On defining ideals and differential algebras of Nichols algebras

TL;DR

This paper investigates the defining ideals of Nichols algebras by analyzing braid group elements, constructing differential algebras, and offering new perspectives on Serre relations, with algorithms and primitive elements introduced.

Contribution

It introduces a novel approach to understanding Nichols algebra ideals through primitive elements, differential algebra construction, and new insights on Serre relations.

Findings

Primitive elements identified in braid group algebra

Algorithms for ideal decomposition proposed

Differential algebra framework established

Abstract

This paper is devoted to understanding the defining ideal of a Nichols algebra from the decomposition of specific elements in the group algebra of braid groups. A family of primitive elements are found and algorithms are proposed. To prove the main result, the differential algebra of a Nichols algebra is constructed. Moreover, another point of view on Serre relations is provided.