Homology, Homotopy and Applications

Volume 24 (2022)

Number 2

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

Pages: 31 – 51

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n2.a2


Valeriy Bardakov (Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk, Russia; Novosibirsk State Agrarian University, Novosibirsk, Russia; and Regional Scientific and Educational Mathematical Center, Tomsk State University, Tomsk, Russia)

Mahender Singh (Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Punjab, India)


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 relate the exact sequence for linear cycle sets with that for their underlying abelian groups via the forgetful functor and also discuss generalities on dynamical 2-cocycles.


brace, cycle set cohomology, linear cycle set, extension, group cohomology, Yang–Baxter equation

2010 Mathematics Subject Classification

Primary 16T25, 81R50. Secondary 20J05, 20N02, 57M27.

Received 3 March 2021

Received revised 12 July 2021

Accepted 12 July 2021

Published 10 August 2022