Related papers: Monoidal 2-structure of Bimodule Categories

For any finite group G, we show that the 2-local G-equivariant stable homotopy category, indexed on a complete G-universe, has a unique equivariant model in the sense of Quillen model categories. This means that the suspension functor,…

We construct an equivalence between the 2-categories VMonCat of rigid V-monoidal categories for a braided monoidal category V and VModTens of oplax braided functors from V into the Drinfeld centers of ordinary rigid monoidal categories. The…

We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group $G$ and when restricted to either a Frobenius kernel $G_r$ or a finite Chevalley group…

Let $G$ be a finite group and $G'$ its commutator subgroup. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered…

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…

We produce a cofibrantly generated simplicial symmetric monoidal model structure for the category of (small unital) C*-categories, whose weak equivalences are the unitary equivalences. The closed monoidal structure consists of the maximal…

The feasibility of a classification-by-rank program for modular categories follows from the Rank-Finiteness Theorem. We develop arithmetic, representation theoretic and algebraic methods for classifying modular categories by rank. As an…

We study the problem of classification of simple transitive 2-representations for the (non-finitary) 2-category of bimodules over the dual numbers. We show that simple transitive 2-representations with finitary apex are necessarily of rank…

The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A…

We prove that for q\in\C* not a nontrivial root of unity the cohomology group defined by invariant 2-cocycles in a completion of Uq(g) is isomorphic to H^2(P/Q;\T), where P and Q are the weight and root lattices of g. This implies that the…

We consider non-trivial irreducible tensor products of modular representations of a symmetric group $S_n$ in characteristic 2 for even $n$ completing the proof of a classification conjecture of Gow and Kleshchev about such products.

It is well-known that the tensor product of two bialgebras constitutes the binary product in the category of cocommutative bialgebras and morphisms of bialgebras between them. In this paper, we extend this result to triangular bialgebras…

We classify braided extensions $C$ of a rank $2$ fusion category. The result shows that $C$ is tensor equivalent to a Deligne's tensor product of some known categories, except $C$ is slightly degenerate and generated by a…

A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant $\mathcal{C}'$ of a fully faithful representation $\mathcal{C}\to\operatorname{Bim}(R)$ of a…

This is the second part of the paper (the first part is published in Jour. of AMS, vol.9, 1135--1170, q-alg/9508017). In the first part, we defined for every modular tensor category (MTC) inner products on the spaces of morphisms and proved…

In this paper, we extend the notion of modular functor and fusion category to what we called $G$ equivariant modular functor and $G$ equivariant fusion category, where $G$ is a finite group, and establish a correspondence between between…

Let $\mathcal{C}$ be a finite tensor category and $\mathcal{M}$ an exact left $\mathcal{C}$-module category. We call $\mathcal{M}$ unimodular if the finite multitensor category ${\sf Rex}_{\mathcal{C}}(\mathcal{M})$ of right exact…

A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1-dimensional representations are the invertible…

We develop the idea of a supersymmetric monoidal supercategory, following ideas of Kapranov. Roughly, this is a monoidal category in which the objects and morphisms are ${\bf Z}/2$-graded, equipped with isomorphisms $X \otimes Y \to Y…

A graded tensor category over a group $G$ will be called a crossed product tensor category if every homogeneous component has at least one multiplicatively invertible object. Our main result is a description of the crossed product tensor…