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

TL;DR

This paper constructs and analyzes the quantum divided power algebra associated with finite-dimensional Nichols algebras of diagonal type, providing explicit bases, presentations, and proving key algebraic properties.

Contribution

It introduces the quantum divided power algebra for these Nichols algebras, offering explicit generators, relations, and structural properties such as noetherianity and finite Gelfand-Kirillov dimension.

Findings

Established bases and presentations for the algebras

Proved the algebras are noetherian

Showed the algebras have finite Gelfand-Kirillov dimension

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 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.