The quantum divided power algebra of a finite-dimensional Nichols algebra of diagonal type

Abstract

Let ${\mathcal{B}}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q}$. We consider the graded dual ${\mathcal{L}}_{\mathfrak{q}}$ of the distinguished pre-Nichols algebra ${\tilde{\mathcal{B}}}_{\mathfrak{q}}$ from [A3] and the divided powers algebra ${\mathcal{U}}_{\mathfrak{q}}$, a suitable Drinfeld double of $\mathcal{L}_{\mathfrak{q}} # \mathbf{k} \mathbb{Z}^θ$. We provide basis and presentations by generators and relations of ${\mathcal{L}}_{\mathfrak{q}}$ and ${\mathcal{U}}_{\mathfrak{q}}$, and prove that they are noetherian and have finite Gelfand-Kirillov dimension.

arXiv comment20 pages; second version has a new title: 'The quantum divided power algebra of a finite-dimensional Nichols algebra of diagonal type'. Accepted in Math. Res. Letters