%0 Journal Article
%T The quantum divided power algebra of a finite-dimensional Nichols algebra of diagonal type
%A Nicol¨˘s Andruskiewitsch
%A Iv¨˘n Angiono
%A Fiorela Rossi Bertone
%J Mathematics
%D 2015
%I arXiv
%X 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 $\widetilde{\mathcal{B}}_{\mathfrak{q}}$ from [A3] and the divided powers algebra $\mathcal{U}_{\mathfrak{q}}$, a suitable Drinfeld double of $\mathcal{L}_{\mathfrak{q}} \# \mathbf{k} \mathbb{Z}^{\theta}$. 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.
%U http://arxiv.org/abs/1501.04518v4