Classification of set-theoretical solutions to the pentagon equation

Abstract: The pentagon equation classically originates from the field of Mathematical Physics. Our attention is placed on the study of set-theoretical solutions of this equation, namely, maps $s: X \times X \to X \times X$ given by $s(x, y)=(xy, \theta_x(y))$, where $X$ is a semigroup and $\theta_x:X \to X$ is a map satisfying two laws. In this talk, we give some recent descriptions of some classes of solutions achieved starting from particular semigroups. Into the specific, we provide a characterization of \emph{idempotent-invariant} solutions on a Clifford semigroup $X$, that are those for which $\theta_a$ remains invariant on the set of idempotents $E(X)$. In addition, we will focus on the classes of \emph{involutive} and \emph{idempotent} solutions, which are solutions fulfilling $s^2=id_{X \times X}$ and $s^2=s$, respectively.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar