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Let $\mathscr {C}$ denote a finite group scheme-theoretical category over an algebraically closed field of characteristic $p\ge 0$ as introduced by the first author. For any indecomposable exact module category over $\mathscr {C}$, we classify its simple objects and provide an expression for their projective covers, in terms of double cosets and projective representations of certain closed subgroup schemes, which upgrades a result by Ostrik for group-theoretical fusion categories. As a byproduct, we describe the simples and indecomposable projectives of $\mathscr {C}$, and parametrize the Brauer-Piccard group of ${\rm Coh}(G)$ for any finite connected group scheme $G$. Finally, we apply our results to describe the blocks of the center of ${\rm Coh}(G)$, which is a group scheme-theoretical category., Comments appreciated

Let $W$ be a finite dimensional purely odd supervector space over $\mathbb{C}$, and let $\sRep(W)$ be the finite symmetric tensor category of finite dimensional superrepresentations of the finite supergroup $W$. We show that the set of equivalence classes of finite non-degenerate braided tensor categories $\C$ containing $\sRep(W)$ as a Lagrangian subcategory is a torsor over the cyclic group $\mathbb{Z}/16\mathbb{Z}$. In particular, we obtain that there are $8$ non-equivalent such braided tensor categories $\C$ which are integral and $8$ which are non-integral., Comment: 25 pages; Sections 2.5, 2.6 were renamed and slightly revised; Lemma 2.6, Proposition 2.7, Remark 2.8, Section 2.7 and Proposition 2.10 are new; Corollary 3.3 and its proof were revised; Section 5 was revised; Theorem 6.1 was reformulated and its proof was slightly revised; new references were added and some minor corrections were made

We extend \cite[Theorem 4.5]{DGNO} and \cite[Theorem 4.22]{LKW} to positive characteristic (i.e., to the finite, not necessarily fusion, case). Namely, we prove that if $\D$ is a finite non-degenerate braided tensor category over an algebraically closed field $k$ of characteristic $p>0$, containing a Tannakian Lagrangian subcategory $\Rep(G)$, where $G$ is a finite $k$-group scheme, then $\D$ is braided tensor equivalent to $\Rep(D^{\omega}(G))$ for some $\omega\in H^3(G,\mathbb{G}_m)$, where $D^{\omega}(G)$ denotes the twisted double of $G$ \cite{G2}. We then prove that the group $\mathcal{M}_{{\rm ext}}(\Rep(G))$ of minimal extensions of $\Rep(G)$ is isomorphic to the group $H^3(G,\mathbb{G}_m)$. In particular, we use \cite{EG2,FP} to show that $\mathcal{M}_{\rm ext}(\Rep(\mu_p))=1$, $\mathcal{M}_{\rm ext}(\Rep(\alpha_p))$ is infinite, and if $\O(\Gamma)^*=u(\g)$ for a semisimple restricted $p$-Lie algebra $\g$, then $\mathcal{M}_{\rm ext}(\Rep(\Gamma))=1$ and $\mathcal{M}_{\rm ext}(\Rep(\Gamma\times \alpha_p))\cong \g^{*(1)}$., Comment: 21 pages; to appear in the Journal of Algebra

We prove that every finite symmetric integral tensor category $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p>2$ admits a symmetric fiber functor to $\text{sVec}$. This proves Ostrik's conjecture \cite[Conjecture 1.3]{o} in this case. Equivalently, we prove that there exists a unique finite supergroup scheme $\mathcal{G}$ over $k$ and a grouplike element $\epsilon\in k[\mathcal{G}]$ of order $\le 2$, whose action by conjugation on $\mathcal{G}$ coincides with the parity automorphism of $\mathcal{G}$, such that $\mathcal{C}$ is symmetric tensor equivalent to $\Rep(\mathcal{G},\epsilon)$. In particular, when $\mathcal{C}$ is unipotent, the functor lands in $\Vect$, so $\mathcal{C}$ is symmetric tensor equivalent to $\Rep(U)$ for a unique finite unipotent group scheme $U$ over $k$. We apply our result and the results of \cite{g} to classify certain finite dimensional triangular Hopf algebras with the Chevalley property over $k$ (e.g., local), in group scheme-theoretical terms. Finally, we compute the Sweedler cohomology of restricted enveloping algebras over an algebraically closed field $k$ of characteristic $p>0$, classify associators for their duals, and study finite dimensional (not necessarily triangular) local quasi-Hopf algebras and finite (not necessarily symmetric) unipotent tensor categories over an algebraically closed field $k$ of characteristic $p>0$. The appendix by K. Coulembier and P. Etingof gives another proof of the above classification results using the recent paper \cite{Co}, and, more generally, shows that the maximal Tannakian and super-Tannakian subcategory of a symmetric tensor category over a field of characteristic $\ne 2$ is always a Serre subcategory., Comment: With an appendix by Kevin Coulembier and Pavel Etingof, 26 pages. Some changes in the appendix. Some changes in Propositions 2.4 and 3.5, Corollaries 2.6 and 5.10, and Section 3.4. 27 pages, Proposition 2.4 and its proof were modified

We prove an analog of Deligne's theorem for finite symmetric tensor categories $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $2$. Namely, we prove that every such category $\mathcal{C}$ admits a symmetric fiber functor to the symmetric tensor category $\mathcal{D}$ of representations of the triangular Hopf algebra $(k[\dd]/(\dd^2),1\ot 1 + \dd\ot \dd)$. Equivalently, we prove that there exists a unique finite group scheme $G$ in $\mathcal{D}$ such that $\mathcal{C}$ is symmetric tensor equivalent to $\Rep_{\mathcal{D}}(G)$. Finally, we compute the group $H^2_{\rm inv}(A,K)$ of equivalence classes of twists for the group algebra $K[A]$ of a finite abelian $p$-group $A$ over an arbitrary field $K$ of characteristic $p>0$, and the Sweedler cohomology groups $H^i_{\rm{Sw}}(\mathcal{O}(A),K)$, $i\ge 1$, of the function algebra $\mathcal{O}(A)$ of $A$., 14 pages, Theorem 2.21 is new

Let k be any field. We consider the Hopf-Schur group of k, defined as the subgroup of the Brauer group of k consisting of classes that may be represented by homomorphic images of Hopf algebras over k. We show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative Hopf algebras over k, respectively. As a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a Hopf algebra over k, revealing so that the Hopf-Schur group can be much larger than the Schur group of k., 12 pages, latex file

Let g be a finite dimensional complex semisimple Lie algebra, and let V be a finite dimensional represenation of g. We give a closed formula for the mth Frobenius-Schur indicator, m>1, of V in representation-theoretic terms. We deduce that the indicators take integer values, and that for a large enough m, the mth indicator of V equals the dimension of the zero weight space of V. For the classical Lie algebras sl(n), so(2n), so(2n+1) and sp(2n), this is the case for m greater or equal to 2n-1, 4n-5, 4n-3 and 2n+1, respectively., 12 pages, to appear in Journal of Algebra

Let p be a prime, and denote the class of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p, by RG(p). The purpose of this paper is to continue the structure theory of finite dimensional quasi-Hopf algebras started in math.QA/0310253 (p=2) and math.QA/0402159 (p>2). More specifically, we completely describe the class RG(p) for p>2. Namely, we show that if H\in RG(p) has a nontrivial associator, then the rank of H[1] over H[0] is \le 1. This yields the following classification of H\in RG(p), p>2, up to twist equivalence: (a) Duals of pointed Hopf algebras with p grouplike elements, classified in math.QA/9806074. (b) Group algebra of Z_p with associator defined by a 3-cocycle. (c) The algebras A(q), introduced in math.QA/0402159. This result implies, in particular, that if p>2 is a prime then any finite tensor category over C with exactly p simple objects which are all invertible must have Frobenius-Perron dimension p^N, N=1,2,3,4,5 or 7. In the second half of the paper we construct new examples of finite dimensional quasi-Hopf algebras H, which are not twist equivalent to a Hopf algebra. They are radically graded, and H/Rad(H)=C[Z_n^m], with a nontrivial associator. For instance, to every finite dimensional simple Lie algebra g and an odd integer n, coprime to 3 if g=G_2, we attach a quasi-Hopf algebra of dimension n^{dim(g)}., Latex; 8 pages

Following the ideas of our previous works math.QA/0008232 (joint with Andruskiewitsch) and math.QA/0101049, we study families of triangular Hopf algebras obtained by twisting finite supergroups by a twist lying entirely in the odd part. These families are parametrized by data (G,V,u,B), where G is a finite group, V its finite dimensional representation, u a central element of G of order 2 acting by -1 on V, and B an element of S^2V. We fix the discrete data G,V,u, and find the set of isomorphism classes of the members of the family as Hopf algebras, in terms of the continuous parameter B. This set is often infinite, which provides examples of nontrivial continuous families of triangular Hopf algebras. The lowest dimension in which such a family occurs is 32, in which case we get 3 families which are dual to the 3 families of pointed Hopf algebras of dimension 32 constructed recently by Grana. Furthermore, we show that if (S^2V)^G=0 then such continuous families are nontrivial not only up to a Hopf algebra isomorphism, but also up to twisting of the multiplication. Thus, they provide counterexamples to Masuoka's weakened Kaplansky's 10th conjecture, which claims that up to twisting, there are finitely many types of Hopf algebras in each dimension. Finally, we study the algebra structure of the duals of our families, and show they are direct sums of Clifford algebras. Since there are finitely many types of Clifford algebras in each dimension, this allows us to construct nontrivial families of rigid tensor structures on the abelian category of modules over a finite dimensional algebra, with a fixed Grothendieck ring., Comment: 9 pages, Latex

One of the most fundamental problems in the theory of finite- dimensional Hopf algebras is their classification over an algebraically closed field k of characteristic 0. This problem is extremely difficult, hence people restrict it to certain classes of Hopf algebras, e.g. to semisimple ones. Semisimple Hopf algebras deserve to be considered as "quantum" analogue of finite groups, but even so, the problem remains extremely hard (even in low dimensions) and very little is known. A great boost to the theory of Hopf algebras was given by Drinfeld who invented the fundamental class of (quasi)triangular Hopf algebras (A,R). Thus, it is natural to first try to classify semisimple triangular Hopf algebras (A,R) over k (i.e. R is unitary). The theory of such Hopf algebras is now essentially closed by a sequence of works by P. Etingof and the author. The key theorem in this theory states that A is obtained from a group algebra of a unique (up to isomorphism) finite group by twisting its usual comultiplication. The proof of this theorem relies on Deligne's theorem on Tannakian categories in an essentail way. The purpose of this paper is to explain the proof in full details. In particular, to discuss all the necessary background from category theory (e.g. symmetric, rigid, and Tannakian categories), and the properties of the category of representations of triangular (semisimple) Hopf algebras., 22 pages, latex; a few minor corrections were made and Section 7 was expended

A fundamental problem in the theory of Hopf algebras is the classification and explicit construction of finite-dimensional quasitriangular Hopf algebras over C. These Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld (e.g. the quantum double of a Hopf algebra). However, this intriguing problem turns out to be extremely hard and it is still widely open. One can hope that resolving the problem in the triangular case would contribute to the understanding of the general case. There is no complte classification of finite-dimensional triangular Hopf algebras yet. However, the problem was solved in the semisimple case by Etingof and the author, in the minimal triangular pointed case by the author, and more generally for triangular Hopf algebras with the Chevalley property by Andruskiewitsch, Etingof and the author. The purpose of this paper is to report on all of this progress, and to explain in full details the mathematics and ideas involved in this theory. The classification in the semisimple case relies on Deligne's theorem on Tannakian categories and on Movshev's theory in an essential way. We explian Movshev's theory in details, and refer the reader to math.QA/0007147 for a detailed discussion of the first aspect. We also discuss the existence of grouplike elements in quasitriangular semisimple Hopf algebras, and the representation theory of cotriangular semisimple Hopf algebras. We conclude the paper with a list of related open problems., Comment: 48 pages, latex; sections 7,8 were added

In a previous paper we prove that any semisimple triangular Hopf algebra A over an algebraically closed field of characteristic 0 (say the field of complex numbers C) is obtained from a finite group after twisting the ordinary comultiplication of its group algebra in the sense of Drinfeld; that is A=C[G]^J for some finite group G and a twist J\in C[G]\ot C[G]. In this paper we explicitly describe the representation theory of co-triangular semisimple Hopf algebras A^*=(C[G]^J)^* in terms of representations of some associated groups. As a corollary we prove that Kaplansky's 6th conjecture from 1975 holds for A^*; that is that the dimension of any irreducible representation of A^* divides the dimension of A., 7 pages, latex

Masuoka proved that for a prime p, semisimple Hopf algebras of dimension 2p over an algebraically closed field k of characteristic 0, are trivial (i.e. are either group algebras or the dual of group algebras). Westreich and the second author obtained the same result for dimension 3p, and then pushed the analysis further and among the rest obtained the same result for semisimple Hopf algebras H of dimension pq so that H and H^* are of Frobenius type (i.e. the dimensions of their irreducible representations divide the dimension of H). They concluded with the conjecture that any semisimple Hopf algebra H of dimension pq over k, is trivial. In this paper we use Theorem 1.4 in our previous paper q-alg/9712033 to prove that both H and H^* are of Frobenius type, and hence prove this conjecture., 5 pages, latex; in this version we slightly changed the proof of Lemma 2 which was originally incomplete. The paper will appear in the Journal of Algebra

In this paper we construct and study two new families of finite dimensional pointed Hopf algebras which generalize Radford's families. We show that over any infinite field which contains a primitive nth root of unity, one of the families contains infinitely many non-isomorphic Hopf algebras of any dimension of the form Nn^2, where 2, Comment: 19 pages, latex. This is an extended revision which will appear in J. Alg

In this paper we consider some properties of semisimple Hopf algebras of dimension pq where p and q are distinct primes. These properties are useful for classification of such Hopf algebras. In particular, we show that for such a Hopf algebra H, if H and H^* are of Frobenius type then H or H^* is a group algebra., 9 pages, latex

We generalize the definition of an exact sequence of tensor categories due to Brugui\`eres and Natale, and introduce a new notion of an exact sequence of (finite) tensor categories with respect to a module category. We give three definitions of this notion and show their equivalence. In particular, the Deligne tensor product of tensor categories gives rise to an exact sequence in our sense. We also show that the dual to an exact sequence in our sense is again an exact sequence. This generalizes the corresponding statement for exact sequences of Hopf algebras. Finally, we show that the middle term of an exact sequence is semisimple if so are the other two terms., Comment: 21 pages