Veronese subalgebras and Veronese morphisms for a class of Yang–Baxter algebras

Abstract

We study $d$-Veronese subalgebras ${\mathcal{A}}^{(d)}$ of Yang–Baxter algebras ${\mathcal{A}}_X=\mathcal{A}(\text{k},X,r)$ related to finite nondegenerate involutive set-theoretic solutions $(X,r)$ of the Yang–Baxter equation, where $\text{k}$ is a field and $d\ge 2$ is an integer. We find an explicit presentation of the $d$-Veronese ${\mathcal{A}}^{(d)}$ in terms of one-generators and quadratic relations. We introduce the notion of a $d$-Veronese solution $(Y,r_Y)$, canonically associated to $(X,r)$ and use its Yang–Baxter algebra ${\mathcal{A}}_Y=\mathcal{A}(\text{k},Y,r_Y)$ to define a Veronese morphism $v_{n,d}:{\mathcal{A}}_Y\to {\mathcal{A}}_X$. We prove that the image of $v_{n,d}$ is the $d$-Veronese subalgebra ${\mathcal{A}}^{(d)}$ and find explicitly a minimal set of generators for its kernel. The results agree with their classical analogues in the commutative case. We show that the Yang–Baxter algebra $\mathcal{A}(\text{k},X,r)$ is a PBW algebra if and only if $(X,r)$ is a square-free solution. In this case, the $d$-Veronese $A^{(d)}$ is also a PBW algebra.