On Nichols algebras of diagonal type

TL;DR

This paper provides a minimal set of defining relations for Nichols algebras of diagonal type with finite root systems, confirming a conjecture for abelian group-like elements and expanding understanding of their structure.

Contribution

It offers an explicit, minimal list of defining relations for these Nichols algebras, including new variations beyond known quantum Serre relations.

Findings

Confirmed the Andruskiewitsch-Schneider conjecture for abelian groups

Provided a minimal defining relations list for Nichols algebras of diagonal type

Extended the understanding of finite-dimensional pointed Hopf algebras

Abstract

We give an explicit and essentially minimal list of defining relations of a Nichols algebra of diagonal type with finite root system. This list contains the well-known quantum Serre relations but also many new variations. A conjecture by Andruskiewitsch and Schneider states that any finite-dimensional pointed Hopf algebra over an algebraically closed field of characteristic zero is generated as an algebra by its group-like and skew-primitive elements. As an application of our main result, we prove the conjecture when the group of group-like elements is abelian.