On finite GK-dimensional Nichols algebras of diagonal type: rank 3 and Cartan type

TL;DR

This paper proves that rank 3 or Cartan type Nichols algebras of diagonal type with finite Gelfand-Kirillov dimension have finite root systems, advancing the classification of these algebraic structures.

Contribution

It confirms the conjecture for rank 3 and Cartan type cases, linking finite GK-dimension to finite root systems in Nichols algebras.

Findings

Confirmed the conjecture for rank 3 Nichols algebras.

Confirmed the conjecture for Nichols algebras of Cartan type.

Established a connection between finite GK-dimension and finite root systems.

Abstract

This paper contributes to the proof of the conjecture posed in arXiv:1606.02521, stating that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system. We prove the conjecture assuming that the rank is 3 or that the braiding is of Cartan type.