Liftings of Nichols algebras of diagonal type III. Cartan type $G_2$

TL;DR

This paper completes the classification of certain Hopf algebras of Cartan type G_2, providing formulas for liftings where quantum Serre relations are satisfied, advancing the understanding of Nichols algebra liftings.

Contribution

It introduces a comprehensive classification of liftings of Nichols algebras of Cartan type G_2, including explicit formulas and a detailed procedure for relation determination.

Findings

Complete classification of Hopf algebras of Cartan type G_2

Development of a general formula for liftings with quantum Serre relations

Detailed methodology based on recent algebraic work

Abstract

We complete the classification of Hopf algebras whose infinitesimal braiding is a principal Yetter-Drinfeld realization of a braided vector space of Cartan type $G_2$ over a cosemisimple Hopf algebra. We develop a general formula for a class of liftings in which the quantum Serre relations hold. We give a detailed explanation of the procedure for finding the relations, based on the recent work of Andruskiewitsch, Angiono and Rossi Bertone.