Related papers: On Examples and Classification of Frobenius Object…
A pair of adjoint functors $(F,G)$ is called a Frobenius pair of the second type if $G$ is a left adjoint of $\beta F\alpha$ for some category equivalences $\alpha$ and $\beta$. Frobenius ring extensions of the second kind provide examples…
Let G be a simple complex algebraic group. By using a notion of a G-category we define invariants of tangles with flat G-connections in their complements. We also show that quantized universal enveloping algebras at roots of unity provide…
The infinitesimal counterpart of a Lie groupoid is its Lie algebroid. As a vector bundle, it is given by the source vertical tangent bundle restricted to the identity bisection. Its sections can be identified with the invariant vector…
Given a finite set of $r$ points in a closed surface of genus $g$, we consider the torsion elements in the mapping class group of the surface leaving the finite set invariant. We show that the torsion elements generate the mapping class…
Using a notation of corner between edges when graph has a fixed rotation, i.e. cyclical order of edges around vertices, we define combinatorial objects - combinatorial maps as pairs of permutations, one for vertices and one for faces.…
We introduce the dual notions of $\mathcal{E}(\mathcal{X},M,\mathcal{Y})$ and $\mathcal{M}(\mathcal{X},M,\mathcal{Y})$, and investigate when they have enough injective objects or projective objects, when they are resolving or co-resolving,…
We give a general method to build categories of combinatorial manifolds, i.e. categories of combinatorial objects satisfying some local property at every "point", as coreflective subcategories of categories of relational presheaves. To do…
In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. In particular, we show that every Frobenius object in a monoidal category M arises from an ambijunction…
Let $p$ be an odd prime, $m\in {\mathbb N}$ and set $q=p^m$, $G=\operatorname{PSL}_n(q)$. Let $\theta$ be a standard graph automorphism of $G$, $d$ be a diagonal automorphism and $\operatorname{Fr}_q$ be the Frobenius endomorphism of…
Let $G$ denote a connected semisimple and simply connected algebraic group over an algebraically closed field $k$ of positive characteristic and let $g$ denote a regular element of $G$. Let $X$ denote any equivariant embedding of $G$. We…
In this chapter we survey some particular topics in category theory in a somewhat unconventional manner. Our main focus will be on monoidal categories, mostly symmetric ones, for which we propose a physical interpretation. These are…
We consider certain quotient algebras of tensor algebras of bimodules $M$ over a finite-dimensional algebra $R$, and we investigate Frobenius type properties of such algebras. Our main interest is in the case where $M=R^*$, the linear dual…
We define a Frobenius algebra over fusion categories of the form Rep$(G)\boxtimes$Rep$(G)$ which generalizes the diagonal subgroup of $G\times G$. This allows us to extend field theoretical constructions which depend on the existence of a…
Group-theoretical fusion categories are defined by data concerning finite groups and their cohomology: A finite group $G$ endowed with a three-cocycle $\omega$, and a subgroup $H\subset G$ endowed with a two-cochain whose coboundary is the…
We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga-Gorenstein ring. We then apply this result to the Frobenius category of special Cohen-Macaulay modules…
A Lie groupoid, called \textit{material Lie groupoid}, is associated in a natural way to any elastic material. The corresponding Lie algebroid, called \textit{material algebroid}, is used to characterize the uniformity and the homogeneity…
We classify indecomposable commutative separable (special Frobenius) algebras and their local modules in (untwisted) group-theoretical modular categories. This gives a description of modular invariants for group-theoretical modular data. As…
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…
We study twisted conjugacy classes of the unit element in different groups. Fel'shtyn and Troitsky showed that the twisted conjugacy class of the unit element of an abelian group is a subgroup for every automorphism. The structure is…
We study the Frobenius problem for certain k-tuplets, which include prime k-tuplets, in particular prime triplets and prime quadruplets. Moreover, we analyze some properties of the numerical semigroups associated with these tuplets.