Applications of self-distributivity to Yang–Baxter operators and their cohomology

Abstract

Self-distributive (SD) structures form an important class of solutions to the Yang–Baxter equation (YBE), which underlie spectacular knot-theoretic applications of self-distributivity (SD). It is less known that one can go the other way around, and construct an SD structure out of any left non-degenerate (LND) set-theoretic YBE solution. This structure captures important properties of the solution: invertibility, involutivity, biquandle-ness, the associated braid group actions. Surprisingly, the tools used to study these associated SD structures also apply to the cohomology of LND solutions, which generalizes SD cohomology. Namely, they yield an explicit isomorphism between two cohomology theories for these solutions, which until recently were studied independently. The whole story is full of open problems. One of them is the relation between the cohomologies of a YBE solution and its associated SD structure. These and related questions are covered in the present survey.