Victor Ostrik

Module categories over the Drinfeld double of a finite group

Let C be a (semisimple abelian) monoidal category. A module category over C is a (semisimple abelian) category M together with a functor C×M → M and an associativity constraint (= natural isomorphism of two composition functors C × C × M → M) satifying natural axioms, see [18]. In physics, one is interested in the case when C is a category of representations of some vertex algebra V and irreducible objects of M are interpreted as boundary conditions for the conformal field theory associated to V, see [3, 9]. Thus, it is interesting from a physical point of view for a given monoidal category C to classify all possible module categories over C (it is known that in many interesting cases, the list of answers is finite, see [18]). This problem is also of mathematical interest, for example, the module categories with just one isomorphism class of irreducible objects are exactly the same as the fiber functors C → Vec, see [18]. It is known that, for a fusion categories of ŝl(2) at positive integer levels, the module categories are classified by ADE Dynkin diagrams, see [12, 17, 18]. In this paper, we consider another class of examples, known in physics as holomorphic orbifold models (see, e.g., [7, 11]). Let G be a finite group. It is well known that the monoidal structures (= associativity constraints) on the category Vec of G-graded vector spaces with the usual functor of tensor product are classified by the group H3(G,C∗), see [13]. For ω ∈ H3(G,C∗), let Vecω denote the corresponding monoidal category. Let D(G,ω) be the Drinfeld center

Fusion categories and homotopy theory

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.

Quiver varieties and Lusztig's algebra

Abstract 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.

On blocks of Deligne's category Rep(St)

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 Delignes category Rep(S_t).

Cohomology of Spaltenstein varieties

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.

Classification of finite-dimensional irreducible modules over -algebras

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.

Level-Rank Duality via Tensor Categories

AbstractWe give a new way to derive branching rules for the conformal embedding

On braided fusion categories I

(\hat{\mathfrak{sl}}_n)_m\oplus(\hat{\mathfrak{sl}}_m)_n\subset(\hat{\mathfrak{sl}}_{nm})_1.