Combinatorial aspects of dynamical Yang-Baxter maps and dynamical braces

TL;DR

This paper introduces dynamical braces, an algebraic structure linked to dynamical Yang-Baxter maps, revealing combinatorial correspondences and providing new examples and interpretations of these mathematical objects.

Contribution

It defines dynamical braces as a new algebraic system and explores their connection to dynamical Yang-Baxter maps and combinatorial structures.

Findings

Established a correspondence between dynamical braces and subsets of A⋉Aut(A)

Provided examples illustrating the structure of dynamical braces

Interpreted dynamical braces within combinatorial frameworks

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 $A\rtimes Aut(A)$. From this viewpoint we give an interpretation and examples of the dynamical brace.