A finite-dimensional Lie algebra arising from a Nichols algebra of diagonal type (rank 2)

TL;DR

This paper explores the structure of a specific finite-dimensional Lie algebra derived from a Nichols algebra of diagonal type, providing explicit computations for the case when the rank is 2.

Contribution

It presents a detailed construction of a Lie algebra associated with a Nichols algebra of diagonal type and computes its structure explicitly for rank 2.

Findings

The Lie algebra ng_{\u00q} is explicitly computed for rank 2.

ng_{} is shown to be isomorphic to a universal enveloping algebra.

The Lusztig algebra al_{} is described as an extension of the Nichols algebra.

Abstract

Let $\mathcal{B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q} \in \mathbf{k}^{\theta \times \theta}$, where $\mathbf{k}$ is an algebraically closed field of characteristic 0. Let $\mathcal{L}_{\mathfrak{q}}$ be the Lusztig algebra associated to $\mathcal{B}_{\mathfrak{q}}$, see http://arxiv.org/abs/1501.04518. We present $\mathcal{L}_{\mathfrak{q}}$ as an extension (as braided Hopf algebras) of $\mathcal{B}_{\mathfrak{q}}$ by $\mathfrak Z_{\mathfrak{q}}$ where $\mathfrak Z_{\mathfrak{q}}$ is isomorphic to the universal enveloping algebra of a Lie algebra $\mathfrak n_{\mathfrak{q}}$. We compute the Lie algebra $\mathfrak n_{\mathfrak{q}}$ when $\theta = 2$.