TY - JOUR

T1 - A Wells type exact sequence for non-degenerate unitary solutions of the Yang–Baxter equation

AU - Bardakov, Valeriy

AU - Singh, Mahender

N1 - Funding Information: The authors are grateful to the referee for the elaborate report which substantially improved the clarity and exposition of the paper. Bardakov is supported by the Ministry of Science and Higher Education of Russia (agreement No. 075-02-2020-1479/1). Singh is supported by the SwarnaJayanti Fellowship grants DST/SJF/MSA-02/2018-19 and SB/SJF/2019-20/04, and the Indo-Russian grant DST /INT/RUS/RSF/P-19. Publisher Copyright: © 2022. International Press. Permission to copy for private use granted

PY - 2022

Y1 - 2022

N2 - 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

AB - 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

KW - Brace

KW - Cycle set cohomology

KW - Extension

KW - Group cohomology

KW - Linear cycle set

KW - Yang–baxter equation

UR - http://www.scopus.com/inward/record.url?scp=85136119308&partnerID=8YFLogxK

U2 - 10.4310/HHA.2022.v24.n2.a2

DO - 10.4310/HHA.2022.v24.n2.a2

M3 - Article

AN - SCOPUS:85136119308

VL - 24

SP - 31

EP - 51

JO - Homology, Homotopy and Applications

JF - Homology, Homotopy and Applications

SN - 1532-0073

IS - 2

ER -