Non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation of multipermutation level 2

Abstract: Set-theoretic solution of the Yang-Baxter equation is a mapping $r:X\times X\to X\times X$ satisfying \[ (r\times 1) (1\times r) (r\times 1) = (1\times r) (r\times 1) (1\times r). \] A solution $r: (x,y)\mapsto (\sigma_x(y),\tau_y(x))$ is called non-degenerate if the mappings $\sigma_x$ and $\tau_y$ are permutations, for all $x,y\in X$. A solution is called involutive if $r^2=1$.

If $(X,r)$ is a non-degenerate involutive solution $(X,r)$ then the relation~$\sim$ defined by $x\sim y\equiv \sigma_x=\sigma_y$ is a congruence. A solution is of multipermutation level 2 if $|(X/\sim)/\sim|=1$.

In our talk we focus on these solutions and we present several constructions and properties.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar