Distinguished Pre-Nichols algebras

TL;DR

This paper introduces the distinguished pre-Nichols algebra, an intermediate algebra between the tensor algebra and Nichols algebra, and explores its properties and applications to Hopf algebras.

Contribution

It formally defines the distinguished pre-Nichols algebra and demonstrates its role in constructing new Noetherian pointed Hopf algebras with finite Gelfand-Kirillov dimension.

Findings

Provides a presentation of the algebra with fewer relations

Shows the algebra yields new Hopf algebra examples

Generalizes results on quantum groups at roots of unity

Abstract

We formally define and study the distinguished pre-Nichols algebra $\widetilde{\mathcal{B}}(V)$ of a braided vector space of diagonal type $V$ with finite-dimensional Nichols algebra $\mathcal{B}(V)$. The algebra $\widetilde{\mathcal{B}}(V)$ is presented by fewer relations than $\mathcal{B}(V)$, so it is intermediate between the tensor algebra $T(V)$ and $\mathcal{B}(V)$. Prominent examples of distinguished pre-Nichols algebras are the positive parts of quantized enveloping (super)algebras and their multiparametric versions. We prove that these algebras give rise to new examples of Noetherian pointed Hopf algebras of finite Gelfand-Kirillov dimension. We investigate the kernel (in the sense of Hopf algebras) of the projection from $\widetilde{\mathcal{B}}(V)$ to $\mathcal{B}(V)$, generalizing results of De Concini and Procesi on quantum groups at roots of unity.