Quivers, Quasi-Quantum Groups and Finite Tensor Categories

Abstract

Abstract. We study finite quasi-quantum groups in their quiver setting developed recently by the first author. 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.

The Green rings of pointed tensor categories of finite type

“…It turns out that the representation category of H n,d is a pointed tensor category of finite type, but not connected. The category can be presented as n/d copies of C(d, 0, q) with d = ord q, see [12,13]. The dual of H n,d is exactly M (n, 0, q) with d = ord q, see [4].…”

“…Recently, the Green rings of the Taft algebras and the generalized Taft algebras were presented by generators and relations in [3] and [14] respectively. The aim of this paper is to compute the Clebsch-Gordan formulae and to present the Green rings of pointed tensor categories of finite type as classified in [13]. Such tensor categories are actually the comodule categories of some pointed coquasi-Hopf algebras of finite corepresentation type, or equivalently the module categories of some elementary quasi-Hopf algebras of finite representation type, in which there are only finitely many iso-classes of indecomposable objects, and as such their Green rings are suitable to study.…”

Quiver bialgebras and monoidal categories

“…admitting only finitely many indecomposable representations up to isomorphism, to describe hereditary pointed monoidal categories in which there are only finitely many iso-classes of indecomposable objects. Follow the terminology of [16], a finite monoidal category is said to be of finite type if it has only finitely many iso-classes of indecomposable objects. Finally we give two examples of quiver monoidal categories.…”

The quasi-Hopf analogue of $\mathrm{u}_q(\mathfrak{sl}_2)$

“…In [7], all pointed Majid algebras M (n, s, q) of finite representation type are classified. Such pointed Majid algebras are indeed the dual of the class of basic quasi-Hopf algebras A(n, s, q) which can be considered as the quasi-Hopf analogues of generalized Taft algebras.…”

“…

In [4], some quasi-Hopf algebras of dimension n 3 , which can be understood as the quasi-Hopf analogues of Taft algebras, are constructed. Moreover, the quasi-Hopf analogues of generalized Taft algebras are considered in [7], where the language of the dual of a quasi-Hopf algebra is used. The Drinfeld doubles of such quasi-Hopf algebras are computed in this paper.

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