Set-theoretic Yang-Baxter (co)homology theory of involutive non-degenerate solutions

TL;DR

This paper develops a set-theoretic Yang-Baxter (co)homology theory for involutive non-degenerate solutions, splitting the homology into normalized and degenerated parts, advancing the algebraic understanding of these solutions.

Contribution

It introduces a normalized homology theory for involutive right non-degenerate solutions and demonstrates the decomposition of the homology into normalized and degenerated components.

Findings

Homology can be split into normalized and degenerated parts.

Normalized homology theory is introduced for involutive solutions.

The decomposition aids in understanding the structure of solutions.

Abstract

W. Rump showed that there exists a one-to-one correspondence between involutive right non-degenerate solutions of the Yang-Baxter equation and Rump right quasigroups. J. S. Carter, M. Elhamdadi, and M. Saito, meanwhile, introduced a homology theory of set-theoretic solutions of the Yang-Baxter equation in order to define cocycle invariants of classical knots. In this paper, we introduce the normalized homology theory of an involutive right non-degenerate solution of the Yang-Baxter equation and prove that the set-theoretic Yang-Baxter homology of certain solutions can be split into the normalized and degenerated parts.