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Mathematical Research Letters
Volume 24 (2017)
Number 3
The quantum divided power algebra of a finite-dimensional Nichols algebra of diagonal type
Pages: 619 – 643
DOI: https://dx.doi.org/10.4310/MRL.2017.v24.n3.a2
Authors
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 $\widetilde{\mathcal{B}}_{\mathfrak{q}}$ from [A3] and the quantum divided power 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.
2010 Mathematics Subject Classification
16Wxx
Received 19 January 2015
Accepted 18 December 2015
Published 1 September 2017