Quivers, Quasi-Quantum Groups and Finite Tensor Categories

TL;DR

This paper classifies finite-dimensional pointed Majid algebras and elementary quasi-Hopf algebras of finite representation type using quiver theory, providing insights into certain finite tensor categories with simple objects of Frobenius-Perron dimension 1.

Contribution

It offers a classification of finite-dimensional pointed Majid algebras of finite corepresentation type via quiver methods, linking to finite tensor categories.

Findings

Classification of elementary quasi-Hopf algebras of finite representation type

Description of finite tensor categories with simple objects of Frobenius-Perron dimension 1

Application of quiver representation theory to finite tensor categories

Abstract

We study finite quasi-quantum groups in their quiver setting developed recently by the first author in arXiv:0902.1620 and arXiv:0903.1472. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently a classification of elementary quasi-Hopf algebras of finite representation type, over the field of complex numbers. By the Tannaka-Krein duality principle, this provides a classification of the finite tensor categories in which every simple object has Frobenius-Perron dimension 1 and there are finitely many indecomposable objects up to isomorphism. Some interesting information of these finite tensor categories is given by making use of the quiver representation theory.