Affine Actions and the Yang-Baxter Equation

TL;DR

This paper investigates the connection between the Yang-Baxter equation and affine actions, classifying solutions via their associated affine actions and C*-dynamical systems, providing new insights into their structure.

Contribution

It introduces a novel classification of Yang-Baxter solutions based on affine actions and C*-dynamical systems, advancing understanding of their algebraic and dynamical properties.

Findings

Classified solutions of the Yang-Baxter equation through affine actions.

Connected solutions to C*-dynamical systems derived from affine actions.

Established new structural results relating affine actions to the Yang-Baxter equation.

Abstract

In this paper, the relations between the Yang-Baxter equation and affine actions are explored in detail. In particular, we classify solutions of the Yang-Baxter equations in two ways: (i) by their associated affine actions of their structure groups on their derived structure groups, and (ii) by the C*-dynamical systems obtained from their associated affine actions. On the way to our main results, several other useful results are also obtained.