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We study the algebraic structure and representation theory of the Hopf algebras ${}_J\mathcal{O}(G)_J$ when $G$ is an affine algebraic unipotent group over $\mathbb{C}$ with $\mathrm{dim}(G) = n$ and $J$ is a Hopf $2$-cocycle for $G$. The cotriangular Hopf algebras ${}_J\mathcal{O}(G)_J$ have the same coalgebra structure as $\mathcal{O}(G)$ but a deformed multiplication. We show that they are involutive $n$-step iterated Hopf Ore extensions of derivation type. The 2-cocycle $J$ has as support a closed subgroup $T$ of $G$, and ${}_J\mathcal{O}(G)_J$ is a crossed product $S \#_{\sigma}U(\mathfrak{t})$, where $\mathfrak{t}$ is the Lie algebra of $T$ and $S$ is a deformed coideal subalgebra. The simple ${}_J\mathcal{O}(G)_J$-modules are stratified by a family of factor algebras ${}_J\mathcal{O}(Z_g)_J$, parametrised by the double cosets $TgT$ of $T$ in $G$. The finite dimensional simple ${}_J\mathcal{O}(G)_J$-modules are all 1-dimensional, so form a group $\Gamma$, which we prove to be an explicitly determined closed subgroup of $G$. A selection of examples illustrate our results., Comment: Preliminary version, comments welcome. Some overlap with arXiv2009.07760 by the second author, parts of which are extended and corrected here

Let $\mathcal{C}:=\mathcal{C}(G,\omega,H,\psi)$ be a finite group scheme-theoretical category over an algebraically closed field of characteristic $p\ge 0$ as defined by the first author. For any indecomposable exact module category over $\mathcal{C}$, 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 of $G$. This upgrades a result of Ostrik for group-theoretical fusion categories in characteristic $0$, and generalizes our previous work for the case $\omega=1$. As a byproduct, we describe the simples and indecomposable projectives of $\mathcal{C}$. Finally, we apply our results to describe the blocks of the center of ${\rm Coh}(G,\omega)$.

Let $\mathscr {C}(G,H,\psi)$ be 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 $\mathscr {C}(G,H,\psi)$, 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 $\mathscr {C}(G,H,\psi)$, 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 ${\rm Coh}(G)$., Comment: Comments appreciated

We extend \cite{G} to the nonsemisimple case. We define and study exact factorizations $\B=\A\bullet \C$ of a finite tensor category $\B$ into a product of two tensor subcategories $\A,\C\subset \B$, and relate exact factorizations of finite tensor categories to exact sequences of finite tensor categories with respect to exact module categories \cite{EG}. We apply our results to study exact factorizations of quasi-Hopf algebras, and extensions of a finite group scheme theoretical tensor category \cite{G2} by another one. We also provide several examples to illustrate our results., Comment: 25 pages

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 use \cite{G} to study the algebra structure of twisted cotriangular Hopf algebras ${}_J\mathcal{O}(G)_{J}$, where $J$ is a Hopf $2$-cocycle for a connected nilpotent algebraic group $G$ over $\mathbb{C}$. In particular, we show that ${}_J\mathcal{O}(G)_{J}$ is an affine Noetherian domain with Gelfand-Kirillov dimension $\dim(G)$, and that if $G$ is unipotent and $J$ is supported on $G$, then ${}_J\mathcal{O}(G)_{J}\cong U(\g)$ as algebras, where $\g={\rm Lie}(G)$. We also determine the finite dimensional irreducible representations of ${}_J\mathcal{O}(G)_{J}$, by analyzing twisted function algebras on $(H,H)$-double cosets of the support $H\subset G$ of $J$. Finally, we work out several examples to illustrate our results., Comment: 27 pages; to appear in IMRN

Let $k$ be an algebraically closed field of characteristic $0$ or $p>2$. Let $\mathcal{G}$ be an affine supergroup scheme over $k$. We classify the indecomposable exact module categories over the tensor category ${\rm sCoh}_{\rm f}(\mathcal{G})$ of (coherent sheaves of) finite dimensional $\mathcal{O}(\mathcal{G})$-supermodules in terms of $(\mathcal{H},\Psi)$-equivariant coherent sheaves on $\mathcal{G}$. We deduce from it the classification of indecomposable {\em geometrical} module categories over $\sRep(\mathcal{G})$. When $\mathcal{G}$ is finite, this yields the classification of {\em all} indecomposable exact module categories over the finite tensor category $\sRep(\mathcal{G})$. In particular, we obtain a classification of twists for the supergroup algebra $k\mathcal{G}$ of a finite supergroup scheme $\mathcal{G}$, and then combine it with \cite[Corollary 4.1]{EG3} to classify finite dimensional triangular Hopf algebras with the Chevalley property over $k$., Comment: 24 pages, added Lemma 5.4, to appear in JPAA. arXiv admin note: text overlap with arXiv:1209.1155

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 $A$., Comment: 14 pages, Theorem 2.21 is new

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 generalize the theory of the second invariant cohomology group $H^2_{\rm inv}(G)$ for finite groups $G$, developed in [Da2,Da3,GK], to the case of affine algebraic groups $G$, using the methods of [EG1,EG2,G]. In particular, we show that for connected affine algebraic groups $G$ over an algebraically closed field of characteristic $0$, the map $\Theta$ from [GK] is bijective (unlike for some finite groups, as shown in [GK]). This allows us to compute $H^2_{\rm inv}(G)$ in this case, and in particular show that this group is commutative (while for finite groups it can be noncommutative, as shown in [GK])., Comment: 21 pages; added references

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 introduce and study the new notion of an {\em exact factorization} $\mathcal{B}=\mathcal{A}\bullet \mathcal{C}$ of a fusion category $\mathcal{B}$ into a product of two fusion subcategories $\mathcal{A},\mathcal{C}\subseteq \mathcal{B}$ of $\mathcal{B}$. This is a categorical generalization of the well known notion of an exact factorization of a finite group into a product of two subgroups. We then relate exact factorizations of fusion categories to exact sequences of fusion categories with respect to an indecomposable module category, which was introduced and studied by P. Etingof and the author in \cite{EG}. We also apply our results to study extensions of a group-theoretical fusion category by another one, provide some examples, and propose a few natural questions., Comment: 14 pages; references added; to appear in the Journal of Algebra

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

We continue the study of twisting of affine algebraic groups G (i.e., of Hopf 2-cocycles J for the function algebra O(G)), which was started in [EG1,EG2], and initiate the study of the associated one-sided twisted function algebras O(G)_J. We first show that J is supported on a closed subgroup H of G (defined up to conjugation), and that O(G)_J is finitely generated with center O(G/H). We then use it to study the structure of O(G)_J for connected nilpotent G. We show that in this case O(G)_J is a Noetherian domain, which is a simple algebra if and only if J is supported on G, and describe the simple algebras that arise in this way. We also use [EG2] to obtain a classification of Hopf 2-cocycles for connected nilpotent G, hence of fiber functors Rep(G)\to Vect. Along the way we provide many examples, and at the end formulate several ring-theoretical questions about the structure of the algebras O(G)_J for arbitrary G., Comment: 21 pages

Let $k$ be an algebraically closed field of characteristic $p\ge 0$. Let $G$ be an affine group scheme over $k$. We classify the indecomposable exact module categories over the rigid tensor category $\text{Coh}_f(G)$ of coherent sheaves of finite dimensional $k-$vector spaces on $G$, in terms of $(H,\psi)-$equivariant coherent sheaves on $G$. We deduce from it the classification of indecomposable {\em geometrical} module categories over $\Rep(G)$. When $G$ is finite, this yields the classification of {\em all} indecomposable exact module categories over the finite tensor category $\Rep(G)$. In particular, we obtain a classification of twists for the group algebra $k[G]$ of a finite group scheme $G$. Applying this to $u(\mathfrak {g})$, where $\mathfrak {g}$ is a finite dimensional $p-$Lie algebra over $k$ with positive characteristic, produces (new) finite dimensional noncommutative and noncocommutative triangular Hopf algebras in positive characteristic. We also introduce and study group scheme theoretical categories, and study isocategorical finite group schemes., Comment: 30 pages

We study good (i.e., semisimple) reductions of semisimple rigid tensor categories modulo primes. A prime p is called good for a semisimple rigid tensor category C if such a reduction exists (otherwise, it is called bad). It is clear that a good prime must be relatively prime to the M\"uger squared norm |V|^2 of any simple object V of C. We show, using the Ito-Michler theorem in finite group theory, that for group-theoretical fusion categories, the converse is true. While the converse is false for general fusion categories, we obtain results about good and bad primes for many known fusion categories (e.g., for Verlinde categories). We also state some questions and conjectures regarding good and bad primes., Comment: 11 pages

We develop a theory of descent and forms of tensor categories over arbitrary fields. We describe the general scheme of classification of such forms using algebraic and homotopical language, and give examples of explicit classification of forms. We also discuss the problem of categorification of weak fusion rings, and for the simplest families of such rings, determine which ones are categorifiable., Comment: 22 pages; minor changes in Example 3.3, Proposition 3.7, Theorem 4.3(iv) and Theorem 4.6(iv)

Let k be any field. J-P. Serre proved that the spectrum of the Grothendieck ring of the k-representation category of a group is connected, and that the same holds in characteristic zero for the representation category of a Lie algebra over k. We say that a tensor category C over k is virtually indecomposable if its Grothendieck ring contains no nontrivial central idempotents. We prove that the following tensor categories are virtually indecomposable: Tensor categories with the Chevalley property; representation categories of affine group schemes; representation categories of formal groups; representation categories of affine supergroup schemes (in characteristic \ne 2); representation categories of formal supergroups (in characteristic \ne 2); symmetric tensor categories of exponential growth in characteristic zero. In particular, we obtain an alternative proof to Serre's Theorem, deduce that the representation category of any Lie algebra over k is virtually indecomposable also in positive characteristic (this answers a question of Serre), and (using a theorem of Deligne in the super case, and a theorem of Deligne-Milne in the even case) deduce that any (super)Tannakian category is virtually indecomposable (this answers another question of Serre)., Comment: 13 pages

Let \lambda be a partition of a positive integer n. Let C be a symmetric rigid tensor category over a field k of characteristic 0 or char(k)>n, and let V be an object of C. In our main result (Theorem 4.3) we introduce a finite set of integers F(\lambda) and prove that if the Schur functor \mathbb{S}_{\lambda} V of V is semisimple and the dimension of V is not in F(\lambda), then V is semisimple. Moreover, we prove that for each d in F(\lambda) there exist a symmetric rigid tensor category C over k and a non-semisimple object V in C of dimension d such that \mathbb{S}_{\lambda} V is semisimple (which shows that our result is the best possible). In particular, Theorem 4.3 extends two theorems of Serre for C=Rep(G), G is a group, and \mathbb{S}_{\lambda} V is \wedge^n V or Sym^n V, and proves a conjecture of Serre (\cite{s1})., Comment: 15 pages, minor corrections in Subsection 4.6 and in the proof of Lemma 4.20

This work is a detailed version of arXiv:0704.0195 [math.QA]. We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also give a comprehensive and self-contained exposition of the known results on braided fusion categories without assuming them pre-modular or non-degenerate. The guiding heuristic principle of our work is an analogy between braided fusion categories and Casimir Lie algebras., Comment: 102 pages, LaTeX, typos corrected

Let C be a fusion category faithfully graded by a finite group G and let D be the trivial component of this grading. The center Z(C) of C is shown to be canonically equivalent to a G-equivariantization of the relative center Z_D(C). We use this result to obtain a criterion for C to be group-theoretical and apply it to Tambara-Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara-Yamagami categories. Finally, we prove a general result about existence of zeroes in S-matrices of weakly integral modular categories., Comment: 27 pages

We prove that the quantum double of the quasi-Hopf algebra A_q(g) of dimension n^{dim g} attached in arXiv:math/0403096 to a simple complex Lie algebra g and a primitive root of unity q of order n^2 is equivalent to Lusztig's small quantum group u_q(g) (under some conditions on n). We also give a conceptual construction of A_q(g) using the notion of de-equivariantization of tensor categories., Comment: latex, 6 pages; proof of Theorem 4.1 added, a few references added

We first show that every group-theoretical category is graded by a certain double coset ring. As a consequence, we obtain a necessary and sufficient condition for a group-theoretical category to be nilpotent. We then give an explicit description of the simple objects in a group-theoretical category (following Ostrik, arXiv:math/0202130) and of the group of invertible objects of a group-theoretical category, in group-theoretical terms. Finally, under certain restrictive conditions, we describe the universal grading group of a group-theoretical category., Comment: Latex, 11 pages

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., Comment: 12 pages, latex file

We classify braided tensor categories over C of exponential growth which are quasisymmetric, i.e., the squared braiding is the identity on the product of any two simple objects. This generalizes the classification results of Deligne on symmetric categories of exponential growth, and of Drinfeld on quasitriangular quasi-Hopf algebras. In particular, we classify braided categories of exponential growth which are unipotent, i.e., those whose only simple object is the unit object. We also classify fiber functors on such categories. Finally, using the Etingof-Kazhdan quantization theory of Poisson algebraic groups, we give a classification of coconnected Hopf algebras, i.e. of unipotent categories of exponential growth with a fiber functor., Comment: 8 pages, latex

We characterize a natural class of modular categories of prime power Frobenius-Perron dimension as representation categories of twisted doubles of finite p-groups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects of C have integral Frobenius-Perron dimensions then C is group-theoretical. As a consequence, we obtain that semisimple quasi-Hopf algebras of prime power dimension are group-theoretical. Our arguments are based on a reconstruction of twisted group doubles from Lagrangian subcategories of modular categories (this is reminiscent to the characterization of doubles of quasi-Lie bialgebras in terms of Manin pairs)., Comment: 23 pages, LaTeX, typos corrected

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., Comment: 12 pages, to appear in Journal of Algebra

In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series of C. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Muger., Comment: 18 pages, AMS-Latex. Theorem 6.11 and 2 references added

Let p be a prime, and let RG(p) denote the set of equivalence classes of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p. The purpose of this paper is to classify finite dimensional quasi-Hopf algebras A whose radical is a quasi-Hopf ideal and has codimension p; that is, A with grA in RG(p), where grA is the associated graded algebra taken with respect to the radical filtration on A. The main result of this paper is the following theorem: Let A be a finite dimensional quasi-Hopf algebra whose radical is a quasi-Hopf ideal of prime codimension p. Then either A is twist equivalent to a Hopf algebra, or it is twist equivalent to H(2), H_\pm(p), A(q), or H(32), constructed in math.QA/0310253, math.QA/0402159 and math.QA/0403096. Note that any finite tensor category whose simple objects are invertible and form a group of order p under tensor is the representation category of a quasi-Hopf algebra A as above. Thus this paper provides a classification of such categories., Comment: 10 pages

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)}., Comment: Latex; 8 pages

It is shown in math.QA/0310253 that a finite dimensional quasi-Hopf algebra over the complex numbers with radical of codimension 2 is twist equivalent to a Nichols Hopf algebra, or to a lifting of one of four special quasi-Hopf algebras of dimensions 2, 8, 8, and 32. The purpose of this paper is to construct new finite dimensional basic quasi-Hopf algebras A(q) of dimension n^3, n>2, parametrized by primitive roots of unity q of order n^2, with radical of codimension n, which generalize the construction of the basic quasi-Hopf algebras of dimension 8 given in math.QA/0310253. These quasi-Hopf algebras are not twist equivalent to a Hopf algebra, and may be regarded as quasi-Hopf analogs of Taft Hopf algebras. By math.QA/0301027, our construction is equivalent to the construction of new finite tensor categories whose simple objects form a cyclic group of order n, and which are not tensor equivalent to a representation category of a Hopf algebra. In a later publication we plan to use our construction to classify finite tensor categories whose simple objects form a cyclic group of prime order n. We also prove that if H is a finite dimensional radically graded quasi-Hopf algebra with H[0]=(C[\Z/n\Z],\Phi), where n is prime and \Phi is a nontrivial associator, such that H[1] is a free left module over H[0] of rank 1 (it is always free) then H is isomorphic to A(q)., Comment: 8 pages, latex; minor corrections. The paper will appear in Journal of Pure and Applied Algebra

It is shown in math.QA/0301027 that a finite dimensional quasi-Hopf algebra with radical of codimension 1 is semisimple and 1-dimensional. On the other hand, there exist quasi-Hopf (in fact, Hopf) algebras, whose radical has codimension 2. Namely, it is known that these are exactly the Nichols Hopf algebras H_{2^n} of dimension 2^n, n\ge 1 (one for each value of n). The main result of this paper is that if H is a finite dimensional quasi-Hopf algebra over C with radical of codimension 2, then H is twist equivalent to a Nichols Hopf algebra H_{2^n}, n\ge 1, or to a lifting of one of the four special quasi-Hopf algebras H(2), H_+(8), H_-(8), H(32) of dimensions 2, 8, 8, and 32, defined in Section 3. As a corollary we obtain that any finite tensor category which has two invertible objects and no other simple object is equivalent to \Rep(H_{2^n}) for a unique n\ge 1, or to a deformation of the representation category of H(2), H_+(8), H_-(8), or H(32). As another corollary we prove that any nonsemisimple quasi-Hopf algebra of dimension 4 is twist equivalent to H_4., Comment: 9 pages, latex

In this paper we provide a complete classification of fusion categories of Frobenius-Perron (FP) dimension pq, where p1 and which have only one non-invertible object of dimension n., Comment: 12 pages, latex; Corollary 7.4 was added

In this paper we contribute to the classification of Hopf algebras of dimension pq, where p,q are distinct prime numbers. More precisely, we prove that if p and q are odd primes with p

We explain that a new theorem of Deligne on symmetric tensor categories implies, in a straightforward manner, that any finite dimensional triangular Hopf algebra over an algebraically closed field of characteristic zero has Chevalley property, and in particular the list of finite dimensional triangular Hopf algebras over such a field given in math.QA/0008232, math.QA/0101049 is complete. We also use Deligne's theorem to settle a number of questions about triangular Hopf algebras, raised in our previous publications, and generalize Deligne's result to nondegenerate semisimple categories in characteristic $p$, by using lifting methods developed in math.QA/0203060., Comment: 8 pages, latex

In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra pl(1,1), provides a noetherian prime counterexample., Comment: AMS-TeX; 8 Pages; no figures

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

Recall (math.QA/9812151) that the exponent of a finite-dimensional complex Hopf algebra H is the order of the Drinfeld element u of the Drinfeld double D(H) of H. Recall also that while this order may be infinite, the eigenvalues of u are always roots of unity (math.QA/9812151, Theorem 4.8); i.e., some power of u is always unipotent. We are thus naturally led to define the quasi-exponent of a finite-dimensional Hopf algebra H to be the order of unipotency of u. The goal of the paper is to create a theory of quasi-exponent, which would be parallel to the theory of the exponent developed in math.QA/9812151. In particular, we give two other equivalent definitions of the quasi-exponent, and prove that it is invariant under twisting. Furthermore, we prove that the quasi-exponent of a finite-dimensional pointed Hopf algebra H is equal to the exponent of the group G(H) of grouplike elements of H. (In particular, the order of the squared antipode of H divides exp(G(H)).) As an application, we find that if H is obtained by twisting the quantum group at root of unity U_q(g) then the order of any grouplike element in H divides the order of q., Comment: 8 pages, latex; new results about twists of U_q(g) were proved in the new version

In this paper we study the properties of Drinfeld's twisting for finite-dimensional Hopf algebras. We determine how the integral of the dual to a unimodular Hopf algebra $H$ changes under twisting of $H$. We show that the classes of cosemisimple unimodular, cosemisimple involutive, cosemisimple quasitriangular finite-dimensional Hopf algebras are stable under twisting. We also prove the cosemisimplicity of a coalgebra obtained by twisting of a cosemisimple unimodular Hopf algebra by two different twists on two sides (such twists are closely related to biGalois extensions), and describe the representation theory of its dual. Next, we define the notion of a non-degenerate twist for a Hopf algebra $H$, and set up a bijection between such twists for $H$ and $H^*$. This bijection is based on Miyashita-Ulbrich actions of Hopf algebras on simple algebras. It generalizes to the non-commutative case the procedure of inverting a non-degenerate skew-symmetric bilinear form on a vector space. Finally, we apply these results to classification of twists in group algebras and of cosemisimple triangular finite-dimensional Hopf algebras in positive characteristic, generalizing the previously known classification in characteristic zero., Comment: 14 pages, AMS-Latex. In Section 5 several results were improved and simplified. Minor errors corrected and several references added

Recall that a triangular Hopf algebra A is said to have the Chevalley property if the tensor product of any two simple A-modules is semisimple, or, equivalently, if the radical of A is a Hopf ideal. There are two reasons to study this class of triangular Hopf algebras: First, it contains all known examples of finite-dimensional triangular Hopf algebras; second, it can be, in a sense, "completely understood". Namely, it was shown in our previous work with Andruskiewitsch that any finite-dimensional triangular Hopf algebra with the Chevalley property is obtained by twisting of a finite-dimensional triangular Hopf algebra with R-matrix of rank at most 2 which, in turn, is obtained by "modifying" the group algebra of a finite supergroup. This provides a classification of such Hopf algebras. The goal of this paper is to make this classification more effective and explicit, i.e. to parameterize isomorphism classes of finite-dimensional triangular Hopf algebras with the Chevalley property by group-theoretical objects, similarly to how it was done in our previous work in the semisimple case. This is achieved in Theorem 2.2, where these classes are put in bijection with certain septuples of data. In the semisimple case, the septuples reduce to the quadruples, and we recover the result of our previous paper. In the minimal triangular pointed case, we recover a classification theorem from a previous paper of the second author., Comment: 7 pages, latex

We say that a Hopf algebra has the Chevalley property if the tensor product of any two simple modules over this Hopf algebra is semisimple. In this paper we classify finite dimensional triangular Hopf algebras with the Chevalley property, over the field of complex numbers. Namely, we show that all of them are twists of triangular Hopf algebras with R-matrix having rank <=2, and explain that the latter ones are obtained from group algebras of finite supergroups by a simple modification procedure. We note that all examples of finite dimensional triangular Hopf algebras which are known to the authors, do have the Chevalley property, so one might expect that our classification potentially covers all finite dimensional triangular Hopf algebras., Comment: 26 pages, latex. The first proof of Theorem 5.2.1, which is given in Section 5, was completed. The original proof missed a step, and we are grateful to A. Davydov for pointing it out

It is well known that if two finite groups have the same symmetric tensor categories of representations over C, then they are isomorphic. We study the following question: when do two finite groups G1,G2 have the same tensor categories of representations over C (without regard for the commutativity constraint). We call two groups with such property isocategorical. We give an example of two groups which are isocategorical but not isomorphic: the affine symplectic group of a vector space over the field of two elements, and an appropriate "affine pseudosymplectic group" introduced by R.Griess (containing the "pseudosymplectic group" of A.Weil). On the other hand, we give a classification of groups isocategorical to a given group. In particular, we show that if G has no nontrivial normal subgroups of order 2^{2m} then any group isocategorical to G must actually be isomorphic to G. The proofs use the theory of triangular Hopf algebras. We also apply the notion of isocategorical groups to studying the question: when are two triangular semisimple Hopf algebras isomorphic as Hopf algebras?, Comment: 17 pages, latex

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

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., Comment: 22 pages, latex; a few minor corrections were made and Section 7 was expended

In 1997 we proved that any triangular semisimple Hopf algebra over an algebraically closed field k of characteristic 0 is obtained from the group algebra k[G] of a finite group G, by twisting its comultiplication by a twist in the sense of Drinfeld. In this paper, we generalize this result to not necessarily finite-dimensional cotriangular Hopf algebras. Namely, our main result says that a cotriangular Hopf algebra A over k is obtained from a function algebra of a pro-algebraic group by twisting by a Hopf 2-cocycle, and possibly changing its R-form by a central grouplike element of A^* of order <=2, IF AND ONLY IF the trace of squared antipode on any finite-dimensional subcoalgebra of A is the dimension of this subcoalgebra. This generalization, like the original theorem, is proved using Deligne's theorem on Tannakian categories. In the second half of the paper, we give examples of twisted function algebras. In particular, we show that in the infinite- dimensional case, the squared antipode may not equal the identity. On the other hand, we show that in all of our examples of twisted function algebras, the squared antipode is unipotent, and conjecture it to be the case for any twisted function algebra. We prove this conjecture in a large number of special cases, using the quantization theory of the first author and D.Kazhdan., Comment: 15 pages, latex

Recall that a finite group is called perfect if it does not have non-trivial 1-dimensional representations (over the field of complex numbers C). By analogy, let us say that a finite dimensional Hopf algebra H over C is perfect if any 1-dimensional H-module is trivial. Let us say that H is biperfect if both H and H^* are perfect. Note that, H is biperfect if and only if its quantum double D(H) is biperfect. It is not easy to construct a biperfect Hopf algebra of dimension >1. The goal of this note is to describe the simplest such example we know. The biperfect Hopf algebra H we construct is based on the Mathiew group of degree 24, and it is semisimple. Therefore, it yields a negative answer to Question 7.5 from a previous paper of the first two authors (math.QA/9905168). Namely, it shows that Corollary 7.4 from this paper stating that a triangular semisimple Hopf algebra over C has a non-trivial group-like element, fails in the quasitriangular case. The counterexample is the quantum double D(H)., Comment: 5 pages, latex