Searcharxiv⌕ Search

arXiv subjects

Viktor Ostrik

Publications and source records attributed to Viktor Ostrik.

15 recordsLinked to original sources
arXiv2017

On fusion categories

Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any modular category (not necessarily hermitian) is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category and classify categories of prime dimension.

math.QA↗
arXiv2004

An analogue of Radford's S^4 formula for finite tensor categories

We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors ?^{**} and D tensor ^{**}? tensor D^{-1}. This provides a categorical generalization of D. Radford's S^4-formula for finite dimensional Hopf algebras and its generalizations for weak Hopf algebras and for quasi-Hopf algebras, and conjectured in general in \cite{EO}. When C is braided, we establish a connection between the above isomorphism and the Drinfeld isomorphism of C. We also show that a factorizable braided tensor category is unimodular (i.e., D=1). Finally, we apply our theory to prove that the pivotalization of a fusion category is spherical, and give a purely algebraic characterization of exact module categories.

math.QA↗
arXiv2004

From Double Hecke algebra to Fourier transform

The paper contains a systematic theory of the one-dimensional Double Hecke algebra, including applications to the difference Fourier transform, Macdonald's polynomials, Gaussian sums at roots of unity, and Verlinde algebras. The main result is the classification of finite-dimensional representations for generic q and at roots of unity.

math.QA↗
arXiv2003

The minimal degeneration singularities in the affine Grassmannians

The minimal degeneration singularities in the affine Grassmannians of simple simply-laced algebraic groups are determined to be either Kleinian singularities of type A, or closures of minimal orbits in nilpotent cones. The singularities for non-simply-laced types are studied by intersection cohomology and equivariant Chow group methods.

math.AG↗
arXiv2003

Finite tensor categories

We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D.Nikshych. In particular, we generalize to the categorical setting the Hopf and quasi-Hopf algebra freeness theorems due to Nichols-Zoeller and Schauenburg, respectively. We also give categorical versions of the theory of distinguished group-like elements in a finite dimensional Hopf algebra, of Lorenz's result on degeneracy of the Cartan matrix, and of the absence of primitive elements in a finite dimensional Hopf algebra in zero characteristic. We also develop the theory of module categories and dual categories for not necessarily semisimple finite tensor categories; the crucial new notion here is that of an exact module category. Finally, we classify indecomposable exact module categories over the simplest finite tensor categories, such as representations of a finite group in positive characteristic, representations of a finite supergroup, and representations of the Taft Hopf algebra.

math.QA↗
arXiv2003

Module categories over representations of $SL_q(2)$ and graphs

We classify module categories over the category of representations of quantum $SL(2)$ in a case when $q$ is not a root of unity. In a case when $q$ is a root of unity we classify module categories over the semisimple subquotient of the same category.

math.QA↗
arXiv2002

Fusion categories of rank 2

We classify semisimple rigid monoidal categories with two isomorphism classes of simple objects over the field of complex numbers. In the appendix written by P.Etingof it is proved that the number of semisimple Hopf algebras with a given finite number of irreducible representations is finite.

math.QA↗
arXiv2001

On q-analog of McKay correspondence and ADE classification of sl^(2) conformal field theories

The goal of this paper is to classify ``finite subgroups in U_q sl(2)'' where $q=e^{\piı/l}$ is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U_q sl(2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to sl^(2) at level k=l-2. We show that ``finite subgroups in U_q sl(2)'' are classified by Dynkin diagrams of types A_n, D_{2n}, E_6, E_8 with Coxeter number equal to $l$, give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in (sl(2))_k conformal field theory.

math.QA↗
arXiv2001

Module categories, weak Hopf algebras and modular invariants

We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and classify module categories over the fusion category of $\hat{sl}(2)$ at a positive integer level where we meet once again ADE classification pattern.

math.QA↗
arXiv2001

Calculating canonical distinguished involutions in the affine Weyl groups

Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. In this note we calculate the canonical distinguished involutions in the affine Weyl groups of rank <8. We also prove some partial results relating canonical distinguished involutions and Dynkin's diagrams of the nilpotent orbits in the Langlands dual group.

math.RT↗
arXiv2001

On tensor categories attached to cells in affine Weyl groups II

George Lusztig conjectured that asymptotic affine Hecke algebra of a simply connected group can be explicitly described in terms of convolution algebras. Main Theorem of this note (which is a continuation of RT/0010089) is a weak version of this Conjecture. This version is strong enough to reprove all previously known results (due to Nanhua Xi) in this direction, for example the case of type $\tilde A_n$, see QA/0010159.

math.RT↗
arXiv2000

Dimensions of quantized tilting modules

Let $U$ be the quantum group with divided powers in $p-$th root of unity for prime $p$. For any two-sided cell $A$ in the corresponding affine Weyl group one associates tensor ideal in the category of tilting modules over $U$. In this note we show that for any cell $A$ there exists tilting module $T$ from the corresponding tensor ideal such that biggest power of $p$ which divides $dim T$ is $p^{a(A)}$ where $a(A)$ is Lusztig's $a-$function. In new version some typos are corrected and exposition is improved following suggestions of the referee.

math.QA↗
arXiv1999

On the equivariant K-theory of the nilpotent cone

Let G be a simple algebraic group over the complex numbers. Let N be the cone of nilpotent elements in the Lie algebra of G. Let K_{G x C^*}(N) denote the Grothendieck group of the category of G x C^*-equivariant coherent sheaves on N. In this note we construct a Kazhdan-Lusztig type canonical basis of K_{G x C^*}(N) over representation ring of C^*. This basis is parametrized by the set of dominant weights for G. On the other hand we conjecture that this basis is close to the basis consisting of irreducible G-equivariant bundles on nilpotent orbits. This would give us a natural construction of Lusztig's bijection between two sets: \{dominant weights for G\} and \{pairs consisting of a nilpotent orbit O and irreducible G-equivariant bundle on O.

math.AG↗
arXiv1999

Cohomology of subregular tilting modules for small quantum groups

Let $U$ be the quantum group with divided powers in $l-$th root of unity and let $u\subset U$ be the Frobenius kernel. V.Ginzburg and S.Kumar proved that the cohomology algebra of $u$ with trivial coefficients is isomorphic to the functions algebra of the nilpotent cone of the corresponding Lie algebra. In this note we show that there exists tilting module $T$ such that the cohomology of $u$ with coefficients in $T$ is isomorphic to the functions algebra of the closure of the subregular nilpotent orbit.

math.QA↗