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

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 $\theta =2$.

19 pages