Combinatorial aspects of dynamical Yang-Baxter maps and dynamical braces

Abstract

In this article we propose an algebraic system, which is an abelian group $(A,+)$ with a family of non-associative and non-(left)distributive multiplications $\{{\cdot }_{\lambda }{\}}_{\lambda \in H}$. We call this algebraic system dynamical brace. The dynamical brace corresponds to a certain dynamical Yang-Baxter map (which is left nondegenerate and satisfy the unitary condition). Combinatorial aspects of the dynamical brace give us a correspondence between the dynamical brace and a certain family of subsets of $A⋊Aut(A)$. From this viewpoint we give an interpretation and examples of the dynamical brace.

21 pages, 6 graphs