c = 0 , a − 2 d = 0 mo dulo q , there is some y ∈ F q that finishe s the claim, unless q = 2 and − cy 2 + ( a − 2 d ) y + b = y 2 + y . But in the latter case, it follow s that det x = 0, whic h con tradicts the inv ertibilit y of x . Therefore, w e ma y assume (5). If w e write γ =  b c d e  , 9 then w e hav e that  b d c e  = γ ∗ = γ x =  b c d e   a 1 − 1 0  =  ab − c b ad − e d  . Therefore, γ =  d ( a − 1) d d d  . But, for any y , t , suc h a map is equiv alen t to  ay + t − y y t  γ  ay + t y − y t  =  d ( y 2 + ay t + t 2 ) ( a − 1) d ( y 2 + ay t + t 2 ) d ( y 2 + ay t + t 2 ) d ( y 2 + ay t + t 2 )  . In order to sho w the statemen t of this lemma, it suffice s to sho w that there are y , t ∈ Z /q n Z so that y 2 + ay t + t 2 = 1 /d . By Hensel’s Lemma, it suffices to prov e there is a solution modulo q suc h that 2 y + at 6 = 0. If q = 2 , this is clear, since w e can tak e y = t = 1. ( d a nd a m ust b oth b e 1, since γ is in v ertible.) If q 6 = 2, then it is equiv alen t to (2 y + at ) 2 + (4 − a 2 ) t 2 = 4 / d . Observ e that w e may choo se 2 y + at and t indep enden tly; that is w e w an t to find z , w ∈ F q so that z 2 + (4 − a 2 ) w 2 = 4 /d . If 4 /d is a quadratic residue, we may c ho ose z 2 = 4 /d, w 2 = 0. Therefore, supp o se that d is not a quadratic residue. Then, if (4 − a 2 ) is also not a quadratic residue, we may tak e z = 0 , w 2 = 4 / ((4 − a 2 ) d ). So, say that 4 − a 2 is a quadratic residue , 4 − a 2 = f 2 . Then, our equ ation b ecomes z 2 + ( f w ) 2 = 4 /d , where f 6 = 0. D efine the sets S i = { 2 /d + i, 2 /d − i } for i = 0 , 1 , 2 , . . . q − 1 2 . By the pigeonhole principle, w e m ust either hav e that 2 /d is a quadratic residue, in whic h case w e are done, or that t w o quadratic residues in the same S i , sa y { z 2 , ( f w ) 2 } = S i , whic h implies that z 2 + ( f w ) 2 = 4 / d . Lemma 3.6. L et C b e an inte gr al fusion c ate gory o f F r ob enius-Perr on dimension pq 2 . Then, either C is faithful ly gr ade d by Z /p Z , o r C is gr oup-the or e tic al. Pr o of. By [ENO2], any fusion category of dimension p m q n is Morita equiv alent to a nilp oten t fusion catego ry . Therefore, ev ery fusion category of dimension pq 2 is either Morita equiv alen t to a category with a faithful Z /q Z grading, o r one with a faithf ul Z /p Z grading. Supp ose that C is Morita equiv alen t to a category D with a f a ithful Z /q Z grading. Let D 0 b e the trivial comp onen t of the grading. Then, D 0 is an in tegral fusion category of dimension pq . Therefore, by [EGO], D 0 is gr o up-theoretical, and t hus Morita equiv alen t to a p oin ted category D ′ 0 , and D is Morit a equiv a lent to some Z /q Z graded category D ′ whose trivial comp onen t is D ′ 0 (see [ENO2], Lemma 3.3). But 10 the p ossible dimensions of ob jects o f D ′ are only 1 , √ q , √ p, √ q p . Th us, since Morita equiv alence preserv es in tegra lit y , D ′ is p ointe d, and therefore C is gro up-theoretical. Next, supp ose that C is not Morita equiv alen t to a category D with a fa it hf ul Z /q Z gr a ding, and that it do es not p ossess a faithf ul Z /p Z grading. Then, since all fusion categories of dimension pq 2 are solv able [ENO2], C is an equiv arian tization of some categor y C 0 of dimension q 2 b y Z /p Z . Since all integral categor ies of dimension q 2 are p oin ted, a nd a n y equiv ariantization of a p oin ted categor y is group-theoretical, the statemen t of this lemma follo ws. Lemma 3.7. Any in te gr al fusion c ate gory of F r ob enius-Perr on dim e nsion pq 2 , which is Z /p Z -gr ade d, such that the trivial c omp onent of the gr ading is V ec Z /q 2 Z ,ξ , is g r oup- the or etic al. Pr o of. Since o ur category is in tegral, it is either p oin ted, in whic h case we are done, or there is an ob ject of dimension q . If that is the case, then b y Lemma 2.6, ξ = 0. Th us, the Picard group of the trivial compo nen t is O ( Z /q 2 Z + ( Z /q 2 Z ) ∗ ). Therefore, the catego ry must b e group-theoretical, since q ( Z /q 2 Z + ( Z /q 2 Z ) ∗ ) is an in v arian t Lagrangian subspace, under an y action, with resp ect to the split quadratic form. Lemma 3.8. L et α and γ b e 2 × 2 matric es, with γ inve rtible , and α ∗ γ skew- symmetric. T hen, α c ommutes w i th γ − 1 γ ∗ . Pr o of. Let ϕ = αγ − 1 . Since α ∗ γ is ske w-symmetric, ϕ is as w ell. Explicit computa- tion rev eals that γ ϕγ ∗ = γ ∗ ϕγ , whic h implies α γ − 1 γ ∗ = γ − 1 γ ∗ α . 4 Pro o f of Theor e m 1.1 W rite A = ( Z /q Z ) 2 . By Lemmas 3.6 and 3.7, either C is gr o up-theoretical, or C is a Z /p Z -graded category with trivial comp onent V ec A,ξ . Since C has all ob jects o f in tegral dimension by a ssumption, either C is p ointed, in whic h case w e are done, or C has an ob ject of dimension q , in whic h case by Lemma 2.6, ξ = 0. Therefore, if p = 2, b y [TY] suc h categories a re parameterize d b y a quadratic form γ , a nd b y [GNN] suc h catego r ies are gro up-theoretical if, and only if, there is a subgroup L ⊂ A , suc h that | L | = p | A | = q , suc h t ha t γ is 0 when restricted to L . Since A is a t w o - dimensional v ector space, this is equiv alen t to the form γ b eing isotropic. This completes the pro of if p = 2 . Th us, w e will assume that p is o dd. In particular, C is a Z /p Z -graded category with trivial comp onen t V ec A suc h that C ǫ con tains a single simple ob j ect, where ǫ is the generator of Z /p Z . F rom Theorem 3.2, suc h C a re parameterize d b y an elemen t of H 3 ( Z /p Z , C ∗ ), to gether 11 with an equiv alence class of a pair o f maps α : A → A, γ : A → A ∗ , where γ is an isomorphism, whic h satisfy γ ∗ α sk ew-symmetric, and (1). W rite x for γ − 1 γ ∗ . F rom Lemma 3.8, we ha v e that α and x comm ute. Consider the matrix M =  α Id x − 1 0  as a tw o-b y-tw o matrix o v er the comm uta tiv e su bring o f matrices generated b y α and x . Then, since M p = Id, we m ust ha v e det( M ) p = Id. Since det( M ) = − x − 1 , w e ha v e that x p = − Id. O v er the a lgebraic closure of F q , we may ch o ose a basis suc h that x =  − µ 0 0 − λ  , where λ p = µ p = 1. Since det x = det γ − 1 det γ ∗ = 1, µ = λ − 1 . By Lemma 3 .5 there is exactly one solution, up to equiv alence, to the equation γ ∗ = γ x , where γ is inv ertible. This equation is a system of four linear equations in t he en tries of the matrix for γ . Solving it yields γ =  0 − λ − 1 g g 0  . W rite α =  a b c d  . Then, w e ha v e that γ ∗ α =  0 g − λg 0   a b c d  =  cg dg − aλ − 1 g − bλ − 1 g  is sk ew-symmetric. In other words, α =  a 0 0 aλ − 1  . W e m ust therefore hav e:     a 0 1 0 0 aλ − 1 0 1 − λ 0 0 0 0 − λ − 1 0 0     p = Id , 12 or equiv alen tly , there exist ζ 1 , ζ 2 distinct p th ro ots of unity such that  a 1 − λ 0  is conjugate to  ζ 1 0 0 ζ 2  , i.e. a = ζ 1 + ζ 2 , λ = ζ 1 ζ 2 . Since p is o dd, x and α hav e the same blo c k form, and therefore the same centralizer. Claim 4.1. There exists a basis so that b oth x and α a re matrices o v er F q if, and only if, p | q 2 − 1. Pr o of. Observ e that p | q 2 − 1 ⇔ | G al( F q [ ζ p ] : F q ) | ≤ 2 ⇔ ζ + ζ − 1 ∈ F q for eac h ζ whic h is a primitive p th ro ot of unit y . When λ = 1, x = − Id, and α = ( ζ 1 + ζ − 1 1 ) Id are central, and are matrices ov er F q ⇔ ζ 1 + ζ − 1 1 ∈ F q ⇔ p | q 2 − 1. When λ 6 = 1 , to see the “only if ” , observ e that since λ + λ − 1 = − tr x ∈ F q , w e kno w that p | q 2 − 1. T o see the “if ”, let ψ =  λ − 1 − 1 λ  , and observ e that ψ ( γ − 1 γ ∗ ) ψ − 1 =  − ( λ + λ − 1 ) − 1 1 0  , (6) ψ α ψ − 1 =  ( λ + λ − 1 + 1) a λ +1 a λ +1 − a λ +1 a λ +1  , b oth of whic h are matrices o ve r F q . It is th us clear that if q 2 − 1 is not divis ible by p , there are no non-p ointed categorifications, and if q 2 − 1 is divisible by p , then up to grading-equiv alence, non- p oin ted categorifications a re determined b y an elemen t of H 3 ( Z /p Z , C ∗ ) together with unordered pairs { ζ 1 , ζ 2 } of distinct p th ro ots of unity under the equiv alence { ζ 1 , ζ 2 } ∼ { ζ − 1 1 , ζ − 1 2 } . Such pairs determine the pair ( γ , α ) uniquely b ecause they determine γ uniquely , a nd γ − 1 γ ∗ and α hav e the same cen tralizer. Claim 4.2. The categor y C is g roup-theoretical if, and o nly if, ζ 1 ζ 2 ∈ F q . 13 Pr o of. The resulting category is group-theoretical if, and only if, there exists a La- grangian subspace L ⊂ A ⊕ A ∗ , with respect to the split quadratic form q ( a ⊕ b ) = ba , whic h is in v arian t under t he action of Z /p Z . Fix the homomorphism ρ . W rite M = ρ ( ǫ ), and let α and γ b e as in (1). Denote the c ho sen basis of A by e 1 , e 2 . This giv es a basis e 1 , e 2 , e ∗ 1 , e ∗ 2 for A ⊕ A ∗ . Because γ − 1 γ ∗ and α ha v e the same centralize r, Lemma 3.5 and equation 6 imply that w e ma y assume γ =  − 1 − ( λ + λ − 1 + 1) 1 − 1  . Th us, M =      a ( λ 2 + λ +1) λ 2 + λ a λ +1 − λ ( λ +1) 2 − λ ( λ +1) 2 − a λ +1 a λ +1 λ 2 + λ +1 ( λ +1) 2 − λ ( λ +1) 2 − 1 − ( λ 2 + λ +1) λ 0 0 1 − 1 0 0      . (7) F or a ny elemen t or subspace a of A ⊕ A ∗ , denote b y π a t he pro jection of a on to A . Fix some Lagrangia n subspace L . W e consider three cases. Case 1: π L has dimension 0. It fo llo ws that L = A ∗ , and b y insp ection, such Lagrangian subspaces are nev er inv ariant subspaces of the action of Z /p Z by the homomorphism ρ . Case 2: π L has dimension 1. In this case, w e prov e that there is an in v arian t Lagrangian subspace if, and only if, λ ∈ F q . First w e will sho w the “only if ”. Since π L has dimension 1, there is some vec tor v 6 = 0 ∈ π L . W e claim that v ∈ L . T o see this, let v ′ b e a lift o f v to L . W r ite v ′ = v + w . Since v + w ∈ L , it suffices to sho w that w ∈ L . T a k e w ′ ∈ L , suc h that w ′ / ∈ h v ′ i . Since π L has dimension 1, we ha v e tha t π w ′ = λv . Consider the ve ctor w ′ − λv ′ . W e hav e π ( w ′ − λv ′ ) = 0, so w ′ − λv ′ ∈ A ∗ . Since w ′ − λv ′ ∈ L , we ha ve that 0 = q (( w ′ − λv ′ ) + v ′ ) = q (( w ′ − λv ′ ) + v + w ) = (( w ′ − λv ′ ) + w )( v ) = ( w ′ − λv ′ )( v ) , 14 since w ( v ) = q ( v ′ ) = 0 . But b ecause w ( v ) = 0, w ∈ h w ′ − λv ′ i ⊂ L , since the subspace of A ∗ whic h ev aluates to 0 o n a non-zero v ector in A is one-dimensional. Therefore, v ∈ L . It follows that M v ∈ L , a nd therefore that αv = π ( M v ) ∈ π L = h v i . Thus , v is an eigen v ector of α . It follows that an eigen v alue of α lies in F q . Since the eigen v a lues of α are − λ and − λ − 1 , w e ha v e that λ ∈ F q . In order to see t he “if ”, consider L = h v , w i , where v = (1 , − λ, 0 , 0) , w = (0 , 0 , λ, 1) . Since v ∈ A, w ∈ A ∗ , and w ( v ) = 0, it is clear that L is a Lagrangian subspace. Th us, it suffices to sho w that L is inv ariant under the action of Z /p Z , or that M v , M w ∈ h v , w i . By (7), M v = a λ v + ( λ + 1) w , and M w = − 1 λ − 1 + 1 v . Case 3: π L has dimension 2. In this case, w e prov e that there is no in v arian t Lagrangian subs pace if λ / ∈ F q . Assume to the contrary . Since π L has dimension 2, e 1 , e 2 ∈ π L . L et v b e an arbitrary lift of e 1 to L , and w b e an arbitrar y lift of e 2 to L . Clearly , w e hav e L = h v , w i . Since q ( v ) = 0, and π v = e 1 , v must hav e the form (1 , 0 , 0 , s ) for some s ∈ F q . Similarly , w m ust hav e the form (0 , 1 , s ′ , 0). Since q ( v + w ) = 0, w e hav e that s ′ = − s . In other w ords, our Lagrang ia n subspac e w o uld ha v e to b e the span o f t w o vec tors in the form v = (1 , 0 , 0 , s ), w = (0 , 1 , − s, 0). W e ha v e M v ∈ h v , w i . Now, w e explicitly compute M v = ( κ, τ , − 1 , 1), where κ = a ( λ + 1)( λ 2 + λ + 1 ) − sλ 2 λ ( λ + 1) 2 , τ = − sλ + a ( λ + 1) ( λ + 1) 2 . Since M v ∈ h v , w i , we ha v e that there exists c v , c w ∈ F q suc h that c v v + c w w − M v = 0. Since π ( c v v + c w w ) = ( c v c w ), w e know c v = κ, c w = τ . Th us, 0 = κv + τ w − M v = (0 , 0 , 1 − τ s, κs − 1 ). As 1 − τ s = ( sζ 2 + ζ 1 ζ 2 + 1)( sζ 1 + ζ 1 ζ 2 + 1) ζ 2 1 ζ 2 2 + 2 ζ 1 ζ 2 + 1 , w e deduce ( sζ 2 + ζ 1 ζ 2 + 1)( sζ 1 + ζ 1 ζ 2 + 1) = 0 . 15 Without loss of generality w e ma y assum e s = − ζ 1 − ζ − 1 2 . Then, 0 = κs − 1 = − ( ζ 1 + ζ 2 )( ζ 1 ζ 2 + 1) 2 ζ 1 ζ 2 2 . But, this is impo ssible, as ζ 1 , ζ 2 are p th ro ots of unity , p is o dd, ζ 1 ζ 2 / ∈ F q , and ζ 1 6 = ζ 2 . (The last t w o a ssumptions are needed only when q = 2.) A t this p oint, w e can count t he num b er of non-group- theoretical catego r ies o f dimension pq 2 up to gra ding-equiv alence, a nd up to general equiv alence. If p do es not divide q 2 − 1, then all categorifications are p oin ted. If p divides q − 1, then all p th ro ots of unity lie in F q . Therefore, non-group-theoretical catego rifications o ccur only when p divides q + 1; w e hav e one suc h categorification f or eac h pair { ζ 1 , ζ 2 } suc h that ζ 1 ζ 2 6 = 1. T o accoun t fo r grading- equiv alences , w e first compute the n um b er ( p − 1)( p − 3) 4 of pairs { ζ 1 , ζ 2 } with ζ i 6 = 1 up to equiv alence { ζ 1 , ζ 2 } ∼ { ζ − 1 1 , ζ − 1 2 } . T o this w e add the n um b er p − 1 2 of pairs { 1 , ζ } up to equiv alence { 1 , ζ } ∼ { 1 , ζ − 1 } , for a to tal of ( p − 1) 2 4 non-group-theoretical categories up to grading-equiv alence. T o accoun t fo r general equiv alences, consider the action of Aut Z /p Z on our cat- egories. An elemen t g ∈ (Aut Z /p Z ) acts b y m ultiplication b y g − 2 on H 3 ( Z /p Z , C ∗ ), and sends ( ζ 1 , ζ 2 ) → ( ζ g 1 , ζ g 2 ). There are three o rbits o n H 3 ( Z /p Z , C ∗ ) under this ac- tion: the quadratic non-residues, the quadratic r esidues, and 0 in an orbit by it self. The stabilizer of the first tw o orbits is ± 1, whic h then a cts on the pairs { ζ 1 , ζ 2 } as in the graded case, giving ( p − 1) 2 4 categorifications eac h, for ( p − 1) 2 2 together. The { 0 } orbit in H 3 ( Z /p Z , C ∗ ) yields tw o types o f orbits on the set o f pairs { ζ 1 , ζ 2 } . Clearly Aut Z /pZ acts t ransitiv ely on pairs { 1 , ζ } . So supp ose that ζ 2 = ζ k 1 . Then the set { k , k − 1 } determines the orbit, and the n um b er of suc h sets whic h do not con tain 0 , ± 1 is p − 3 2 . This give s a to tal of p 2 − p 2 categorifications. 5 Pro o f of Coro llary 1.2 By Lemma 3.6, a ll categories of dimension pq 2 without a faithful Z /p Z -grading are group-theoretical. Let us supp ose that C o f dimension pq 2 is faithfully Z /p Z -graded, and is the category of represen ta t ions of some semi-simple Hopf algebra H , and demonstrate that C is group-theoretical. The faithful Z /p Z gra ding on C induces a faithful Z /p Z -grading on H ∗ as follow s. Since C is faithfully Z /p Z -graded, there exists a cen tral group-lik e elemen t c ∈ H , suc h that c p = 1, defining the grading. This elemen t define s the decomp osition 16 H ∗ = L k H ∗ k , where H ∗ k = { f ∈ H ∗ s.t. f ( c x ) = ζ k f ( x ) } and ζ = e 2 πi p . Clearly H ∗ k 6 = 0 for a ll k . W e consider the sub-algebra H ∗ 0 of H ∗ , and we let D denote the category of H ∗ 0 -bimo dules in C ; D is Morita equiv alen t to C . Because H ∗ 0 ∈ C 0 , D is also Z /p Z - graded, and w e hav e | D 0 | = q 2 , so D 0 is pointed. F urthermore, H ∗ is an algebra in D , whose 0-comp onen t is H ∗ 0 , the unit in D . Claim 5.1. The multiplication map µ : H ∗ k ⊗ H ∗ 0 H ∗ l → H ∗ k + l is an isomorphism for all k and l . Pr o of. W e claim first tha t the map ∆ l = ( π l ⊗ Id) ◦ ∆ : H → H / ( c − ζ l ) ⊗ H is injectiv e. Indeed, supp ose a ∈ H s.t. ∆ l ( a ) = 0. Then f o r all V ∈ C l , U ∈ C , w e ha v e that a | V ⊗ U = 0. T aking U = H , w e ha v e V ⊗ H ∼ = (dim V ) H , so a m ust b e zero. By duality , µ : H ∗ ⊗ H ∗ 0 H ∗ l → H ∗ (8) is surjectiv e. By the Nic hols-Z o eller theorem [NZ], H ∗ is free ov er H ∗ 0 , of rank p . Therefore, t he left hand side of (8) has dimension p · dim H ∗ l = dim H ∗ . Th us (8) is an isomorphism. Restricting t o t he graded comp onen ts yields the claim. The claim implies that eac h H ∗ k is a n inv ertible ob ject in D , and so in particular there are in v ertible ob jects in each D k , in addition to the q 2 in v ertible ob jects in D 0 . As the num b er of inv ertible ob j ects m ust divide the ov erall dimension pq 2 of D , w e conclude that D is p oin ted. 6 Categorifi cations of R 3 ,A Lemma 6.1. I f | A | is c op ri m e to 3 , Z / 3 Z -gr ad e d c ate gories C with trivial c omp onent V ec A such that C ǫ c ontains a single simp l e obje ct, ar e al l c ate gorific ations of R 3 ,A , and ar e, up to gr ading-e quivalen c e, p ar ameterize d by p a irs ( ξ , γ ) , wher e ξ is an ele m ent of H 3 ( Z / 3 Z , C ∗ ) ≃ Z / 3 Z , and γ is a map A → A ∗ such that γ ∗ γ − 1 γ ∗ is skew-symmetric. If o ur e quivalenc e is n ot r e quir e d to pr eserve gr ading , we m ust add itional ly ide ntify γ with γ ∗ . Pr o of. Clearly , categorifications o f R 3 ,A are Z / 3 Z -graded categories C with trivial comp onen t V ec A suc h that C ǫ con tains a single simple ob ject. T o s ee the rev erse inclusion, recall that C g is contains a unique simple ob j ect if, and only if , ρ ( g ) has its 17 upp er rig h t entry an isomorphism. Th us, it suffices to sho w that ρ ( g ) has its upper righ t en try an isomorphism if, a nd o nly if, ρ ( g − 1 ) do es. But this is clear b ecause ρ is a homomorphism into the split orthogonal group, so the upp er righ t en t ry of ρ ( g ) is the dual of the upp er r igh t en tr y of ρ ( g − 1 ). Therefore, b y Theorem 3.2, C is determined up to grading - equiv alence b y an elemen t of H 3 ( Z / 3 Z , C ∗ ) ≃ Z / 3 Z , together with a map α : A → A , and an isomorphism γ : A → A ∗ , satisfying the relations γ ∗ α is sk ew-symmetric and (1). W rite x for γ − 1 γ ∗ . T o solv e equation (1), w e explicitly compute:  Id 0 0 Id  =  α Id x − 1 0  3 =  α 3 + x − 1 α + αx − 1 α 2 + x − 1 x − 1 α 2 + x − 2 x − 1 α  . In particular, α = x . The condition that γ ∗ α is sk ew-symmetric then b ecomes that γ ∗ γ − 1 γ ∗ is sk ew-symmetric, from which it follo ws that ( γ − 1 γ ∗ ) 3 = − Id. When α = x , and x 3 = − Id, it is not hard to che c k that (1 ) is satisfie d. In other words, t he conditions on ( α, γ ) giv en by The orem 3.2 are equiv alent to α = γ − 1 γ ∗ , and γ ∗ γ − 1 γ ∗ sk ew-symmetric. Therefore, the c hoice of maps α and γ is equiv alen t to the ch oice o f a single map γ suc h tha t γ ∗ γ − 1 γ ∗ is sk ew-symmetric. Finally , in the case where w e do not require that o ur equiv alence preserv es grading, w e m ust figure out what happ ens under the action of Aut Z / 3 Z . In or- der to do this, w e mus t consider what happ ens to γ under the transforma t ion M → M − 1 = M ∗ . In our case, M =  γ − 1 γ ∗ γ ∗ − 1 γ 0  , M ∗ =  0 γ − 1 γ ∗ γ γ − 1 ∗  . W e find M ∗ ∼  1 0 γ ∗ γ − 1 γ ∗ 1   0 γ − 1 γ ∗ γ γ − 1 ∗   1 0 − γ ∗ γ − 1 γ ∗ 1  =  γ − 1 ∗ γ γ − 1 γ ∗ 0  , whic h is the same as the matrix M with γ replaced with γ ∗ . Lemma 6.2. L et q 6 = 3 , A b e an ab elian q -gr oup, a nd γ a non-de gener ate map A → A ∗ such that γ ∗ γ − 1 γ ∗ is skew-symmetric. Then A may b e de c omp ose d as L i ( C i ⊕ C i ) , for cyclic gr oups C i , wher e the C i ⊕ C i ar e mutual ly o rtho gonal with r e- sp e ct to γ , and on e ach c omp onent C i ⊕ C i , either γ is skew-symmetric, or ( γ − 1 γ ∗ ) 2 = γ − 1 γ ∗ − Id . 18 Pr o of. W rite x = γ − 1 γ ∗ . Since γ ∗ γ − 1 γ ∗ is sk ew-symmetric, it follo ws that x 3 = − Id. W rite A ′ = Ker( x + Id ) , A ′′ = Im ( x + Id). First, w e claim that Ker( x + Id) = Ker( x + Id) 2 . T o see this, o bserv e that if ( x + Id ) 2 g = 0, then since x 3 = − Id, 0 = ( x − 2 Id)( x + Id) 2 g = ( x 3 − 3 x − 2 Id) g = − 3( x + Id) g . Since q 6 = 3, multiplic ation b y − 3 is inv ertible o n A , and therefore, ( x + Id) g = 0. It follow s that A = A ′ ⊕ A ′′ . Obviously , x restricts to eac h comp onen t. It is clear that on A ′ , x is − Id. Since on A ′′ , x + Id is in v ertible, a nd 0 = x 3 + Id = ( x + Id)( x 2 − x + Id ) , w e ha v e t ha t x 2 = x − Id on A ′′ . No w, w e claim that γ restricts to each comp onent. Since γ : A → A ∗ , restricting to Im( x + Id) means that γ : Im( x + Id) → Im( x + Id) ∗ , and restricting to Ker( x + Id) means that γ : Ker( x + Id) → Ker( x + Id) ∗ . The first follow s from the fact tha t γ ( x + Id) = ( x + Id) ∗ γ ∗ , and the second fo llows fr o m ( γ − γ ∗ )( x + Id) = ( x + Id) ∗ γ . W e ha v e sho wn that A = A ′ ⊕ A ′′ where x and γ restrict to A ′ and A ′′ , x is − Id on A ′ , and x 2 = x − Id on A ′′ . D enote b y n the unique natural num b er so that q n A ′′ = 0, but q n − 1 A ′′ 6 = 0. W e will show that A ′′ decomp oses as a direct sum L i ( C i ⊕ C i ) whic h resp ects γ , by strong induction o n | A ′′ | . The base case, where | A ′′ | = 1, is trivial. In order to do the inductiv e step, first suppose that q n − 1 γ ( g , g ) = 0 for an y g suc h that there do es not exist a g ′ where g = q g ′ . Since any elemen t of A ′′ is a multiple of some suc h g , w e w ould ha ve that q n − 1 γ ( g , g ) = 0 for an y g ∈ A ′′ . F rom this, we would hav e that q n − 1 γ ∗ = − q n − 1 γ ; therefore, b y the defi nition of x , we w ould hav e that γ ∗ = γ x , w e w o uld hav e q n − 1 x = − q n − 1 Id. Because x 2 = x − Id, q n − 1 Id = − ( q n − 1 x ) = x ( − q n − 1 Id) = q n − 1 x 2 = q n − 1 x − q n − 1 Id = − 2 q n − 1 Id. Therefore, 3 q n − 1 Id = 0. Since q 6 = 3, w e would hav e q n − 1 Id = 0, a contradiction. Th us, w e ha v e that there is some g such that there does not exist a g ′ with g = q g ′ and such that q n − 1 γ ( g , g ) 6 = 0. W rite B = Ker γ g ∩ Ker γ ∗ g . W e claim that A ′′ = B ⊕h g i ⊕ h xg i . T o verify this, it suffices to sho w that the map ( a, b ) → ( γ ( g , ag + bxg ) , γ ∗ ( g , ag + bxg )) whic h maps Z /q n Z × Z /q n Z → ( ∪ g ∈ A ∗ Im g ) 2 is an isomorphism. This is clear from the explicit computat io n that ( γ ( g , ag + bxg ) , γ ∗ ( g , ag + bxg )) = (( a + b ) s, as ), where s = γ ( g , g ) is a generator of ∪ g ∈ A ∗ Im g . Since Ker γ xg = Ker γ ∗ g , and Ker γ ∗ xg = Ker( γ ∗ − γ ) g , it is clear that B is orthogonal to h g i ⊕ h xg i . Applying the inductiv e h yp othesis to B completes the proo f. The pro of that A ′ decomp oses as a direct sum L i ( C i ⊕ C i ) whic h r esp ects γ , under the assumption that γ is sk ew-symme tric, is the standard pro of that ev ery sk ew-form has a sy mplectic basis ov er a v ector space, where instead of splitting off h v , v ′ i so that γ ( v , v ′ ) 6 = 0, we split off h g , g ′ i suc h that the order of the cyclic subgroup generated by g is q n , and so that q n − 1 γ ( g , g ′ ) 6 = 0 . 19 7 Pro o f of Theor e m 1.4 By Lemma 6.1, the categorifications are in one to one corresp o ndence with an elemen t of H 3 ( Z / 3 Z , C ∗ ) ≃ Z / 3 Z together with a map γ : A → A ∗ satisfying ( γ − 1 γ ∗ ) 3 = − Id. In o rder to classify such f orms up to equiv alence, it suffices to classify suc h forms on the q -pa r ts o f A for eac h prime q . By our Lemmas 3.4, 3.5, a nd 6.2, there are Q ( a i / 2 + 1) c hoices fo r γ , as o n eac h C i ⊕ C i , there are t wo choice s, dep ending on whether or not γ is sk ew-symmetric. Since there a re t hree choice s for the elemen t of H 3 ( Z / 3 Z , C ∗ ) ≃ Z / 3 Z , the statemen t of this corollar y fo llo ws, pro vided that w e can sho w that Aut( Z / 3 Z ) acts trivially o n H 3 ( Z / 3 Z , C ∗ ) and our solutions for γ . T o see that it a cts trivially on t he cohomolog y gr o up, recall that H 3 ( Z / 3 Z , C ∗ ) = ( Z / 3 Z ) ⊗ ( − 2) . T o see that it acts trivially on our solutions, observ e tha t γ b eing sk ew- symmetric is the same as γ ∗ b eing sk ew-symme tric. As γ is determined b y on ho w man y of eac h t yp e of C i ⊕ C i it is sk ew-symmetric, γ and γ ∗ are equiv alent. Th us, the statemen t of the theorem follo ws. References [DGNO] V. D r inf eld, S. Gelaki, D. Niksh yc h, and V. Ostrik. Gr oup-the or etic al pr o p - erties of nil p otent mo dular c ate gories . http://ar xiv.org/abs/0 704.0195v2 [math.QA] 2 Apr 20 0 7. [EGO] P . Etingof, S. Gelaki, and V. Ostrik. Classific ation of F usion Cate gories of Dimension pq . In t. Math. Res. Not. 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