Homology of left non-degenerate set-theoretic solutions to the Yang-Baxter equation

TL;DR

This paper develops and compares two (co)homology theories for left non-degenerate set-theoretic solutions to the Yang-Baxter equation, unifying known cases and introducing new constructions, with applications to cycle sets and quandles.

Contribution

It introduces an explicit isomorphism between two (co)homology theories for LND solutions and extends classical (co)homology to new algebraic structures like cycle sets.

Findings

Classical (co)homology recovered for groups and racks

New (co)homology constructions for cycle sets

Connections established between 2-cocycles and group cohomology

Abstract

This paper deals with left non-degenerate set-theoretic solutions to the Yang-Baxter equation (=LND solutions), a vast class of algebraic structures encompassing groups, racks, and cycle sets. To each such solution is associated a shelf (i.e., a self-distributive structure) which captures its major properties. We consider two (co)homology theories for LND solutions, one of which was previously known, in a reduced form, for biracks only. An explicit isomorphism between these theories is described. For groups and racks we recover their classical (co)homology, whereas for cycle sets we get new constructions. For a certain type of LND solutions, including quandles and non-degenerate cycle sets, the (co)homologies split into the degenerate and the normalized parts. We express 2-cocycles of our theories in terms of group cohomology, and, in the case of cycle sets, establish connexions with extensions. This leads to a construction of cycle sets with interesting properties.