Finite quasi-quantum groups of diagonal type

TL;DR

This paper classifies finite-dimensional pointed Majid algebras of diagonal type, linking them to finite-dimensional pointed Hopf algebras over abelian groups, and provides methods for their explicit construction.

Contribution

It offers a classification of elementary quasi-Hopf algebras of diagonal type using a connection to Hopf algebras and Nichols algebras, with new construction techniques.

Findings

Classification of finite-dimensional graded pointed Majid algebras of diagonal type.

Explicit connection established between Majid algebras and Hopf algebras over abelian groups.

Efficient methods for constructing these algebras provided.

Abstract

The goal of the present paper is to classify an interesting class of elementary quasi-Hopf algebras, or equivalently, finite-dimensional pointed Majid algebras. By a Tannaka-Krein type duality, this determines a big class of pointed finite tensor categories. Based on some interesting observations of normalized 3-cocycles on finite abelian groups, we elucidate an explicit connection between our objective pointed Majid algebras and finite-dimensional pointed Hopf algebras over finite abelian groups. With a help of this connection and the successful theory of diagonal Nichols algebras over abelian groups, we provide a conceptual classification of finite-dimensional graded pointed Majid algebras of diagonal type. Some efficient methods of construction are also given.