A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems

TL;DR

This paper provides a finite presentation of Nichols algebras of diagonal type with finite root systems, establishing their defining ideals are finitely generated and exploring their basis structure.

Contribution

It introduces a presentation by generators and relations for these Nichols algebras and proves the convexity of lexicographic order on Lyndon words within their PBW basis.

Findings

The defining ideal of such Nichols algebras is finitely generated.

The lexicographic order on Lyndon words is convex for PBW generators.

The PBW basis is orthogonal with respect to the canonical form.

Abstract

We obtain a presentation by generators and relations of any Nichols algebra of diagonal type with finite root system. We prove that the defining ideal is finitely generated. The proof is based in Kharchenko's theory of PBW basis of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for such PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.