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

We study d -Veronese subalgebras \mathcal{A}^{(d)} of Yang–Baxter algebras \mathcal{A}_{X}= \mathcal{A}(\textbf{k}, X, r) related to finite nondegenerate involutive set-theoretic solutions (X, r) of the Yang–Baxter equation, where \textbf{k} is a field and d\geq 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}(\textbf{k}, Y, r_{Y}) to define a Veronese morphism v_{n,d}:\mathcal{A}_{Y}\rightarrow \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}(\textbf{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.