Anyons in an exactly solved model and beyond

We develop a theory of module categories over monoidal categories (this is a straightforward categorization 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 we classify module categories over the fusion category of sl(2) at a positive integer level where we meet once again the ADE classification pattern.

Module categories, weak Hopf algebras and modular invariants

We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk
part is constructed using a unitary tensor category C as in the Levin-Wen model, whereas the boundary is associated with amodule category over C.We also consider domainwalls
(or defect lines) between different bulk phases.Adomainwall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of
higher codimension will also be studied. In summary, we give a dictionary between physical ingredients of lattice models and tensor-categorical notions.

/pdf/models-for-gapped-boundaries-and-domain-walls-4cgf0tpisu.pdf

Models for Gapped Boundaries and Domain Walls

https://www.math.uni-hamburg.de/home/runkel/PDF/tft1.pdf

TFT construction of RCFT correlators I: partition functions

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.

On braided fusion categories I

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 (not necessarily hermitian) modular category 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. At the end of the paper we generalize some of these results to positive characteristic.

http://annals.math.princeton.edu/wp-content/uploads/annals-v162-n2-p01.pdf

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.

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.

/pdf/finite-tensor-categories-1lchb4j72c.pdf

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 δ:V**→D⊗**V⊗D−1. This provides a categorical generalization of Radford's S 4 formula for finite-dimensional Hopf algebras (presented in 1976), which was proved by Nikshych (2002) for weak Hopf algebras and by Hausser and Nill (1999) for quasi-Hopf algebras, and was conjectured in general by Etingof and Ostrik (2003). When C is braided, we establish a connection between δ and the Drinfeld isomorphism of C, extending the result of Radford (1992). 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 defined by Etingof and Ostrik (2003).

An analogue of Radford's S4 formula for finite tensor categories

We provide a complete classification of fusion categories of Frobenius-Perron (FP) dimension pq over complex numbers, where p 1 and which have only one noninvertible object of dimension n.

Viktor Ostrik

Papers

On fusion categories

On fusion categories

Finite Tensor Categories

An analogue of Radford's S4 formula for finite tensor categories

Classification of fusion categories of dimension pq