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

We study d -Veronese subalgebras A⁽ᵈ⁾ of Yang–Baxter algebras AX= 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≥ 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, rY) , canonically associated to (X,r) and use its Yang–Baxter algebra AY= A(k, Y, rY) to define a Veronese morphism vn,d:AY→ AX . We prove that the image of vn,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.