Related papers: Modular invariants for group-theoretical modular d…
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…
We develop a theory of \emph{locally Frobenius algebras} which are colimits of certain directed systems of Frobenius algebras. A major goal is to obtain analogues of the work of Moore \& Peterson and Margolis on \emph{nearly Frobenius…
We study exact module categories over the representation categories of finite-dimensional quasi-Hopf algebras. As a consequence we classify exact module categories over some families of pointed tensor categories with cyclic group of…
We develop a new method for analyzing moduli problems related to the stack of pure coherent sheaves on a polarized family of projective schemes. It is an infinite-dimensional analogue of geometric invariant theory. We apply this to two…
We study the geometry of algebraic monoids. We prove that the group of invertible elements of an irreducible algebraic monoid is an algebraic group, open in the monoid. Moreover, if this group is reductive, then the monoid is affine. We…
We introduce a complete set of combinatorial data that encode the category $2\mathfrak{Cob}$ of all $2$-cobordisms. As an application, we show that the local monoids of $2\mathfrak{Cob}$ do not have finitely axiomatizable equational…
We describe how dagger-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional 'quantum algebras'. We develop the concept of an involution monoid, and use it to construct a correspondence between…
We study representations of the double affine Lie algebra associated to a simple Lie algebra. We construct a family of indecomposable integrable representations and identify their irreducible quotients. We also give a condition for the…
In this paper, we classify irreducible representations of affine group superschemes over fields $F$ of characteristic not two in terms of those over a separable closure $F^{\mathrm{sep}}$ and their Galois twists. We also compute the…
Given a morphism of (small) groupoids with injective object map, we provide sufficient and necessary conditions under which the induction and co-induction functors between the categories of linear representations are naturally isomorphic. A…
Given an oligomorphic group $G$ and a measure $\mu$ for $G$ (in a sense that we introduce), we define a rigid tensor category $\underline{\mathrm{Perm}}(G; \mu)$ of "permutation modules," and, in certain cases, an abelian envelope…
Ng and Schauenburg generalized higher Frobenius-Schur indicators to pivotal fusion categories and showed that these indicators may be computed utilizing the modular data of the Drinfel'd center of the given category. We consider two classes…
In Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum $\mathbb{A}$ in a modular fusion category $\mathcal{C}$ was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper,…
We characterize Frobenius and separable monoidal algebra extensions $i: R\ra S$ in terms given by $R$ and $S$. For instance, under some conditions, we show that the extension is Frobenius, respectively separable, if and only if $S$ is a…
In a previous paper we showed that the category of cocommutative color Hopf algebras is semi-abelian in case the group $G$ is abelian and finitely generated and the characteristic of the base field is different from 2 (not needed if $G$ is…
Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or…
We study the structure of the category of graded, connected, countable-dimensional, commutative and cocommutative Hopf algebras over a perfect field $k$ of characteristic $p$. Every $p$-torsion object in this category is uniquely a direct…
Let p be an odd prime, F the field of p elements and G a finite abelian p-group with an arbitrary involutory automorphism. Extend this automorphism to the group algebra FG and consider the unitary and the symmetric normalized units of FG.…
An algebraic classification is given for spaces of holomorphic vector-valued modular forms of arbitrary real weight and multiplier system, associated to irreducible, T-unitarizable representations of the full modular group, of dimension…
In this paper, we classify certain subcategories of modules over a ring R. A wide subcategory of R-modules is an Abelian subcategory of R-Mod that is closed under extensions. We give a complete classification of wide subcategories of…