Date:	Fri, November 28, 2014
Time:	11:15
Place:	Seminar Room (120), Research I

Abstract: Set-theoretic solutions of the Yang--Baxter equation form a meeting ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set \(X\) and a bijective map \(r:X\times X\to X\times X\) which satisfies the braid relations. Associated to each set-theoretic solution are several algebraic constructions: the monoid \(S(X, r)\), the group \(G(X, r)\), the semigroup algebra \(k\,S(X, r)\) over a field \(k\), generated by \(X\) and with quadratic relations \(xy = .r(x, y)\), and also a special permutation group. In this talk, I shall discuss some of the remarkable algebraic properties of these object. One of the measures of a solution is its multipermutation level of complexity. I shall consider the close relation between the multipermutation level of a solution and the properties of the associated groups. I shall present some recent developments in the study of a conjecture of the author that every finite square free-solution is a multipermutation solution.