Examples of Nichols algebras associated to upper triangular solutions of the Yang-Baxter equation in rank 3

TL;DR

This paper classifies certain Nichols algebras linked to specific upper triangular solutions of the Yang-Baxter equation in three dimensions, highlighting their algebraic relations and structure.

Contribution

It identifies and characterizes Nichols algebras with quadratic relations associated to upper triangular Yang-Baxter solutions in rank 3, expanding understanding of their algebraic properties.

Findings

Identification of Nichols algebras with quadratic relations

Connection between Nichols algebras and upper triangular Yang-Baxter solutions

Classification results for rank 3 cases

Abstract

We determine some Nichols algebras that admit a non-trivial quadratic relation associated to some families of upper triangular solutions of the Yang-Baxter equation of dimension 3.