Extensions of Yang-Baxter sets

TL;DR

This paper extends the concept of Yang-Baxter sets to vector spaces, exploring their morphisms and extensions, and relates these structures to solutions of the Yang-Baxter equation and the virtual pure braid group.

Contribution

It introduces a framework for extending Yang-Baxter sets in vector spaces and connects these extensions to solutions of the Yang-Baxter equation and virtual braid groups.

Findings

Describes a family of solutions for the Yang-Baxter equation on product spaces.

Establishes a relation between Yang-Baxter set extensions and the virtual pure braid group.

Provides structural insights into morphisms and extensions of Yang-Baxter sets.

Abstract

The paper extends the notion of braided set and its close relative - the Yang-Baxter set - to the category of vector spaces and explore structure aspects of such a notion as morphisms and extensions. In this way we describe a family of solutions for the Yang-Baxter equation on the product of B and C if given B and C correspond to two linear (set-theoretic) solutions of the Yang-Baxter equation. One of the key observation is the relation of this question with the virtual pure braid group.