Frobenius-Perron dimensions of integral Z + -rings and applications
aa r X i v : . [ m a t h . R A ] D ec FROBENIUS-PERRON DIMENSIONS OF INTEGRAL Z + -RINGS AND APPLICATIONS PAVEL ETINGOF
Abstract.
We introduce the notion of the Frobenius-Perron di-mension of an integral Z + -ring and give some applications of thisnotion to classification of finite dimensional quasi-Hopf algebraswith a unique nontrivial simple module, and of quasi-Hopf andHopf algebras of prime dimension p . To Nicol´as Andruskiewitsch on his 60th birthday with admiration Introduction
The goal of this paper is to define and study the notion of theFrobenius-Perron dimension of an integral Z + -ring A which is not nec-essarily a fusion ring (i.e., does not necessarily have a *-structure).Namely, if N i are the matrices of multiplication by the basis vectors of A and d i are their largest positive eigenvalues, then we let p be the lefteigenvector of N i with eigenvalues d i , normalized to have positive inte-ger entries with greatest common divisor 1. Then we set FPdim( A ) := P i p i d i . For fusion rings p i = d i so FPdim( A ) = P i d i , which coincideswith the standard definition in [EGNO]. Also, if A is the Grothendieckring of a finite tensor category C then FPdim( A ) = FPdim( C ) /D , where D is the greatest common divisor of the dimensions of the indecompos-able projectives of C .We prove a number of properties of FPdim( A ). In particular, weshow that if X ∈ A is a Z + -generator of A and r the rank of A then(2 d ) r − ≥ FPdim( A ), where d is the Frobenius-Perron dimension of a Z + -generator of A .We then give two applications of this notion. The first application isto classification of quasi-Hopf and Hopf algebras with two simple mod-ules. We give a number of examples of such quasi-Hopf and Hopf alge-bras and prove a number of restrictions on their structure. The secondapplication is to quasi-Hopf and Hopf algebras H of prime dimension p . It is conjectured that any such quasi-Hopf algebra is commutativeand cocommutative, but this is not known even in characteristic zero, nd we prove some restrictions on the structure of H if it is not com-mutative and cocommutative; e.g., we show that H has to have at leastfour simple modules. In particular, it follows that a quasi-Hopf alge-bra of prime dimension ≤
31 over any field is necessarily commutativeand cocommutative. In the case of Hopf algebras, it is known thatif H is not commutative and cocommutative then q > pq > pq +2 > (Corollary 8.2). Also, weshow that pq +2 > min(9 , φ ( p )), where φ is a certain function such that φ ( p ) ∼ √ (cid:16) log p log log p (cid:17) / as p → ∞ (Theorem 8.6). This means that if p is large enough, we must have pq +2 >
9. Moreover, we show that if H has a nontrivial simple module X with X ∼ = X ∗∗ then pq +2 > φ ( p ).The organization of the paper is as follows. In Section 2 we definethe Frobenius-Perron dimension of an integral Z + -ring A and provebasic properties of this notion. In Section 3 we prove the lower boundon the dimension of a Z + -generator of A . In Section 4 we prove someauxiliary lemmas. In Section 5 we give applications to quasi-Hopf alge-bras with two simple modules. In Section 6 we prove some bounds fordimensions of tensor categories with no nontrivial invertible objects. InSection 7 we give applications to classification of quasi-Hopf algebrasof prime dimension. Finally, in Section 8 we give applications to classi-fication of Hopf algebras of prime dimension, improving the bound onthe characteristic from [NW]. Acknowledgements.
The author is grateful to C. Negron for usefuldiscussions and to V. Ostrik, S.-H. Ng and X. Wang for correctionsand comments on the draft of this paper. The work of the author waspartially supported by the NSF grant DMS-1502244.2.
The Frobenius-Perron dimension of an integral Z + -ring Dimensions in integral Z + -rings. Let A be a transitive unital Z + -ring of finite rank ([EGNO], Definitions 3.1.1, 3.3.1). This meansthat A is a free finitely generated abelian group with basis b i , i ∈ I anda unital associative ring structure such that b i b j = P k N kij b k with N kij ∈ Z + and b = 1 for some element 0 ∈ I , and for each j, k there exists i with N kij >
0. In this case we have a homomorphism FPdim : A → R ,determined uniquely by the condition that d i := FPdim( b i ) >
0, calledthe
Frobenius-Perron dimension ([EGNO], Subsection 3.3).Let N i be the matrix with entries N kij . Let d = ( d i ) be the columnvector with entries d i . Then we have N i d = d i d , i.e., d is a common(right) eigenvector of the matrices N i . The Frobenius-Perron theorem mplies that we also have a left eigenvector p = ( p i ) with p i > p N i = d i p . Lemma 2.1.
For each i, j, k one has p k ≥ N kji d j p i . In particular, onehas p k ≥ p /d k .Proof. One has d j p k = P i N kji p i , which implies the statement. (cid:3) If A is a fusion ring then N i ∗ = N Ti , so p is proportional to d andit is natural to normalize p by setting p = d ; in this case the numberFPdim( A ) := p · d = | d | = P i d i is called the Frobenius-Perrondimension of A ([EGNO], Section 3.3). However, in general p doesnot come with a natural normalization, which is why in [EGNO] theFrobenius-Perron dimension of a general Z + -ring of finite rank is notdefined.Note that when A is the Grothendieck ring of a finite tensor cate-gory C then one may take p i to be the Frobenius-Perron dimensionsof the indecomposable projectives, and p · d = P i p i d i is then calledthe Frobenius-Perron dimension of C . However, this normalization de-pends on the choice of C : e.g., the categories C of representations of Z / C of representations of Sweedler’s 4-dimensional Hopf algebracategorify the same Z + -ring but give different normalizations of p .Now recall ([EGNO]) that A is said to be integral if FPdim lands in Z . In this case, the vector p can be uniquely normalized in such a waythat p i are positive integers whose greatest common divisor is 1. Letus call this normalization the canonical normalization . Definition 2.2.
The Frobenius-Perron dimension of A is FPdim( A ) := p · d = P i p i d i , where p i are normalized canonically.In particular, we see that FPdim( A ) ≥ P i d i .Note that for fusion rings the canonical normalization just gives p = d as above, so our definition of FPdim( A ) agrees with the onein [EGNO]. However, in general p = 1, as shown by the followingexample. Example 2.3.
Consider the ring A with basis 1 , X with X = 4. Thus, A is the Grothendieck ring of the representation category C = Rep( U )of the restricted enveloping algebra U over a field k of characteristic2 generated by primitive elements x, y, z with z central, [ x, y ] = z and x = y = 0 , z = z . Then d = 1 , d = 2, p = 2 , p = 1. This showsthat the normalization with p = 1 is not very natural: this would give p = 1 /
2, which is not an algebraic integer. .2. Grothendieck rings of integral finite tensor categories.
Now assume that A is the Grothendieck ring of a finite tensor cate-gory C . Then C is integral, i.e., of the form Rep H , where H is a finitedimensional quasi-Hopf algebra ([EGNO]). Thus the Frobenius-Perrondimensions of objects of C are their usual vector space dimensions, andFPdim( C ) = dim H . In this case p i divide the dimensions dim( P i ) ofthe indecomposable projectives P i but aren’t necessarily equal to them.E.g., if H is Sweedler’s 4-dimensional Hopf algebra, then p = p = 1but the indecomposable projectives P , P are 2-dimensional. It is easyto see that in general we have the following proposition. Proposition 2.4.
Let C be an integral finite tensor category with Grothendieckring A , and D be the greatest common divisor of dimensions of the pro-jective objects of C . Then FPdim( C ) = D · FPdim( A ) . We also have
Proposition 2.5.
Let X i be the simple objects of C . Then one has p i D ≥ d i , and the equality holds for a given i if and only if the object X i is projective. Otherwise, one has p i D ≥ d i .Proof. The first statement follows from the fact that the projectivecover P i of X i contains X i in its composition series. For the secondstatement, it is enough to note that by the Frobenius property of C wehave dim P i ≥ X i if X i is not projective. (cid:3) Example 2.6.
A tensor category C is called unipotent if its only simpleobject is the unit object, i.e. A = Z or, equivalently, FPdim( A ) = 1([EG1]). In other words, C = Rep H , where H is a local finite dimen-sional quasi-Hopf algebra. By [EGNO], Corollary 4.4.2, for dim H > p >
0. Moreover, in this casethe associated graded gr( H ) of H with respect to the radical filtrationis cocommutative , i.e. the group algebra of a finite unipotent groupscheme over k , so dim H = FPdim( C ) is a power of p (and clearly anypower can arise this way).3. Lower bound on the dimension of a Z + -generator Let A be an integral Z + -ring of rank r > b = P i m i b i ∈ A with m i ≥
0. Let us say that b is a Z + - generator of A if for some n all coefficients of 1 + b + ... + b n are positive. Let d = FPdim( b ) and N = FPdim( A ). This is a straightforward generalization of [W], Proposition 2.2(7) to the quasi-Hopf case, and I thank S. Gelaki for pointing this out. heorem 3.1. (i) Let χ be the characteristic polynomial of b on A ,and Q ( z ) = ( z − d ) − χ ( z ) . Then Q ( d ) is a positive integer divisible by N . In particular, Q ( d ) ≥ N .(ii) Let s be the number of roots of Q with positive imaginary part(counted with multiplicities). Then d ≥ N r − (cid:18) − s + 1 r (cid:19) − sr − , (iii) We have d ≥ N r − (cid:18) ⌈ r − ⌉ r (cid:19) ⌈ r − ⌉ r − . In particular, d ≥ N r − .Proof. (i) First note that by the Frobenius-Perron theorem, d is a sim-ple eigenvalue of b , so Q ( d ) is a positive integer. It remains to showthat Q ( d ) is divisible by N . Let v be an integral row vector such that p · v = 1 (it exists since p i are relatively prime). We have Q ( b ) v = m d ,where m = ( Q ( b ) v ) is an integer. Therefore Q ( d ) = Q ( d ) p · v = p Q ( b ) v = m p · d = mN. This implies that N divides Q ( d ).(ii) Let λ i , i = 1 , ..., r − s − Q , and µ , ...µ s the roots with positive imaginary part. We have Q ( d ) = r − s − Y i =1 ( d − λ i ) s Y j =1 | d − µ j | ≥ N. Let α i = λ i /d and β j = µ j /d . Then r − s − Y i =1 (1 − α i ) s Y j =1 | − β j | ≥ Nd r − . This can be written as r − s − Y i =1 (1 − α i ) s Y j =1 (1 − β j + | β j | ) ≥ Nd r − . By the Frobenius-Perron theorem, | β j | ≤
1, so we get r − s − Y i =1 (1 − α i ) s Y j =1 (2 − β j ) ≥ Nd r − . hus by the arithmetic and geometric mean inequality, r − s − X i =1 (1 − α i ) + s X j =1 (2 − β j ) ≥ ( r − s − (cid:18) Nd r − (cid:19) r − s − . But the left hand side is the trace of 1 − b/d , so it is ≤ r , as Tr( b ) ≥ r ≥ ( r − s − (cid:18) Nd r − (cid:19) r − s − . This implies that d ≥ N r − (cid:18) − s + 1 r (cid:19) − sr − , as desired.(iii) Analyzing the function of s in (ii) and using that 2 s ≤ r −
1, wefind easily that the worst case scenario is the maximal possible valueof s , i.e., s = ⌊ r − ⌋ . This gives the result. The last statement followsfrom the fact that (cid:18) ⌈ r − ⌉ r (cid:19) ⌈ r − ⌉ r − ≥ . (cid:3) Auxiliary lemmas
Lemma 4.1.
Let H be a local finite dimensional quasi-Hopf algebra, M a finite dimensional H -module, and a : M → M ∗∗ a homomorphism.Then generalized eigenspaces of a (regarded as a linear map M → M )are H -submodules of M . In particular, if M is indecomposable, thenall eigenvalues of a are the same.Proof. The associated graded gr( H ) of H with respect to the radicalfiltration is a Hopf quotient of an enveloping algebra, so it has S = 1.Thus the operator S − ah = S ( h ) a , so( a − λ ) hv = h ( a − λ ) v + ( S − h ) av We claim that( a − λ ) N hv = N X i =0 (cid:18) Ni (cid:19) ( S − i ( h )( a − λ ) N − i a i v. he proof is by induction in N . Namely, the case N = 0 is clear, andfor the induction step note that( a − λ ) N X i =0 (cid:18) Ni (cid:19) ( S − i ( h )( a − λ ) N − i a i v == N X i =0 (cid:18) Ni (cid:19) ( S − i ( h )( a − λ ) N +1 − i a i v + N X i =0 (cid:18) Ni (cid:19) ( S − i +1 ( h )( a − λ ) N − i a i +1 v, so the induction step follows from the identity (cid:0) Ni (cid:1) + (cid:0) Ni − (cid:1) = (cid:0) N +1 i (cid:1) .Therefore, since ( S − i ( h ) is zero for sufficiently large i , the gen-eralized eigenspaces of a are H -submodules of M . This implies thestatement. (cid:3) Remark 4.2.
Note that in Lemma 4.1, the ordinary eigenspaces of a (as opposed to generalized eigenspaces) don’t have to be submodulesof M , e.g. M = H and a = S when S = 1. An example of a localHopf algebra with S = 1 is given in [NWW], Theorem 1.3, case B3. Lemma 4.3.
Let V be a simple object of a tensor category C , and a : V → V ∗∗ be an isomorphism. Then the morphism a ⊗ ( a ∗ ) − commutes with the composition of a ⊗ , evaluation and coevaluation V ⊗ V ∗ → V ∗∗ ⊗ V ∗ → → V ⊗ V ∗ Proof.
It is easy to see that the morphism a ⊗ ( a ∗ ) − preserves theevaluation and the coevaluation moprhism, so it commutes with theircomposition. It also obviously commutes with a ⊗ (cid:3) Now let H be a finite dimensional Hopf algebra over an algebraicallyclosed field k with antipode S such that S = 1. Lemma 4.4.
Let M be a finite dimensional H -module which tensorgenerates Rep( H ) and a : M → M ∗∗ be an isomorphism. Then a isnot a scalar (when regarded as a linear map M → M ).Proof. Assume the contrary. Then for any h ∈ H we have h | M = S ( h ) | M . Hence the same holds for any tensor product of copies of M and M ∗ . But since M tensor generates Rep( H ), some such productcontains the regular representation of H , which implies that S = 1 on H , a contradiction. (cid:3) Recall that a Hopf algebra H is called simple if it has no nontrivialHopf quotients. Lemma 4.5.
Let H be a simple finite dimensional Hopf algebra overan algebraically closed field k of characteristic = 2 with S = 1 . Let be an irreducible 2-dimensional H -module such that V ∼ = V ∗∗ . Then S = 1 . In particular, the distinguished grouplike element of H isnontrivial.Proof. Since H is simple and has a 2-dimensional simple module, H cannot have nontrivial 1-dimensional modules: otherwise H wouldhave a nontrivial commutative Hopf quotient. For the same reason,Ext ( k , k ) = 0, since otherwise H would have a nontrivial local Hopfquotient. Also it is clear that V tensor generates Rep( H ).Assume the contrary, i.e. that S = 1. Pick an isomorphism a : V → V ∗∗ such that a = 1 (this can be done since S = 1). Then a is semisimple with eigenvalues ± k ) = 2) and not a scalarby Lemma 4.4, so Tr( a ) = 0. Since H has no nontrivial 1-dimensionalmodules and Ext H ( k , k ) = 0, this implies that the Loewy series of V ⊗ V ∗ must be , W, , where W is an irreducible 2-dimensional mod-ule. By Lemma 4.3, the morphism a ⊗ ( a ∗ ) − commutes with thecomposition of a ⊗
1, evaluation and coevaluation V ⊗ V ∗ → V ⊗ V ∗ .Hence the trace of the block of the morphism a ⊗ ( a ∗ ) − on the con-stituent W is −
2, which implies that a ⊗ ( a ∗ ) − must act by = − W . But since H is simple, W must tensor generate H , which gives acontradiction with Lemma 4.4. (cid:3) Example 4.6.
Let n ≥ q be a primitive n -th root of unity, and H = u q ( sl ) be the corresponding small quantum group over C , generated bya grouplike element K and skew-primitive elements E , F with K n = 1, E n = 0, F n = 0 and the usual commutation relations KE = q EK , KF = q − EK , [ E, F ] = K − K − q − q − . This Hopf algebra has a 2-dimensionaltautological representation V , which is simple, self-dual and tensorgenerates Rep( H ), Also S ( a ) = KaK − , so the order of S is theorder of q , which is n for odd n and n/ n . Thus S = 1 for n = 4. However, H is simple if and only if n is odd: otherwise it has a1-dimensional representation ψ such that ψ ( E ) , ψ ( F ) = 0, ψ ( K ) = − n = 4 is that therepresentation W is not irreducible but rather isomorphic to ψ ⊕ ψ andtherefore does not tensor generate Rep( H ). Lemma 4.7.
Let X be a simple object of a finite tensor category C ,such that X ⊗ X ∗ is projective. Then X is projective.Proof. Assume the contrary. Then we have P X ⊗ X ∗ = X ⊗ X ∗ ⊕ Y. ence Hom( X, P X ) = Hom( , P X ⊗ X ∗ ) = 0. Thus P X is the injectivehull of X , i.e. the socle of P X is X . Hence P X ⊗ X ∗ = 2 X ⊗ X ∗ ⊕ Z ,so dim Hom( X, P X ) = dim Hom( , P X , ⊗ X ∗ ) ≥
2, a contradiction. (cid:3) Finite dimensional quasi-Hopf algebras with twosimple modules
In this section we consider integral finite tensor categories of rank r = 2 unless otherwise specified.5.1. The lower bound for the Frobenius-Perron dimension ofa Z + -generator. For r = 2 the Z + -ring A is Z + -generated by thenontrivial basis element X , so we may take d = FPdim( X ). Theorem3.1 then gives the inequality d ≥ N/
2. More explicitly, let b = 1 , b = X . We have X = aX + b for some a ≥ b ≥
1. Hence d = ad + b . It iseasy to see that p = d − a and p = 1, so FPdim( A ) = p + p d = 2 d − a .Thus we get Lemma 5.1.
Let A be a transitive Z + -ring of rank , and let the di-mension of the nontrivial basis element X be d . Then N − ≥ d ≥ N/ , where N = FPdim( A ) . The case of large characteristic.
Now let C = Rep H be therepresentation category of a quasi-Hopf algebra H over k with two sim-ple modules. If C is semisimple, it is easy to see that C = Vec( Z / , ω ) isthe category of Z / ω . Sofrom now till the end of the section we assume that C is non-semisimple.Let C be the Cartan matrix of C . Lemma 5.2. | det C | ≤ FPdim( C ) d .Proof. Using the arithmetic and geometric mean inequality, we have | det C | = | c c − c c | ≤ max( c c , c c ) ≤≤ d max(( c + c d ) , ( c + c ) d ) ≤≤ d ( c + ( c + c ) d + c d ) = FPdim( C ) d . (cid:3) Let A be the Grothendieck ring of C . roposition 5.3. Suppose that the characteristic of k is or p > FPdim( C ) d , C is not pointed, and Id ∼ = ∗∗ as an additive functor.Then FPdim( C ) = m FPdim( A ) , where m > is an integer. In par-ticular, FPdim( C ) cannot be square free.Proof. By [EGNO], Theorem 6.6.1 and Lemma 5.2, P and P must beproportional in the Grothendieck ring of C , so P = ( d − a ) P . Thisimplies that P = m (( d − a ) + X ), where m is a positive integer.Hence FPdim( C ) = dim P + d dim P = m (2 d − a ) .Assume that m = 1. Since C is not pointed, it is unimodular. Alsowe have [ P : X ] = 1. By the Frobenius property of C this implies that P = X , so d − a = 0, a contradiction. Thus, m >
1, as desired. (cid:3)
Remark 5.4.
1. Note that the statement that m > H where H isa Nichols Hopf algebra of dimension 2 n ([N]) or exceptional quasi-Hopfalgebras of dimension 8 or 32.5.3. Arbitrary characteristic.Corollary 5.5.
In any characteristic one has
FPdim( C ) ≥ d · FPdim( A ) ,i.e., D ≥ d , with the equality if and only if X is projective. In partic-ular, FPdim( C ) ≥ FPdim( A ) . Moreover, if X is not projective then D ≥ d and FPdim( C ) ≥ FPdim( A ) .Proof. This follows from Proposition 2.5, since p = 1. (cid:3) Minimal categories.
Let us say that C is minimal if X is pro-jective. Proposition 5.6.
Suppose C is minimal and d > . Then d = p r where p = char( k ) , and a = p r − p s for some s ≤ r , so FPdim( A ) = p s ( p r − s +1) , FPdim( C ) = p r + s ( p r − s +1) , D = p r and dim P = b = p r + s .Proof. We have X = aX ⊕ P , and P is an iterated extension of , i.e.it coincides with the projective cover of in the unipotent subcategoryof C . This implies the statement, since the Frobenius-Perron dimensionof any unipotent category is the power of the characteristic (Example2.6). (cid:3) Corollary 5.7.
Let C be a minimal category of Frobenius-Perron di-mension pn > where p is a prime and p > n . Then p = n + 1 and n is a power of , i.e., p = 2 m + 1 is a Fermat prime. Moreover, char( k ) = 2 , d = p − and X = ( p − X + p − . roof. By Proposition 5.6 we have pn = q r + s ( q r − s + 1), where q is aprime. Since p > n , we must have s = 0, n = q r and p = q r + 1. Thus q is a power of 2 and p is a Fermat prime, as desired. (cid:3) Examples.Example 5.8.
Minimal categories as in Corollary 5.7 exist for everyFermat prime p . Namely, let G = Aff( p ) be the group of affine lineartransformations of the field F p , then we can take C = Rep k ( G ). Thisshows that Proposition 5.3 fails in small characteristic. Example 5.9.
Consider the case a = 0, i.e., X = d for some positiveinteger d . We claim that for d > A admits a categorifi-cation by a finite tensor category in characteristic p if and only if d isa power of p , and in particular it never admits such categorification incharacteristic 0.Indeed, let C be a finite tensor category with Grothendieck ring A .We may assume that C is tensor generated by X . Then C = C ⊕ C is Z / ( , X ) = Ext ( X, ) = 0. Thus P is an iteratedextension of and dim P = FPdim( C ) is a power of p , since C isunipotent (Example 2.6). Also, P X is an iterated extension of X , and X ⊗ P X = P . This implies that d [ P X : X ] = p m for some m , whichimplies that d = p s for some s .On the other hand, such categories exist for every s . Indeed, it isenough to construct such a category C for s = 1, then for any s wecan take the subcategory C s in C ⊠ s generated by X ⊠ s . For p = 2,an example of C is given in Example 2.3. If p >
2, let H be the Hopfalgebra Z / ⋉ k [ x, y ] / ( x p , y p ), where x, y are primitive and the generator g ∈ Z / gxg − = − x, gyg − = − y . Let J = P p − j =0 x j ⊗ y j j ! be atwist for H , and H J be the corresponding twisted triangular Hopfalgebra. Then we can take C to be the category of comodules over H J .5.6. Categorifications of X = X + d ( d − . Suppose a = 1, i.e., X = X + d ( d − C be a finite tensor category with Grothendieckring A . Since X is self-dual, this implies that any morphism a : X → X ∗∗ has zero trace. Indeed, otherwise X ⊗ X = ⊕ Y where Y hasonly one copy of X in the composition series. Thus Hom( Y, ) = 0 (as Y is self-dual), so dim Hom( X ⊗ X, ) ≥
2, a contradiction.
Proposition 5.10.
The characteristic of k is a prime p dividing d ( d − . Moreover, if H is a Hopf algebra then p divides d .Proof. Consider the Loewy series of X ⊗ X . Exactly one of its termscontains X . So we have a 3-step filtration of X ⊗ X with compo-sition factors M, X, N invariant under any automorphism of X ⊗ X , here M, N are indecomposable iterated self-extensions of (in fact, M and N ∗ are cyclic). The H -modules M and N factor through a localquasi-Hopf algebra H ′ . Therefore, by Lemma 4.1, for any isomorphism a : X → X ∗∗ the morphism a ⊗ a on X ⊗ X has only one eigenvalueon M and only one on N . Moreover, these two coincide by Lemma4.3; we will normalize a in such a way that this eigenvalue is 1. Then,computing the trace of a ⊗ a , we get d ( d −
1) = 0 in k , as claimed.Now assume that H is a Hopf algebra and let a , ..., a d be the eigen-values of a . We may assume that d >
2. Then the eigenvalues of a ⊗ a are a i a j . At least d ( d −
1) of these numbers are 1, so at most d of themare = 1. We claim that a i = 1 for all i or a i = − i . Indeed,if none of a i are 1 or − a i = 1 for all i , so a i a j = 1 for i = j ,hence a i = a − j , which is impossible since d >
2. So at least one a i is1 or −
1, and we can assume it is 1 by multiplying a by − n > i such that a i = 1. If a i = 1 , a j = 1 then a i a j , a j a i = 1, so 2 n ( d − n ) ≤ d . Thus for d > n = d , i.e. a i = 1 for all i . Thus, d = Tr( a ) = 0 in k , as desired. (cid:3) Categorifications of X = X + 2 . Now consider the case d = 2,i.e. the fusion rule X = X + 2. Let C X be the tensor subcategory of C generated by X . Proposition 5.11.
One has X ⊗ X = P ⊕ X , where P is a nontriv-ial extension of by . Moreover, P and X are the indecomposableprojectives of C X , so FPdim( C X ) = 6 .Proof. Let us first show that X ⊗ X = P ⊕ X . Assume the con-trary. Since X ⊗ X is self-dual and dim Hom( , X ⊗ X ) = 1, we getthat X ⊗ X must be indecomposable with Loewy series , X, . Thendim Hom( X, X ⊗ X ⊗ X ) = dim End( X ⊗ X ) = 2. But X ⊗ X ⊗ X isself-dual, has composition series X, X ⊗ X, X , and Hom(
X, X ⊗ X ) = 0.This means that X ⊗ X ⊗ X = 2 X ⊕ X ⊗ X . Thus for each n , X ⊗ n is a direct sum of copies of X and X ⊗ X . But for some n , the object X ⊗ n has a direct summand which is projective in C X . Hence X ⊗ X isprojective. Therefore, so is X (as it is a direct summand in X ⊗ X ⊗ X ).But then X must be a direct summand in X ⊗ X , a contradiction.Note that dim Hom( X, P ⊗ X ) = dim Hom( X ⊗ X, P ) = dim Hom(
P, P ) =2. Thus, P ⊗ X = 2 X . Thus, for any n , X ⊗ n is a direct sum of copiesof P and X . This implies that P and X are the indecomposable pro-jectives of C X . (cid:3) We note that such a category of dimension 6 does exist, e.g. Rep k ( S ),with char( k ) = 2. It can also be realized as the Z / ec k ( Z / Z / of Vec k ( Z / k ( Z / , ω ) Z / , where ω ∈ H ( Z / , k × ).5.8. Categories of dimension sn , where s is square free andcoprime to n .Proposition 5.12. Let char( k ) be zero or p > FPdim( C ) d , and FPdim( C ) = sn , where s is a square free number coprime to n . Assume that C isnot pointed, and Id ∼ = ∗∗ as additive functors. Then n = mℓ , where ℓ ≥ and m ≥ , so n ≥ .Proof. It follows from Proposition 5.3 that n = mℓ where ℓ = FPdim( A )and m ≥
2. So it remains to show that ℓ ≥
4. To do so, note that ℓ ≥
2, and if ℓ = 2 then C is pointed. Also, if ℓ = 3 then X = X + 2,so char( k ) = 2 (Proposition 5.10). (cid:3) Example 5.13. (see [Ne], Subsection 9.4) The bound n ≥
32 in Propo-sition 5.12 is sharp. Namely, let char( k ) = 2 and let B be the smallquantum group u q ( sl ) at a primitive 8-th root of unity q . It is a Hopfalgebra of dimension 2 = 128 generated by E, F, K with relations KE = q EK , KF = q − EK , [ E, F ] = K − K − q − q − , E = F = K − B does not admit an R -matrix, but itadmits a quasi-Hopf structure Φ with a factorizable R-matrix ([CGR]).Let D = Rep( B Φ ) be the corresponding nondegenerate braided cate-gory. This category contains a unique 1-dimensional nontrivial repre-sentation ψ such that ψ ( E ) = ψ ( F ) = 0, ψ ( K ) = − E generated by and ψ has trivial braiding, i.e., it isequivalent to Rep( Z / C := E ⊥ / E be the de-equivariantizationof the centralizer E ⊥ of E by Z /
2. The centralizer consists of repre-sentations with even highest weights (0 , , , ψ gives a category C with just two simple objects, and X (the quantumadjoint representation).The category C has Frobenius-Perron dimension 32 and is nondegen-erate braided. Its Grothendieck ring is generated by a basis element X with X = 2 X + 3. The indecomposable projectives P , P X have theLoewy series , X ⊕ X, and X, ⊕ , X , and we have X ⊗ X = ⊕ P X .Also C does not admit a pivotal structure: indeed, if g is a pivotalstructure and u : Id → ∗∗ the Drinfeld isomorphism then g − u is anautomorphism of the identity functor, hence acts by 1 on X (as X islinked to 1). But then the squared braiding c : X ⊗ X → X ⊗ X is unipotent, which is a contradiction with formula (4.10) of [CGR],as this formula implies that − c on the tensorproduct of highest weight vectors. oreover, any pivotal finite nondegenerate braided tensor category C over C with two simple objects is equivalent to Vec( Z /
2) with braidingdefined by the quadratic form Q on Z / Q (1) = ± i . Indeed,by the result of [GR], a pivotal finite nondegenerate braided tensorcategory over C must contain a simple projective object, so X is pro-jective, which implies that C is semisimple. Thus, the above exampleshows that the pivotality assumption in [GR] is essential. Remark 5.14.
I don’t know any other finite dimensional quasi-Hopfalgebras over C with a unique nontrivial simple module X , such thatdim X >
1. In particular, I don’t know if there exist Hopf algebraswith this property.5.9.
Non-minimal categories.Corollary 5.15. (i) Suppose C is non-minimal and FPdim( C ) = pn ,where p is a prime and p > n . Then FPdim( A ) divides n and thedimension of every projective object in C is divisible by p .Proof. By Corollary 5.5 we have FPdim( A ) ≤ p FPdim( C ) < p , sosince FPdim( A ) divides pn , it must divide n . Hence D is divisible by p , as desired. (cid:3) Corollary 5.15 has strong implications for the structure of C if n issmall. First of all note that if C is pointed then its Frobenius-Perrondimension has to be even. Proposition 5.16.
Let C be an integral category with two simple ob-jects of Frobenius-Perron dimension pn where p is a prime. If n ≤ then n = 2 and C is pointed or FPdim( C ) = 6 and C is a minimalcategory which categorifies the fusion rule X = X + 2 in characteristic .Proof. By Corollary 5.7, we may assume that C is non-minimal.Suppose n = 1. If p >
2, Corollary 5.15 implies that FPdim( A ) = 1,which is impossible, and if p = 2, the category is semisimple, hencepointed.Suppose n = 2. If p >
2, Corollary 5.15 implies that FPdim( A ) = 2,i.e., C is pointed. If p = 2 then d <
2, so d = 1 and again C is pointed.Suppose n = 3. If p >
3, Corollary 5.15 implies that FPdim( A ) =2 d − a = 3 > d , which implies that d = 2, a = 1, so b = 2 and X = X + 2. In the latter case, char( k ) = 2 (Subsection 5.7). Consideran isomorphism g : P → P ∗∗ , and let w ∈ H be the element realizingthe canonical isomorphism V → V ∗∗∗∗ such that ε ( w ) = 1. Then g − w : P → P is an automorphism, and we may normalize g so hat this automorphism is unipotent. Then the eigenvalues of g on theinvariant part of the graded pieces of the Loewy filtration of P are allequal to 1. Since the trace of any morphism X → X ∗∗ is zero, we seethat Tr( g ) cannot be zero since dim P = p and hence [ P : ] is odd.This is a contradiction, since this implies that is projective so C issemisimple. (cid:3) Representations of finite groups.
Recall that Aff( q ) denotesthe group of affine transformation x → αx + β over the finite field F q . Theorem 5.17.
Let G be a finite group. Then:(i) If G has exactly two irreducible representations over k then either G = Z / or char( k ) = p > .(ii) Let char( k ) = p > and let N be the largest normal p -subgroupof G . Then G has exactly two irreducible representations over k if andonly if G/N is one of the following:(1) G = Aff( q ) where q is a Fermat prime, and p = 2 ;(2) G = Aff( p + 1) where p is a Mersenne prime.(3) G = Aff(9) or G = Z / ⋉ Aff(9) (where the generator of Z / acts by the Galois automorphism of F ) and p = 2 .(4) G = Z / , p = 2 .Proof. (i) Let char( k ) = 0 and X be the nontrivial irreducible repre-sentation of G . Then X ⊗ X has to contain dim X copies of the trivialrepresentation. Thus dim X = 1 and G = Z / G is trivial and using the easy factthat consecutive prime powers are q − , q for a Fermat prime q , or p, p + 1 for a Mersenne prime p , or 8 ,
9. See e.g. [CL] for a short proofof this statement. (cid:3) Lower bounds for integral finite tensor categorieswith no nontrivial invertible objects
Proposition 6.1.
Let C = Rep( H ) be a non-semisimple integral finitetensor category of rank r such that Ext ( , ) = 0 (e.g., char( k ) = 0 )and C has no nontrivial invertible objects. Then:(i) FPdim
C ≥ r + 3 ;(ii) If r = 2 then FPdim( C ) ≥ .In particular, these bounds hold if C is not pointed and simple (hasno nontrivial tensor subcategories).Proof. (i) It is clear that dim P ≥
4, since it has head and socle .Assume first that dim P = 4. Then P is uniserial with Loewyseries , V, for some 2-dimensional simple module V , so V ∗ ∼ = V and im Ext ( , V ) = 1, while Ext ( , Y ) = 0 for any simple Y = V . So if V ⊗ V has Loewy series , Y, , then Y must be simple, and we musthave Y ∼ = V (as otherwise Ext ( , Y ) = 0). Then V ⊗ V = P , so V isprojective by Lemma 4.7, a contradiction. Thus, V ⊗ V = ⊕ X , where X is a self-dual simple 3-dimensionalmodule. Consider the tensor product P ⊗ V = P V ⊕ T , which hascomposition factors V, , X, V . Note that P V necessarily includes com-position factors V, , V . If P V contains no other composition factors,i.e. it is 5-dimensional, then T = X , so X = P X is projective. Butthen X ⊗ V = P V ⊕ Q (as Hom( X ⊗ V, V ) = 0), so Q is projectiveand dim Q = 1, which is impossible. Thus, P V is indecomposable 8-dimensional with Loewy series V, ⊕ X, V .Let W be a 2-dimensional projective module. Then W ∗ ⊗ W = P ,so End( W ⊗ V ) = Hom( W ∗ ⊗ W, V ⊗ V ) = Hom( P , ⊕ X ) = k . But C is unimodular, so the head and socle of any indecomposable projectivecoincide, and hence if P is a non-simple indecomposable projectivethen dim End( P ) ≥
2. Thus, W ⊗ V = Q is a 4-dimensional simpleprojective module. Moreover, Q ⊗ V = W ⊗ V ⊗ V = W ⊕ W ⊗ X ,andEnd( W ⊗ X ) = Hom( W ∗ ⊗ W, X ⊗ X ) = Hom( P , X ⊗ X ) = Hom( X ⊗ P , X ) . But Ext ( V, X ) = 0, so P X includes composition factors X, V, X , hencedim P X ≥
8. Thus X ⊗ P , which has dimension 12, cannot containmore than one copy of P X . Hence End( W ⊗ X ) = Hom( X ⊗ P , X ) = k . Thus W ⊗ X is a 6-dimensional simple projective module. So W canbe recovered from Q as the unique 2-dimensional summand in Q ⊗ V ,i.e., the assignment W Q is injective.Now suppose we have s projective 2-dimensional objects. Then theycontribute 4 s to FPdim( C ), and the corresponding modules Q con-tribute 16 s . The rest of the simple modules except , V, X contributeat least 8 each, so in total 8( r − s − P = 4, dim P V = 8, dim P X ≥ C ) ≥ · · s +16 s +8( r − s −
3) = 8 r +20+4 s ≥ r +20 , as claimed.Now assume that dim P ≥
5. Then by Lemma 2.1, for each 2-dimensional simple module W we have dim P W ≥ /
2, so W is notprojective, and dim P W ≥ Another way to get a contradiction is to note that V = V + 2 in theGrothendieck ring and apply Proposition 5.11. f dim P ≥ X be a simple submodule of P / . If dim X ≥ P X ≥ X, , X ). HenceFPdim( C ) ≥ · r −
2) = 8 r + 10, as claimed.It remains to consider the case dim X = 2, in which case we musthave dim P ≥
6. Then dim P X ≥
5. If dim P X = 5 then theLoewy series of P X is X, , X , so X ∗ ⊗ P X has composition series X ∗ ⊗ X, X ∗ , X ∗ ⊗ X . But X ∗ ⊗ P X = P ⊕ Q , and P has compo-sition series , X, X ′ , where dim X ′ = 2, and Q is a 4-dimensionalprojective, so cannot involve . Thus X ∗ ⊗ X = ⊕ Y , where Y is a3-dimensional simple, i.e. we have only one 2-dimensional simple con-stituent in X ∗ ⊗ P X , a contradiction. Thus, dim P X ≥
6. If dim P X > P > C ), soFPdim( C ) ≥
19 + 8( r −
2) = 8 r + 3, as claimed.Thus it remains to consider the case dim P = dim P X = 6. Then theLoewy series of P X must have the form X, ⊕ , X (so Ext ( , X ) = k )and the Loewy series of P therefore looks like , X ⊕ X, , where X is self-dual. Now consider the module X ⊗ X . If it has Loewy series , X ′ , where X ′ is simple 2-dimensional then Hom( P , X ⊗ X ) = k (as then X ⊗ X is not a quotient of P ), which is a contradiction sincewe have [ X ⊗ X, ] = 2. Thus, X ⊗ X = ⊕ Y , where Y is a 3-dimensional simple. So Y contributes at least 9 into FPdim( C ). Thuswe get FPdim( C ) ≥ · r −
3) = 8 r + 3, as claimed. Thisproves (i).To prove (ii), note that dim P is divisible by dim P X . Let X be thenontrivial simple object. If dim X ≥ P X ≥
9, so dim P ≥ C ) ≥ · X = 3 thendim P X ≥
7, so dim P ≥
7, which means that [ P : X ] ≥
2, sodim P ≥
8, dim P X ≥ C ) ≥
32, as claimed.Finally, the case dim X = 2 is impossible by Proposition 5.11, asExt ( , ) = 0. (cid:3) Example 6.2.
1. The bound FPdim( C ) ≥
27 for r = 3 in Proposition6.1 is sharp in any characteristic. Namely, it is attained for the category C of representations of the small quantum group H = u q ( sl ) where q is a primitive cubic root of 1 if char( k ) = 3 and of the restrictedenveloping algebra H = u ( sl ) in characteristic 3. In this case there arethree simple modules: , the 2-dimensional tautological module V andthe 3-dimensional Steinberg module X , which is projective. Moreover,the projective covers of and V have dimension 6 and Loewy series , V ⊕ V, and V, ⊕ , V .2. The bound FPdim( C ) ≥
32 for r = 2 in Proposition 6.1 is alsosharp, as shown by Example 5.13. . There exists a category as in Proposition 6.1 with dim P =4. Namely, let k = C , G be the binary icosahedral group (of order120), and V its 2-dimensional tautological representation. Let z be thenontrivial central element of G . Consider the supergroup G ⋉ V and thecategory C = Rep( G ⋉ V, z ) of representations of G on superspaces suchthat z acts by parity. Then P = ∧ V = Ind G ⋉ VG . The simple objects ofthis category are just simple G -modules, which are labeled by vertices ofthe affine Dynkin diagram e E and have dimensions 1 , , , , , , , , C ) = 480.4. Examples of categories of small dimension FPdim( C ) = 60 satis-fying the assumptions of Proposition 6.1 are given by the representa-tion categories Rep p ( A ) of the alternating group A in characteristics p = 2 , ,
5. Namely:Rep ( A ) has simple objects of dimension 1 , , , , , , ( A ) has simple objects of dimension 1 , , , , , ,
9; The Loewy series of P is , X, , where X is the 4-dimensional simple module.Rep ( A ) has simple objects of dimension 1 , , , ,
5. The Loewy series of P is , X, , where X isthe 3-dimensional simple module.In particular, these examples show that P can have Loewy series , V, with V of dimensions 2 , , Tensor categories of dimension p Now let C be a finite tensor category over an algebraically closedfield k of any characteristic q ≥ p > C is pointed then either it is semisimple (hence C =Vec( Z /p, ω )) or C is unipotent and char( k ) = p .It is conjectured that C is always pointed. The goal of this sectionis to prove it under some assumptions.Suppose C is not pointed, and let A be the Grothendieck ring of C .Let r be its rank, and d the smallest dimension of a nontrivial simpleobject (call it X ). Clearly d >
1. Let d i be the dimensions of sim-ple objects and p i the dimensions of the corresponding indecomposableprojectives. So P i p i d i = p . Also, since FPdim( A ) > p ,we must have FPdim( A ) = p and D = 1. Also we have Ext ( , ) = 0,since otherwise the tensor subcategory of C formed by iterated exten-sions of is nontrivial, and its dimension must divide p by [EGNO],Theorem 7.17.6. roposition 7.1. (i) C has at least simple objects.(ii) p ≥ .Proof. (i) Suppose C has two simple objects. Then by Theorem 3.1, d ≥ p/
2, so p ≥ d ≥ p /
4, hence p ≤
2, a contradiction.Now assume that C has three simple objects, and X, Y of Frobenius-Perron dimensions d, m . By Theorem 3.1, we have d ≥ ( p/ / , so3 d ≥ p . If P X = X then dim P X ≥ d so p ≥ d + m > d ≥ p, a contradiction. Similarly, if P Y = Y then dim P Y ≥ m so p ≥ d + 2 m > d ≥ p, again a contradiction. Thus X and Y are projective. This means thatthe block of is a tensor subcategory of C . The Frobenius-Perrondimension of this subcategory must divide p , so this subcategory mustbe trivial. Hence C is semisimple. Then X = 1 + aX + bY with b = 0,so d = 1 + ad + bm , so ( d, m ) = 1. Now X ⊗ Y = sX + tY , so dm = sd + tm , hence sd = ( d − t ) m . But s, t > X occurs in Y ⊗ Y and Y occurs in X ⊗ X . This gives a contradiction.Thus C has at least 4 simple objects.(ii) It follows from (i) and Proposition 6.1 that p ≥ · p ≥ (cid:3) Corollary 7.2.
One has d ≥ − / p r − . Proof.
This follows from Proposition 7.1, Theorem 3.1(iii) and the factthat the function f ( r ) := (cid:18) ⌈ r − ⌉ r (cid:19) ⌈ r − ⌉ r − satisfies f ( r ) ≥ f (4) = 2 − / if r ≥ (cid:3) Proposition 7.3.
Let H be a quasi-Hopf algebra of prime dimension p with r simple modules, where r > is not of the form k + 1 with k ∈ Z + , k ≥ (e.g., r ≤ ). Then there exists a simple H -module X = such that X ∼ = X ∗∗ .Proof. Assume the contrary. Then the distinguished invertible object χ of H must be (as χ ∗∗ ∼ = χ ), so X ∗∗∗∗ ∼ = X for all X by categoricalRadford’s formula ([EGNO], Subsection 7.19). Thus the group Z / H -modules by taking the dual. Thisimplies that r = 4 k + 1. t remains to show that r = 5. Indeed, for r = 5 the nontrivialsimple objects are X = X , X = X ∗ , X = X ∗∗ , X = X ∗∗∗ , so d j = d and p j = p for 1 ≤ j ≤
4. Consider the product P ⊗ X ∗ , of dimension dp . It decomposes in a direct sum of P j , 1 ≤ j ≤
4. Thus we have dp = np , where n is the number of summands. But p and p arecoprime (since p + 4 dp = p ), so d is divisible by p . Hence d = p and X j are projective for 1 ≤ j ≤
4. Thus Ext ( , ) cannot be zero,a contradiction. (cid:3) Hopf algebras of dimension p Now let H be a Hopf algebra of dimension p (a prime) over a field k of characteristic q . It was recently proved by R. Ng and X. Wang [NW]that if 4 q ≥ p or q = 2 then H = Fun( Z /p ) or q = p and H = k [ x ] /x p with ∆( x ) = x ⊗ ⊗ x or ∆( x ) = x ⊗ ⊗ x + x ⊗ x . Thisis conjectured to hold for any q , and our goal is to improve the bound4 q ≥ p .Assume that q > H is not of this form. If H is semisimpleand cosemisimple then by [EG3], Theorem 3.4, H is commutative andcocommutative, so this is impossible. Thus, by replacing H with H ∗ if needed (as done in [NW]), we may (and will) assume that H isnot semisimple. Also, by a standard argument, H and H ∗ have nonontrivial 1-dimensional modules (as their order must divide p ). Inparticular, the distinguished group-like elements of H and H ∗ must betrivial , so S = 1. Thus S = 1 (as dim H = 0 in k ) and defines aninvolution on the simple H -modules X i , sending X j to X ∗∗ j . Also, H issimple, so Rep( H ) is tensor generated by any nontrivial simple moduleand Ext ( , ) = 0.Let r be the number of simple H -modules, d j be the dimension of X j and p j be the dimension of the projective cover P j of X j . Proposition 8.1. (i) (cf. [EG3] , Lemma 2.10) There exists i suchthat X ∗∗ i = X i and p i and d i are both odd. Moreover, if there exists j = 0 with X j ∼ = X ∗∗ j then there exists i = 0 with p i odd.(ii) ( [NW] , Lemma 2.3) Suppose p i is odd and X ∗∗ i = X i . Then p i ≥ q + 2 .(iii) Let d ∗ be the maximum of all d j , and let i be such that X i ∼ = X ∗∗ i and p i is odd, with largest possible d i . Then we have p ≥ ( q + 2) d i + 1 d i X k = i max( d i d k /d ∗ , ! . iv) If d i = d ∗ then p ≥ ( q + 2) (cid:18) d i + P k = i d k d i (cid:19) . In particular, p ≥ ( q + 2) (cid:18) d i + 2 r − d i (cid:19) . If d i < d ∗ then p ≥ ( q + 2) (cid:18) d i + 1 + r − d i (cid:19) . (v) One has p ≥ q + 2) √ r − .(vi) If d i > then d i ≥ .Proof. (i) Let J be the set of j such that X ∗∗ j ∼ = X j . Since double dualis an involution, we have that P j / ∈ J p j d j is even. Hence P j ∈ J p j d j isodd (as they add up to p ). So there exists i such that p i d i is odd, whichproves the first statement.To prove the second statement, we may assume that d j or p j is even,otherwise we can take i = j . We may also assume that p is odd;otherwise i = 0 automatically. Now consider the even-dimensionalobject X ∗ j ⊗ P j = P ⊕ ⊕ k =0 m k P k . We see that P k =0 m k p k is odd.So there exists k such that p k is odd, and moreover, there is an oddnumber of such k . Hence the involution ∗∗ has a fixed point on the setof such k , which gives the statement.(ii) Let a : P i → P ∗∗ i be an isomorphism. We may choose a so that a = 1. Then Tr( a ) = 0, otherwise is a direct summand in P i , henceis projective and H is semisimple, a contradiction. Let m + , m − be themultiplicities of the eigenvalues 1 and − a . Then m + + m − = p i , m + − m − = sq for some integer s . It is clear that s has to be odd, so | m + − m − | ≥ q , and m − , m + > p i ≥ q + 2.(iii) By Lemma 2.1, p k ≥ N kji p i /d j ≥ N kji ( q + 2) /d j for any j, k . It isclear that for each k there exists j such that X k is a composition factorof X j ⊗ X i (so that N kji ≥
1) and d j ≤ d i d k . Thus p k d k ≥ ( q + 2)max( d i d k /d ∗ , /d i . Hence p = p i d i + X k = i p k d k ≥ ( q + 2) d i + 1 d i X k = i max( d i d k /d ∗ , ! , as desired.(iv) The first inequality follows immediately from (iii). The secondinequality follows from the first one, as d k ≥ k = 0. To prove he third inequality, take j such that d j = d ∗ , and separate the corre-sponding term in the sum in (iii), while replacing all the other termsby 1. This gives the statement.(v) It follows from (iv) that p ≥ ( q + 2) (cid:18) d i + r − d i (cid:19) , so the statement follows by the arithmetic-geometric mean inequality.(vi) This follows from Lemma 4.5. (cid:3) The following corollary improves the bound p > q from [NW]. Corollary 8.2.
One has p > ( q + 2) .Proof. By Proposition 7.1, r ≥
4. Assume first that r = 4. It isclear that X ∗∗ = X for any simple object X (otherwise we wouldhave objects X, X ∗ , X ∗∗ , X ∗∗∗ which are pairwise nonisomorphic, whichwould give r ≥ d i ≥
2, henceby Proposition 8.1(vi) d i ≥
3. Thus, by Proposition 8.1(iv) we get p > q + 2) / r ≥
5. In this case, if d i ≥ d i ≥
3, so if d i = d ∗ then by Proposition 8.1(iv)we have p > q + 2) /
3, and if d i = d ∗ then Proposition 8.1(iv) yields p > q + 2). On the other hand, if d i = 1 then Proposition 8.1(iv)yields p > r ( q + 2) ≥ q + 2).So in all cases we have p > q + 2) /
3, which proves the statement. (cid:3)
Proposition 8.3.
Suppose that for some i we have X i ∼ = X ∗∗ i , p i isodd and d i > . Then pq + 2 > − / p r − . Proof.
By Corollary 7.2, we have d i ≥ − / p r − . Thus by Proposition 8.1(ii) we have p > p i d i ≥ ( q + 2) d i ≥ − / ( q + 2) p r − , which implies the statement. (cid:3) Corollary 8.4. (i) Let H be a non-semisimple Hopf algebra of di-mension p in characteristic q for which there exists i = 0 such that ∗∗ i ∼ = X i (for instance, ∗∗ is isomorphic to Id as an additive functoron Rep( H ) , i.e., S is an inner automorphism). Then if pq + 2 ≤ − / p r − . then H is commutative and cocommutative.(ii) Let W be the Lambert W -function (the inverse to the function f ( z ) = ze z ), and φ ( x ) := 2 − / exp (cid:18) W (2 / log x ) (cid:19) . If pq + 2 ≤ φ ( p ) then H is commutative and cocommutative.Proof. (i) By Proposition 8.3, it suffices to show that there exists i suchthat X ∗∗ i ∼ = X i , d i > p i is odd. But this follows from Proposition8.1(i).(ii) Fix p and note that the function 2 √ x is increasing and 2 − / p /x is decreasing in x . Thus if s is the solution of the equation2 √ s = 2 − / p /s then by Proposition 8.1(v) and part (i), pq + 2 > max(2 √ x, − / p /x ) | x = r − ≥ √ s. Let y = 2 / √ s . We have y = p / /y . Thus 2 / log p = y log( y ) . So log( y ) = W (2 / log p ). Thus we get y = exp (cid:18) W (2 / log p ) (cid:19) , so 2 √ s = 2 − / exp (cid:18) W (2 / log p ) (cid:19) . This implies (ii). (cid:3)
Remark 8.5.
It is easy to show that φ ( x ) ∼ √ (cid:18) log x log log x (cid:19) / as x → ∞ . heorem 8.6. Let H be a Hopf algebra of prime dimension p over afield k characteristic q > , which is not commutative and cocommuta-tive. Then pq + 2 > max (cid:18) , min(9 , φ ( p )) (cid:19) . Proof.
If there exists simple X = with X ∼ = X ∗∗ then Corollary 8.4applies. Otherwise, Proposition 7.3 and Proposition 8.1(v) imply that p > r ( q + 2) ≥ q + 2). This together with Corollary 8.2 implies thetheorem. (cid:3) References [CL] S. Chebolu and K. Lockridge, Fields with indecomposable multiplicativegroups, arXiv:1407.3481, Expo. Math. 34 (2016) 237–242.[CGR] T. Creutzig, A. M. Gainutdinov, I. Runkel, A quasi-Hopf algebra for thetriplet vertex operator algebra, arXiv:1712.07260.[EGNO] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor categories, AMS,Providence, 2015.[EG1] P. Etingof, S. Gelaki, Quasisymmetric and unipotent tensor categories,Math. Res. Lett. 15 (2008), no. 5, 857–866.[EG2] P. Etingof, S. Gelaki, Finite dimensional quasi-Hopf algebras with radical ofcodimension 2, MRL, v. 11 (2004), issue 5, p. 685–696.[EG3] P. Etingof, S. Gelaki, On finite-dimensional semisimple and cosemisimpleHopf algebras in positive characteristic, IMRN, 1998, Issue 16, p. 851–864.[DN] S. Dolfi and G. Navarro, Finite groups with only one nonlinear irreduciblerepresentation, Communications in Algebra, volume 40, issue 11, 2012, pp.4324–4329.[GR] A. M. Gainutdinov, I. Runkel, Projective objects and the modified trace infactorisable finite tensor categories, arXiv:1703.00150.[Ne] C. Negron, Log-modular quantum groups at an even root of unity and quan-tum Frobenius I, arXiv:1812.02277.[N] W. Nichols, Bialgebras of type one, Comm. Algebra 6 (1978), no. 15, 1521–1552.[NW] R. Ng and X. Wang, Hopf algebras of prime dimension in positive charac-teristic, arXiv:1810.00476.[NWW] V. C. Nguyen, L. Wang and X. Wang, Classification of connected Hopfalgebras of dimension p3, I, arXiv:1309.0286.[W] X. Wang, Connected Hopf algebras of dimension p , arXiv:1208.2280. Department of Mathematics, Massachusetts Institute of Technol-ogy, Cambridge, MA 02139, USA
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