Related papers: Rank 4 Premodular Categories
In the context of protomodular categories, several additional conditions have been considered in order to obtain a closer group-like behavior. Among them are locally algebraic cartesian closedness and algebraic coherence. The recent notion…
We define a diagrammatic monoidal category, together with a full and essentially surjective monoidal functor from this category to the category of modules over the exceptional Lie algebra of type $F_4$. In this way, we obtain a set of…
We define a notion on preadditive categories which plays a role similar to the notion of a Grothendieck pretopology on an unenriched category. Each such additive pretopology defines an additive Grothendieck topology and suffices to define…
The classification problem for simple $\mathfrak{sl}(2)$-modules leads in a natural way to the study of the category of finite rank torsion free $\mathfrak{sl}(2)$-modules and its subcategory of rational $\mathfrak{sl}(2)$-modules. We prove…
Let $\C$ be a self-dual fusion category of rank $4$ which has a nontrivial proper fusion subcategory. We identify three new families of Grothendieck rings for $\C$: one of them is completely determined, the other two are parameterized by…
A general procedure is presented which associates to a finite crossed module a premodular category, generalizing the representation categories of a finite group and of its double, and the extent to which the resulting category fails to be…
We define the affine Frobenius Brauer categories associated to each symmetric involutive Frobenius superalgebra $A$. We then define an action of these categories on the categories of finite-dimensional supermodules for orthosymplectic Lie…
We classify irreducible SL(2,K)-modules of low Morley rank (at most 4.rk(K)) as a first step towards a more general conjecture.
A pretorsion theory for the category of all categories is presented. The associated prekernels and precokernels are calculated for every functor.
We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the…
We introduce analogues of Soergel bimodules for complex reflection groups of rank one. We give an explicit parametrization of the indecomposable objects of the resulting category and give a presentation of its split Grothendieck ring by…
Frobenius-Schur indicators (or indicators for short) of objects in pivotal monoidal categories were defined and formulated by Ng and Schauenburg in 2007. In this paper, we introduce and study an analogous formula for indicators in the dual…
The main goal of this paper is to classify $\ast$-module categories for the $SO(3)_{2m}$ modular tensor category. This is done by classifying $SO(3)_{2m}$ nimrep graphs and cell systems, and in the process we also classify the $SO(3)$…
We classify integral modular categories of dimension pq^4 and p^2q^2 where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q^2. In these cases there are well-known…
We construct four series of modular categories from the two-variable Kauffman polynomial, without use of the representation theory of quantum groups at roots of unity. The specializations of this polynomial corresponding to quantum groups…
In this paper, we present a formulation of the moduli problem for rank-2 algebras over general base rings in functorial terms, providing presentations as presheaf quotients of affine schemes by group scheme actions.
We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This…
Grothendieck-Verdier categories (also known as $\ast$-autonomous categories) generalize rigid monoidal categories, with notable representation-theoretic examples including categories of bimodules, modules over Hopf algebroids, and modules…
We give an elementary classification and presentation of the finite quaternionic reflection groups of rank two, based on the notion of a``reflection system''. This simplifies the existing classification, which is shown to be incomplete,…
Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi-Yau triangulated categories with a choice of cluster-tilting object. The grading on the category corresponds to a grading on the…