Lie algebras arising from Nichols algebras of diagonal type

TL;DR

This paper investigates the structure of Lie algebras associated with finite-dimensional Nichols algebras of diagonal type, showing they are either trivial or positive parts of semisimple Lie algebras, based on specific braiding matrices.

Contribution

It establishes a classification of the Lie algebras arising from Nichols algebras of diagonal type, linking them to semisimple Lie algebras via explicit braiding data.

Findings

Lie algebra is either zero or the positive part of a semisimple Lie algebra

Classification based on braiding matrices from arXiv:math/0605795

Connection between Nichols algebras and classical Lie theory

Abstract

Let ${\mathcal B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type with braiding matrix $\mathfrak{q}$, let $\mathcal{L}_{\mathfrak{q}}$ be the corresponding Lusztig algebra as in arXiv:1501.04518 and let $\operatorname{Fr}_{\mathfrak{q}}: \mathcal{L}_{\mathfrak{q}} \to U(\mathfrak{n}^{\mathfrak{q}})$ be the corresponding quantum Frobenius map as in arXiv:1603.09387. We prove that the finite-dimensional Lie algebra $\mathfrak{n}^{\mathfrak{q}}$ is either 0 or else the positive part of a semisimple Lie algebra $\mathfrak{g}^{\mathfrak{q}}$ which is determined for each $\mathfrak{q}$ in the list of arXiv:math/0605795.