# Finite-dimensional quasi-Hopf algebras of Cartan type

## TL;DR

This paper introduces a general method for constructing and classifying finite-dimensional quasi-Hopf algebras of Cartan type, leading to new explicit examples and a better understanding of their structure over abelian groups.

## Contribution

It provides a novel construction method for finite-dimensional quasi-Hopf algebras of Cartan type and classifies certain radically graded cases with explicit examples.

## Key findings

- Classified finite-dimensional radically graded basic quasi-Hopf algebras over abelian groups.
- Constructed many new explicit examples of genuine quasi-Hopf algebras.
- Introduced and studied small quasi-quantum groups.

## Abstract

In this paper, we present a general method for constructing finite-dimensional quasi-Hopf algebras from finite abelian groups and braided vector spaces of Cartan type. The study of such quasi-Hopf algebras leads to the classification of finite-dimensional radically graded basic quasi-Hopf algebras over abelian groups with dimensions not divisible by $2,3,5,7$ and associators given by abelian $3$-cocycles. As special cases , the small quasi-quantum groups are introduced and studied. Many new explicit examples of finite-dimensional genuine quasi-Hopf algebras are obtained.