Arne Van Antwerpen (University of Ghent) – On groups and algebras related to the Yang-Baxter equation

Recall that a combinatorial 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 on $X^3$ it holds that $$ (r \times \operatorname{id}_X) (\operatorname{id}_X \times r) (r \times \operatorname{id}_X) = (\operatorname{id}_X \times r)(r \times \operatorname{id}_X) (\operatorname{id}_X \times r).$$ One can then define the structure group $G(X,r)$/monoid $M(X,r)$ of a solution as the group/monoid generated by X with defining relations $xy = uv$ if $r(x,y)=(u,v)$. In this talk we talk we zoom in on the relation between $G(X,r)$ and a second group structure on this set, stemming from the behaviour of $r^2$. This led to the introduction of skew braces by Rump, and Guarnieri and Vendramin. Recall that a skew brace is a set $B$ with two group structures $(B,+)$ and $(B,\circ)$ that interact via a skew left distributivity condition, i.e. for any $a,b,c \in B$ one has that $a\circ (b+c) = (a \circ b) – a + (a\circ c)$. It turns out that these structures both generate and govern solutions. We will report in some recent advancements relating properties of skew braces to properties of its associated solution.

In the second part of the talk we will focus on the subclass of finite non-degenerate solutions. In the first part we discuss recent work on the structure of the monoid $M(X,r)$ and its monoid algebra $KM(X,r)$, where $K$ is an arbitrary field. In particular, we highlight the importance of the divisibility structure of $M(X,r)$ on the prime ideals of its algebra. Furthermore, we discuss how the homological properties of $KM(X,r)$ are akin to those of the polynomial algebra in several commuting variables. Concretely, a bound on the Gelfand-Kirillov and classical Krull dimension will be discussed. Moreover, some further homological properties can be shown to be equivalent to $r$ being an involution.