A Wells type exact sequence for non-degenerate unitary solutions of the Yang--Baxter equation

TL;DR

This paper develops a Wells-type exact sequence for linear cycle sets, connecting cohomology, automorphisms, and extensions, thereby advancing the algebraic understanding of solutions to the Yang--Baxter equation.

Contribution

It introduces a new exact sequence for linear cycle sets, extending Wells' classical sequence to this algebraic context and relating it to cohomology and automorphisms.

Findings

Derived a four-term exact sequence for linear cycle sets.

Compared the sequence with that for underlying abelian groups.

Discussed implications for dynamical 2-cocycles.

Abstract

Cycle sets are known to give non-degenerate unitary solutions of the Yang--Baxter equation and linear cycle sets are enriched versions of these algebraic systems. The paper explores the recently developed cohomology and extension theory for linear cycle sets. We derive a four term exact sequence relating 1-cocycles, second cohomology and certain groups of automorphisms arising from central extensions of linear cycle sets. This is an analogue of a similar exact sequence for group extensions known due to Wells. We also compare the exact sequence for linear cycle sets with that for their underlying abelian groups via the forgetful functor and discuss generalities on dynamical 2-cocycles.