On finite GK-dimensional Nichols algebras over abelian groups

TL;DR

This paper classifies Nichols algebras over abelian groups with finite Gelfand-Kirillov dimension, identifying specific conditions for their finiteness and providing new examples, including some that are domains.

Contribution

It offers a classification of finite GK-dimensional Nichols algebras over abelian groups, including new examples and a conjecture relating diagonal type to finite root systems.

Findings

Finite GK dimension occurs only for certain Jordan blocks with eigenvalues ±1.

Classified Nichols algebras with finite GK dimension in a broad class, assuming a conjecture about root systems.

Presented new examples of Nichols algebras with finite GK dimension, some being domains.

Abstract

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, $\operatorname{GKdim}$ for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over $\mathbb Z$ with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite $\operatorname{GKdim}$ if and only if the size of the block is 2 and the eigenvalue is $\pm 1$; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that contains those of diagonal type. We conjecture that a Nichols algebra of diagonal type has finite $\operatorname{GKdim}$ if and only if the corresponding generalized root system is finite. Assuming the validity of this conjecture, we classify all braided vector spaces in the mentioned class whose Nichols algebra has finite $\operatorname{GKdim}$. Consequently we present several new examples of Nichols algebras with finite $\operatorname{GKdim}$, including two not in the class alluded to above. We determine which among these Nichols algebras are domains.