Indecomposable and simple solutions of the Yang-Baxter equation

Abstract: Recall that a set-theoretic solution of the Yang-Baxter equation is a tuple $(X,r)$, where $X$ is a non-empty set and $r: X \times X \rightarrow X \times X$ a bijective map such that $$(r \times id_X ) (id_X \times r) (r \times id_X) = (id_X \times r) (r \times id_X ) (id_X \times r),$$ where one denotes $r(x,y)=(\lambda_x(y), \rho_y(x))$. Attention is often restricted to so-called non-degenerate solutions, i.e. $\lambda_x$ and $\rho_y$ are bijective. We will call these solutions for short in the remainder of this abstract. To understand more general objects, it is an important technique to study 'minimal' objects and glue these together. For solutions both indecomposable and simple solutions fit the bill for being a minimal object. In this talk we will report on recent work with I. Colazzo, E. Jespers and L. Kubat on simple solutions. In particular, we will discuss an extension of a result of M. Castelli that allows to identify whether a solution is simple, without having to know or calculate all smaller solutions. This method employs so-called skew braces, which were constructed to provide more examples of solutions, but also govern many properties of general solutions. In the latter part of the talk, we discuss the extension of a method to construct new indecomposable or simple solutions from old ones via cabling, originally introduced by V. Lebed, S. Ramirez and L. Vendramin to unify the known results on indecomposability of solutions.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar