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

https://hu-berlin.zoom.us/j/61686623112 [3]
Contact: Gaetan Borot (gaetan.borot@hu-berlin.de)

We study $d$-Veronese subalgebras $A^{(d)}$ of quadratic algebras $A_X=A(K, X, r)$ 
related to finite nondegenerate involutive set-theoretic solutions $(X, r)$ of the Yang-Baxter equation,
where $K$ is a field and $d> 1$ 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  \rightarrow 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. Finally, 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.