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

TL;DR

This paper explores the connections between self-distributive structures and Yang-Baxter equation solutions, revealing new insights into their cohomology theories and raising open questions about their relationships.

Contribution

It introduces a method to construct SD structures from LND YBE solutions and establishes an explicit isomorphism between their cohomology theories, unifying previously separate frameworks.

Findings

Constructed SD structures from YBE solutions capture key properties.

Established an explicit isomorphism between two cohomology theories.

Identified open questions about the relationship between cohomologies.

Abstract

Self-distributive (SD) structures form an important class of solutions to the Yang--Baxter equation, which underlie spectacular knot-theoretic applications of self-distributivity. It is less known that one go the other way round, 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 leaves numerous open questions. 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.