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

We study $d$-Veronese subalgebras $A^{(d)}$ of Yang-Baxter algebras $A_X= A(K, X, r)$relatedtofinitenondegenerateinvolutiveset-theoreticsolutions$(X, r)$oftheYang-Baxterequation,where$K$isafieldand$d\geq 2$ is an integer. We find an explicit presentation of the $d$-Veronese $A^{(d)}$ 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 $A_Y=A(K,Y,r_Y)$ to define a Veronese morphism $v_{n,d}:A_Y\to A_X$. We prove that the image of $v_{n,d}$ is the $d$-Veronese subalgebra $A^{(d)}$, 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 $A(K,X,r)$ 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.