Retraction problem for quasi-linear cycle sets

Involutive set-theoretic solutions to the Yang-Baxter equation have been widely studied in the last two decades, and various connections with algebraic structures have been introduced and explored. Among these structures, in 2005, Rump introduced cycle sets which are in one-to-one correspondence with left non-degenerate involutive solutions.

One of the questions yet to be solved in the context of involutive set-theoretic solutions to the Yang-Baxter equation is the following: when is such a solution retractable?

 

In this talk, we will focus on cycle sets with an underlying compatible abelian group structure called quasi-linear cycle set, introduced by Rump. We will state basic properties of this structure and discuss some conjectures about the retraction problem for the class of solutions associated with a quasi-linear cycle set.