Remarks on rational solutions of Yang-Baxter equations

TL;DR

This paper explores the geometric origins of unitary rational solutions to the associative Yang-Baxter equation with three spectral parameters and examines their connections to quantum and classical Yang-Baxter equations.

Contribution

It introduces a geometric framework linking vector bundles on cuspidal cubic curves to solutions of the associative Yang-Baxter equation with multiple spectral parameters.

Findings

Solutions arise from vector bundle geometry on cubic curves

Connections established between associative, quantum, and classical Yang-Baxter equations

Provides a geometric interpretation of rational solutions

Abstract

In this article, we study unitary rational solutions of the associative Yang-Baxter equation with three spectral parameters. We explain how such solutions arise from the geometry of vector bundles on a cuspidal cubic curve. Moreover, we investigate how these solutions are related to the quantum and classical Yang-Baxter equations.