Retractability of set theoretic solutions of the Yang-Baxter equation

It is shown that square free set theoretic
involutive non-de\-ge\-nerate solutions of the
Yang-Baxter equation whose associated permutation
group (referred to as an involutive Yang-Baxter
group) is abelian are retractable in the sense of
Etingof, Schedler and Soloviev. This solves a
problem of Gateva-Ivanova in the case of abelian
IYB groups. It also implies that the
corresponding finitely presented
abelian-by-finite groups (called the structure
groups) are poly-${\mathbb Z}$ groups. Secondly,
an example of a solution with an abelian
involutive Yang-Baxter group which is not a
generalized twisted union is constructed. This
answers in the negative another problem of
Gateva-Ivanova. The constructed solution is of
multipermutation level $3$. Retractability of
solutions is also proved in the case where the
natural generators of the IYB group are cyclic
permutations. Moreover, it is shown that such
solutions are generalized twisted unions.

Jaar van publicatie:2010

Trefwoorden:Yang-Baxter equation, set theoretic solutoin, multipermutation solution, permutation group, group of I-type