Segre products and Segre morphisms in a class of Yang-Baxter algebras

TL;DR

This paper studies the structure of Segre products and Segre maps within quadratic Yang-Baxter algebras derived from set-theoretic solutions of the Yang-Baxter equation, providing explicit presentations and analogues of classical geometric maps.

Contribution

It introduces explicit presentations of Segre products and Segre maps for Yang-Baxter algebras, extending classical geometric concepts to a noncommutative algebraic setting.

Findings

Explicit presentation of Segre product in terms of generators and relations

Definition and analysis of Segre maps in Yang-Baxter algebras

Identification of kernels and images of these maps

Abstract

Let $(X,r_X)$ and $(Y,r_Y)$ be finite nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation, and let $A_X = A(\textbf{k}, X, r_X)$ and $A_Y= A(\textbf{k}, Y, r_Y)$ be their quadratic Yang-Baxter algebras over a field $\textbf{k}.$ We find an explicit presentation of the Segre product $A_X\circ A_Y$ in terms of one-generators and quadratic relations. We introduce analogues of Segre maps in the class of Yang-Baxter algebras and find their images and their kernels. The results agree with their classical analogues in the commutative case.