Finite GK-dimensional Nichols algebras of diagonal type and finite root systems

TL;DR

This paper proves that for finite-dimensional diagonal braided vector spaces, the Nichols algebra has finite Gelfand-Kirillov dimension if and only if its root system is finite, confirming a conjecture and extending previous results.

Contribution

It establishes the equivalence between finite Gelfand-Kirillov dimension and finite root systems for Nichols algebras of diagonal type, generalizing prior specific cases.

Findings

Finite GK-dimension corresponds to finite root systems.

Confirmed conjecture from arXiv:1606.02521.

Extended results beyond dimensions 2 and 3.

Abstract

Let $(V,c)$ be a finite-dimensional braided vector space of diagonal type. We show that the Gelfand Kirillov dimension of the Nichols algebra $\mathfrak{B}(V)$ is finite if and only if the corresponding root system is finite, that is $\mathfrak{B}(V)$ admits a PBW basis with a finite number of generators. This had been conjectured in arXiv:1606.02521 and proved for $\dim V=2,3$ in arXiv:1803.08804, arXiv:2106.10143 respectively.