相关论文: On the notion of exact sequence: from Hopf algebra…

Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be…

We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf…

We construct certain tensor categories that are dominated by finitely many simple objects. Objects in these categories are modules over rings of algebra integers. We show how to obtain TQFTs defined over algebra integers from these…

We define fully exact module categories, a subclass of exact module categories over a finite braided tensor category that is stable under the relative Deligne product. In contrast, we demonstrate with examples in both zero and non-zero…

The study of the pentagon (fusion) equation leds to the Structure and the Classification theorem for finite dimenasional Hopf algebras: there exists a one to one correspondence between the set of types of n-dimensional Hopf algebtras and…

These are the lecture notes for a short course on tensor categories. The coverage in these notes is relatively non-technical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of…

We prove that both, the embedding of the category of Hopf algebras into that of bialgebras and the forgetful functor from the category of Hopf algebras to the category of algebras, have right adjoints; in other words: every bialgebra has a…

Connections between heaps of modules and (affine) modules over rings are explored. This leads to explicit, often constructive, descriptions of some categorical constructions and properties that are implicit in universal algebra and…

In this paper we generalize Tannakian formalism to fiber functors over general tensor categories. We will show that (under some technical conditions) if the fiber functor has a section, then the source category is equivalent to the category…

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not…

Torsors under affine groups are generalized in the super context by super-torsors under affine super-groups. We investigate those super-torsors by using Hopf-algebra language and techniques. It is explicitly shown, under suitable…

We give the explicit construction of the product of an arbitrary family of coalgebras, bialgebras and Hopf algebras: it turns out that the product of an arbitrary family of coalgebras (resp. bialgebras, Hopf algebras) is the sum of a family…

Study of the quotient module of a finite-dimensional Hopf subalgebra pair in order to compute its depth yields a relative Maschke Theorem, in which semisimple extension is characterized as being separable, and is therefore an ordinary…

One of the most fundamental problems in the theory of finite- dimensional Hopf algebras is their classification over an algebraically closed field k of characteristic 0. This problem is extremely difficult, hence people restrict it to…

We define left and right kernels of representations of Hopf algebras. In the case of group algebras, left and right kernels coincide and they are the usual kernels of modules. In the general case we show that these kernels coincide with the…

We define model structures on exact categories which we call exact model structures. We look at the relationship between these model structures and cotorsion pairs on the exact category. In particular, when the underlying category is weakly…

In this survey article we propose the notion of a bound quiver for an exact category generalising the classical concept of the Gabriel quiver and its relation for a module category as certain ring extension. The notion is motivated by joint…

This tutorial is intended to give an accessible introduction to Hopf algebras. The mathematical context is that of representation theory, and we also illustrate the structures with examples taken from combinatorics and quantum physics,…

We study the de-equivariantization of a Hopf algebra by an affine group scheme and we apply Tannakian techniques in order to realize it as the tensor category of comodules over a coquasi-bialgebra. As an application we construct a family of…

We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects…