Quasi-Quantum Planes and Quasi-Quantum Groups of Dimension $p^3$ and $p^4$

TL;DR

This paper classifies finite-dimensional graded Majid algebras, a type of quasi-quantum group, of dimensions p^3 and p^4, providing new examples within the quiver framework and linking to nonassociative geometry.

Contribution

It offers explicit classifications of graded pointed Majid algebras of dimensions p^3 and p^4, expanding the understanding of quasi-quantum groups in the quiver framework.

Findings

Explicit classification of Majid algebras of dimension p^3

Explicit classification of Majid algebras of dimension p^4

New examples of finite pointed quasi-quantum groups

Abstract

The aim of this paper is to contribute more examples and classification results of finite pointed quasi-quantum groups within the quiver framework initiated in \cite{qha1, qha2}. The focus is put on finite dimensional graded Majid algebras generated by group-like elements and two skew-primitive elements which are mutually skew-commutative. Such quasi-quantum groups are associated to quasi-quantum planes in the sense of nonassociative geomertry \cite{m1, m2}. As an application, we obtain an explicit classification of graded pointed Majid algebras with abelian coradical of dimension $p^3$ and $p^4$ for any prime number $p.$