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

Iván Ezequiel Angiono

doi:10.4171/jems/567

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 gen- erators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.

Highlights

The consideration of pointed Hopf algebras has grown since the appearance of quantized enveloping algebras [Dr, Ji]
The Lifting Method of Andruskiewitsch and Schneider is the leading method for the classification of finite-dimensional pointed Hopf algebras. Such method depends on the answers to some questions, including the following one: Question 0.1. [And, Question 5.9]: Given a braided vector space of diagonal type, determine if the associated Nichols algebra is finite-dimensional, and in such case compute its dimension
The first part of this question has been answered by Heckenberger in [H2], where the author gives a list of all diagonal braidings whose associated Nichols algebra has a finite root system, but neither an explicit formula for the dimension nor a finite set of defining relations are given

Summary

Introduction

The consideration of pointed Hopf algebras has grown since the appearance of quantized enveloping algebras [Dr, Ji]. Andruskiewitsch and Schneider [AS3] have classified finite-dimensional pointed Hopf algebras whose group of group-like elements is abelian of order not divisible by some small primes using the Lifting method; all the possible such braidings are of finite Cartan type They answered positively the following conjecture for H0 = kΓ, Γ an abelian group as above: Conjecture 0.2. Our main result is Theorem 3.9: we obtain a presentation by generators and relations for any Nichols algebra of diagonal type whose root system is finite. In Subsection 2.3 we recall some results from [HS] involving coideal subalgebras of Nichols algebras of diagonal type with finite root systems and use these results to characterize PBW bases of hyperletters.

Preliminaries

Root systems and coideal subalgebras

Presentation by generators and relations of Nichols algebras of diagonal type