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

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 = mathcal {A}({{textbf {k}}}, X, r_X) and A_Y= mathcal {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_Xcirc 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.