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We study the number of indecomposable summands in tensor powers of the vector representation of SL2. Our main focus is on positive characteristic where this sequence of numbers and its generating function show fractal behavior akin to Mahler functions., Comment: 40 pages, many figures, comments welcome

A symmetric tensor category $\mathcal D$ over an algebraically closed field $k$ is incompressible if every tensor functor out of $\mathcal D$ is an embedding. E.g., the categories $Vec$ and $sVec$ of (super)vector spaces are incompressible. Moreover, by Deligne's theorem, if char$(k)=0$ then any tensor category of moderate growth uniquely fibres over $sVec$, so $Vec$ and $sVec$ are the only incompressible categories in this class. Similarly, in characteristic $p>0$, we have the incompressible Verlinde category $Ver_p$, and any Frobenius exact category of moderate growth uniquely fibres over $Ver_p$. More generally, the Verlinde categories $Ver_{p^n}$, $Ver_{p^n}^+$ are incompressible, and a key conjecture is that every tensor category of moderate growth uniquely fibres over $Ver_{p^\infty}$. This would make the above the only incompressible categories in this class. We prove a part of this conjecture, showing that every tensor category of moderate growth fibres over an incompressible one. So it remains to understand incompressible categories. We say that $\mathcal D$ is subterminal if it every tensor category admits at most one fibre functor to it, and a Bezrukavnikov category if the class of tensor categories that fibre over $\mathcal D$ is closed under quotients. Clearly, a subterminal Bezrukavnikov category is incompressible, and we conjecture the converse. We prove that $Ver_p$ is Bezrukavnikov, generalizing the result of Bezrukavnikov for $Vec$. We also find intrinsic sufficient conditions for incompressibility and subterminality. Namely, $\mathcal D$ is maximally nilpotent if the growth rates of symmetric powers are minimal. We show that a finite maximally nilpotent category is incompressible, and also subterminal if it satisfies an additional geometric reductivity condition. Then we verify these conditions for $Ver_{2^n}$.

This paper clarifies basic definitions in the universal construction of topological theories and monoidal categories. The definition of the universal construction is given for various types of monoidal categories, including rigid and symmetric. It is also explained how to set up the universal construction for non-monoidal categories. The second part of the paper explains how to associate a rigid symmetric monoidal category to a small category, a sort of the Brauer envelope of a category. The universal construction for the Brauer envelopes generalizes some earlier work of the first two authors on automata, power series and topological theories. Finally, the theory of pseudocharacters (or pseudo-representations), which is an essential tool in modern number theory, is interpreted via one-dimensional topological theories and TQFTs with defects. The notion of a pseudocharacter is studied for Brauer categories and the lifting property to characters of semisimple representations is established in characteristic 0 for Brauer categories with at most countably many objects. The paper contains a brief discussion of pseudo-holonomies, which are functions from loops in a manifold to real numbers similar to traces of the holonomies along loops of a connection on a vector bundle on the manifold. It concludes with a classification of pseudocharacters (pseudo-TQFTs) and their generating functions for the category of oriented two-dimensional cobordisms in the characteristic 0 case., Comment: 59 pages, many figures

Let G be a group and V a finite dimensional representation of G over an algebraically closed field k of characteristic p>0. Let $d_n(V)$ be the number of indecomposable summands of $V^{\otimes n}$ of nonzero dimension mod p. It is easy to see that there exists a limit $\delta(V):=\lim_{n\to \infty}d_n(V)^{1/n}$, which is positive (and $\ge 1$) iff V has an indecomposable summand of nonzero dimension mod p. We show that in this case the number $$ c(V):=\liminf_{n\to \infty} \frac{d_n(V)}{\delta(V)^n}\in [0,1] $$ is strictly positive and $$ \log (c(V)^{-1})=O(\delta(V)^2), $$ and moreover this holds for any symmetric tensor category over k of moderate growth. Furthermore, we conjecture that in fact $$ \log(c(V)^{-1})=O(\delta(V)) $$ (which would be sharp), and prove this for p=2,3; in particular, for p=2 we show that $c(V)\ge 3^{-\frac{4}{3}\delta(V)+1}$. The proofs are based on the characteristic p version of Deligne's theorem for symmetric tensor categories obtained in earlier work of the authors. We also conjecture a classification of semisimple symmetric tensor categories of moderate growth which is interesting in its own right and implies the above conjecture for all $p$, and illustrate this conjecture by describing the semisimplification of the modular representation category of a cyclic p-group. Finally, we study the asymptotic behavior of the decomposition of $V^{\otimes n}$ in characteristic zero using Deligne's theorem and the Macdonald-Mehta-Opdam identity.

In this paper we study the asymptotic behavior of the number of summands in tensor products of finite dimensional representations of affine (semi)group (super)schemes and related objects., Comment: 21 pages, many figures, revision, comments welcome

We study invariant theory of the general linear supergroup in positive characteristic. In particular, we determine when the symmetric group algebra acts faithfully on tensor superspace and demonstrate that the symmetric group does not always generate all invariants.

A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper we prove a characteristic p version of this theorem. Namely we show that a pre-Tannakian category over an algebraically closed field of characteristic p>0 admits a fiber functor into the Verlinde category Ver_p (i.e., is the representation category of an affine group scheme in Ver_p) if and only if it has moderate growth and is Frobenius exact. This implies that Frobenius exact pre-Tannakian categories of moderate growth admit a well-behaved notion of Frobenius-Perron dimension. It follows that any semisimple pre-Tannakian category of moderate growth has a fiber functor to Ver_p (so in particular Deligne's theorem holds on the nose for semisimple pre-Tannakian categories in characteristics 2,3). This settles a conjecture of the third author from 2015. In particular, this result applies to semisimplifications of categories of modular representations of finite groups (or, more generally, affine group schemes), which gives new applications to classical modular representation theory. For example, it allows us to characterize, for a modular representation V, the possible growth rates of the number of indecomposable summands in V^{\otimes n} of dimension prime to p., Comment: With an appendix by Alexander Kleshchev

We show that any slightly degenerate weakly group-theoretical fusion category admits a minimal non-degenerate extension. Let $d$ be a positive square-free integer, given a weakly group-theoretical non-degenerate fusion category $\mathcal{C}$, assume that $\text{FPdim}(\mathcal{C})=nd$ and $(n,d)=1$. If $(\text{FPdim}(X)^2,d)=1$ for all simple objects $X$ of $\mathcal{C}$, then we show that $\mathcal{C}$ contains a non-degenerate fusion subcategory $\mathcal{C}(\mathbb{Z}_d,q)$. In particular, we obtain that integral fusion categories of FP-dimensions $p^md$ such that $\mathcal{C}'\subseteq \text{sVec}$ are nilpotent and group-theoretical, where $p$ is a prime and $(p,d)=1$., Comment: 15 pages, final version

We study abelian envelopes for pseudo-tensor categories with the property that every object in the envelope is a quotient of an object in the pseudo-tensor category. We establish an intrinsic criterion on pseudo-tensor categories for the existence of an abelian envelope satisfying this quotient property. This allows us to interpret the extension of scalars and Deligne tensor product of tensor categories as abelian envelopes, and to enlarge the class of tensor categories for which all extensions of scalars and tensor products are known to remain tensor categories. For an affine group scheme G, we show that pseudo-tensor subcategories of RepG have abelian envelopes with the quotient property, and we study many other such examples. This leads us to conjecture that all abelian envelopes satisfy the quotient property.

In arXiv:1910.12059 Liu, Palcoux and Wu proved a remarkable necessary condition for a fusion ring to admit a unitary categorification, by constructing invariants of the fusion ring that have to be positive if it is unitarily categorifiable. The main goal of this note is to provide a somewhat more direct proof of this result. In the last subsection we discuss integrality properties of the Liu-Palcoux-Wu invariants., Comment: 9 pages, latex; the paper is updated with information that the first half of Question 5.8 has been answered (by other authors). Dedication to the memory of Earl J. Taft added

We study generalized Deligne categories and related tensor envelopes for the universal two-dimensional cobordism theories described by rational functions, recently defined by Sazdanovic and one of the authors., Comment: 60 pages

We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $\bf k$. If ${\rm char}({\bf k})=p>0$, we use this method to construct generalizations ${\rm Ver}_{p^n}$, ${\rm Ver}_{p^n}^+$ of the incompressible abelian symmetric tensor categories defined in arXiv:1807.05549 for $p=2$ and by Gelfand-Kazhdan and Georgiev-Mathieu for $n=1$. Namely, ${\rm Ver}_{p^n}$ is the abelian envelope of the quotient of the category of tilting modules for $SL_2(\bf k)$ by the $n$-th Steinberg module, and ${\rm Ver}_{p^n}^+$ is its subcategory generated by $PGL_2(\bf k)$-modules. We show that ${\rm Ver}_{p^n}$ are reductions to characteristic $p$ of Verlinde braided tensor categories in characteristic zero, which explains the notation. We study the structure of these categories in detail, and in particular show that they categorify the real cyclotomic rings $\mathbb{Z}[2\cos(2\pi/p^n)]$, and that ${\rm Ver}_{p^n}$ embeds into ${\rm Ver}_{p^{n+1}}$. We conjecture that every symmetric tensor category of moderate growth over $\bf k$ admits a fiber functor to the union ${\rm Ver}_{p^\infty}$ of the nested sequence ${\rm Ver}_{p}\subset {\rm Ver}_{p^2}\subset\cdots$. This would provide an analog of Deligne's theorem in characteristic zero and a generalization of the result of arXiv:1503.01492, which shows that this conjecture holds for fusion categories, and then moreover the fiber functor lands in ${\rm Ver}_p$., Comment: 60 pages, latex; in v.2 new subsections 3.6, 3.7, 3.8, 4.14, 4.19, small error fixed in Corollary 4.7; in v.3 added Remark 5.3, Proposition 5.4; v.4: a gap in Cor. 2.60(iii) and 2.61 fixed by adding an extra assumption. New proof of Theorem 4.2(i) added

We give two proofs of a level-rank duality for braided fusion categories obtained from quantum groups of type $C$ at roots of unity. The first proof uses conformal embeddings, while the second uses a classification of braided fusion categories associated with quantum groups of type $C$ at roots of unity. In addition we give a similar result for non-unitary braided fusion categories quantum groups of types $B$ and $C$ at odd roots of unity.

We describe the semisimplification of the monoidal category of tilting modules for the algebraic group GL_n in characteristic p > 0. In particular, we compute the dimensions of the indecomposable tilting modules modulo p., Comment: This version corrects a minor mistake in the proof of Lemma 3.4 in the published version

We develop a theory of Frobenius functors for symmetric tensor categories (STC) $\mathcal{C}$ over a field $\bf k$ of characteristic $p$, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor $F: \mathcal{C}\to \mathcal{C}\boxtimes {\rm Ver}_p$, where ${\rm Ver}_p$ is the Verlinde category (the semisimplification of ${\rm Rep}_{\bf k}(\mathbb{Z}/p)$). This generalizes the usual Frobenius twist functor in modular representation theory and also one defined in arXiv:1503.01492, where it is used to show that if $\mathcal{C}$ is finite and semisimple then it admits a fiber functor to ${\rm Ver}_p$. The main new feature is that when $\mathcal{C}$ is not semisimple, $F$ need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor $\mathcal{C}\to {\rm Ver}_p$. We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of $F$, and use it to show that for categories with finitely many simple objects $F$ does not increase the Frobenius-Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which $F$ is exact, and define the canonical maximal Frobenius exact subcategory $\mathcal{C}_{\rm ex}$ inside any STC $\mathcal{C}$ with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius-Perron dimension is preserved by $F$. We prove that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to ${\rm Ver}_p$. We also show that a sufficiently large power of $F$ lands in $\mathcal{C}_{\rm ex}$. Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category and show that a STC with Chevalley property is (almost) Frobenius exact., Comment: 36 pages, latex; small changes in v2; better proof of Prop 7.6

We give a nontrivial lower bound for global dimension of a spherical fusion category., Comment: 11 pages, contains an Appendix joint with Pavel Etingof

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic $p$ in terms of representations of the normnalizer of its Sylow $p$-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group $S_{n+p}$ in characteristic $p$, where $0\le n\le p-1$, and of the Deligne category $\underline{\rm Rep}^{\rm ab}S_t$, where $t\in \Bbb N$. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of $\mathfrak{sl}_2$. We also study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). Finally, we determine the semisimplifications of the tilting categories of $GL(n)$, $SL(n)$ and $PGL(n)$ in characteristic $2$. In the appendix, we classify categorifications of the Grothendieck ring of representations of $SO(3)$ and its truncations., Comment: 32 pages, latex; in v2 minor changes have been made, and the end of Section 7 as well as Section 8 are new; in v3 the nonsymmetric version of Andre-Kahn results on p.11-12 has been added

We study properties of symmetric fusion categories in characteristic $p$. In particular, we introduce the notion of a super Frobenius-Perron dimension of an object $X$ of such a category, and derive an explicit formula for the Verlinde fiber functor $F(X)$ of $X$ (defined by the second author) in terms of the usual and super Frobenius-Perron dimension of $X$. We also compute the decomposition of symmetric powers of objects of the Verlinde category, generalizing a classical formula of Cayley and Sylvester for invariants of binary forms. Finally, we show that the Verlinde fiber functor is unique, and classify braided fusion categories of rank two and triangular semisimple Hopf algebras in any characteristic., Comment: 19 pages, latex; corrected misprints in proof of Proposition 6.1 and added reference to [Ma] in Section 8

To every object $X$ of a symmetric tensor category over a field of characteristic $p>0$ we attach $p$-adic integers $\text{Dim}_+(X)$ and $\text{Dim}_-(X)$ whose reduction modulo $p$ is the categorical dimension $\text{dim}(X)$ of $X$, coinciding with the usual dimension when $X$ is a vector space. We study properties of $\text{Dim}_{\pm}(X)$, and in particular show that they don't always coincide with each other, and can take any value in $\mathbb{Z}_p$. We also discuss the connection of $p$-adic dimensions with the theory of $\lambda$-rings and Brauer characters., Comment: 20 pages, latex; added Proposition 2.9, Remark 2.11 in the new version; subsection 3.4 rewritten, small error pointed out by K. Coulembier corrected in proof of Proposition 3.6 and a another small error in Remark 3.3

We propose a conjectural extension to positive characteristic case of a well known Deligne's theorem on the existence of super fiber functors. We prove our conjecture in the special case of semisimple categories with finitely many isomorphism classes of simple objects., Comment: 15 pages

We survey some results on tensor products of irreducible Harish-Chandra bimodules. It turns out that such tensor products are semisimple in suitable Serre quotient categories. We explain how to identify the resulting semisimple tensor categories and describe some applications to representation theory., Comment: Contribution to the ICM 2014

We classify all fusion categories of rank 3 that admit a pivotal structure over an algebraically closed field of characteristic zero. Also in the Appendix (joint with D.Nikshych) we give some restrictions on Grothendieck rings of near-group categories., Comment: 27 pages

We prove a universal property of Deligne's category $\uRep^{ab}(S_d)$. Along the way, we classify tensor ideals in the category $\uRep(S_d)$., Comment: 19 pages

We give a new way to derive branching rules for the conformal embedding $$(\asl_n)_m\oplus(\asl_m)_n\subset(\asl_{nm})_1.$$ In addition, we show that the category $\Cc(\asl_n)_m^0$ of degree zero integrable highest weight $(\asl_n)_m$-representations is braided equivalent to $\Cc(\asl_m)_n^0$ with the reversed braiding., Comment: 16 pages, to appear in Communications in Mathematical Physics. Version 2 changes: Proof of main theorem made explicit, example 4.11 removed, references updated

Finite W-algebras are certain associative algebras arising in Lie theory. Each W-algebra is constructed from a pair of a semisimple Lie algebra g (our base field is algebraically closed and of characteristic 0) and its nilpotent element e. In this paper we classify finite dimensional irreducible modules with integral central character over W-algebras. In more detail, in a previous paper the first author proved that the component group A(e) of the centralizer of the nilpotent element under consideration acts on the set of finite dimensional irreducible modules over the W-algebra and the quotient set is naturally identified with the set of primitive ideals in U(g) whose associated variety is the closure of the adjoint orbit of e. In this paper for a given primitive ideal with integral central character we compute the corresponding A(e)-orbit. The answer is that the stabilizer of that orbit is basically a subgroup of A(e) introduced by G. Lusztig. In the proof we use a variety of different ingredients: the structure theory of primitive ideals and Harish-Chandra bimodules of semisimple Lie algebras, the representation theory of W-algebras, the structure theory of cells and Springer representations, and multi-fusion monoidal categories., Comment: 52 pages, preliminarly version, comments welcome; v2 53 pages, small corrections

We analyze the structure of the Witt group W of braided fusion categories introduced in the previous paper arXiv:1009.2117v2. We define a "super" version of the categorical Witt group, namely, the group sW of slightly degenerate braided fusion categories. We prove that sW is a direct sum of the classical part, an elementary Abelian 2-group, and a free Abelian group. Furthermore, we show that the kernel of the canonical homomorphism S: W --> sW is generated by Ising categories and is isomorphic to Z/16Z. Finally, we give a complete description of etale algebras in tensor products of braided fusion categories., Comment: 27 pages

Using truncated convolution of perverse sheaves on a flag variety Lusztig associated a monoidal category to a two sided cell in the Weyl group. We identify this category in the case which was not decided previously., Comment: 9 pages

We give a presentation for the cohomology algebra of the Spaltenstein variety of all partial flags annihilated by a fixed nilpotent matrix, generalizing the description of the cohomology algebra of the Springer fiber found by De Concini, Procesi and Tanisaki., Comment: 28 pages

We give a characterization of Drinfeld centers of fusion categories as non-degenerate braided fusion categories containing a Lagrangian algebra. Further we study the quotient of the monoid of non-degenerate braided fusion categories modulo the submonoid of the Drinfeld centers and show that its formal properties are similar to those of the classical Witt group., Comment: 37 pages; many improvements in exposition thanks to the referee suggestions; new section 3.6; new remarks 3.2, 4.12, 5.2, 5.29

Recently P. Deligne introduced the tensor category Rep(S_t) (for t not necessarily an integer) which in a certain precise sense interpolates the categories Rep(S_d) of representations of the symmetric groups S_d. In this paper we describe the blocks of Deligne's category Rep(S_t)., Comment: 43 pages, final version, includes an added appendix as well as other modifications recommended by the referee

We apply the yoga of classical homotopy theory to classification problems of G-extensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG to classifiying spaces of certain higher groupoids. In particular, to every fusion category C we attach the 3-groupoid BrPic(C) of invertible C-bimodule categories, called the Brauer-Picard groupoid of C, such that equivalence classes of G-extensions of C are in bijection with homotopy classes of maps from BG to the classifying space of BrPic(C). This gives rise to an explicit description of both the obstructions to existence of extensions and the data parametrizing them; we work these out both topologically and algebraically. One of the central results of the paper is that the 2-truncation of BrPic(C) is canonically the 2-groupoid of braided autoequivalences of the Drinfeld center Z(C) of C. In particular, this implies that the Brauer-Picard group BrPic(C) (i.e., the group of equivalence classes of invertible C-bimodule categories) is naturally isomorphic to the group of braided autoequivalences of Z(C). Thus, if C=Vec(A), where A is a finite abelian group, then BrPic(C) is the orthogonal group O(A+A^*). This allows one to obtain a rather explicit classification of extensions in this case; in particular, in the case G=Z/2, we rederive (without computations) the classical result of Tambara and Yamagami. Moreover, we explicitly describe the category of all (Vec(A1),Vec(A2))-bimodule categories (not necessarily invertible ones) by showing that it is equivalent to the hyperbolic part of the category of Lagrangian correspondences., Comment: 70 pages, latex; in the new version, a new Appendix written by Ehud Meir has been added

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

The category of character D-modules is realized as Drinfeld center of the abelian monoidal category of Harish-Chandra bimodules. Tensor product of Harish-Chandra bimodules is related to convolution of D-modules via the long intertwining functor (Radon transform) by a result of Beilinson and Ginzburg. Exactness property of the long intertwining functor on a cell subquotient of the Harish-Chandra bimodules category shows that the truncated convolution category can be realized as a subquotient of the category of Harish-Chandra bimodules. Together with the description of the truncated convolution category arXiv:math/0605628v3 this allows us to derive classification of irreducible character sheaves over $\mathbb C$ obtained by Lusztig by a different method. We also give a simple description for the top cohomology of convolution of character sheaves over $\mathbb C$ in a given cell modulo smaller cells and relate the so-called Harish-Chandra functor to Verdier specialization in the De Concini-Procesi compactification., Comment: 28 pages. This is a post-publication revision of the article. The statements (and their numbering) are identical to the published version but some proofs are corrected and/or clarified

We prove a general result which implies that the global and Frobenius-Perron dimensions of a fusion category generate Galois invariant ideals in the ring of algebraic integers., Comment: 7 pages; example is added in the Introduction

We introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups - weakly group-theoretical categories and solvable categories. These are fusion categories that are Morita equivalent to iterated extensions (in the world of fusion categories) of arbitrary, respectively solvable finite groups. Weakly group-theoretical categories have integer dimension, and all known fusion categories of integer dimension are weakly group theoretical. Our main results are that a weakly group-theoretical category C has the strong Frobenius property (i.e., the dimension of any simple object in an indecomposable C-module category divides the dimension of C), and that any fusion category whose dimension has at most two prime divisors is solvable (a categorical analog of Burnside's theorem for finite groups). This has powerful applications to classification of fusion categories and semsisimple Hopf algebras of a given dimension. In particular, we show that any fusion category of integer dimension <84 is weakly group-theoretical (i.e. comes from finite group theory), and give a full classification of semisimple Hopf algebras of dimensions pqr and pq^2, where p,q,r are distinct primes., Comment: 28 pages, latex; added many references and details in proofs

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

We prove a weak version of Lusztig's conjecture on explicit description of the asymptotic Hecke algebras (both finite and affine), and explain its relation to Lusztig's classification of character sheaves., Comment: 19 pages, mistake in Section 5.4 is corrected, other minor corrections; Appendix added

We classify semisimple module categories over the tensor category of representations of quantum SL(2) extending previous results to the roots of unity and positive characteristic cases., Comment: 10 pages, title is changed, proofs of Lemma 2.6 and Proposition 2.7 are expanded, other minor changes

We classify ribbon semisimple monoidal categories with three isomorphism classes of simple objects over the field of complex numbers., Comment: 7 pages, reference added

We study preprojective algebras of graphs and their relationship to module categories over representations of quantum SL(2). As an application, ADE quiver varieties of Nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by Lusztig., Comment: 23 pages, exposition improved

These notes give an exposition of Deligne's theorem on the existense of super fiber functor., Comment: 7 pages

In this note we show that all reductive groups are clean in odd characteristic. In characteristic 2 there are two cuspidal local systems (one for $F_4$ and one for $E_8$) which can not be handled by our method., Comment: 5 pages

We classify the module categories over the double (possibly twisted) of a finite group., Comment: 10 pages, mistake in Section 4 from published version is fixed