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

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 there is associated a shelf (i.e., a self-distributive structure) which captures its...

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Autores principales: Lebed, V., Vendramin, L.
Formato: JOUR
Materias:
Birack
Braided homology
Cubical homology
Cycle set
Extension
Quandle
Rack
Shelf
Yang–Baxter equation
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00018708_v304_n_p1219_Lebed
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_00018708_v304_n_p1219_Lebed2023-10-03T13:52:21Z Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation Lebed, V. Vendramin, L. Birack Braided homology Cubical homology Cycle set Extension Quandle Rack Shelf Yang–Baxter equation 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 there 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. © 2016 Elsevier Inc. Fil:Vendramin, L. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00018708_v304_n_p1219_Lebed
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Birack
Braided homology
Cubical homology
Cycle set
Extension
Quandle
Rack
Shelf
Yang–Baxter equation
spellingShingle Birack
Braided homology
Cubical homology
Cycle set
Extension
Quandle
Rack
Shelf
Yang–Baxter equation
Lebed, V.
Vendramin, L.
Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation
topic_facet Birack
Braided homology
Cubical homology
Cycle set
Extension
Quandle
Rack
Shelf
Yang–Baxter equation
description 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 there 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. © 2016 Elsevier Inc.
format JOUR
author Lebed, V.
Vendramin, L.
author_facet Lebed, V.
Vendramin, L.
author_sort Lebed, V.
title Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation
title_short Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation
title_full Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation
title_fullStr Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation
title_full_unstemmed Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation
title_sort homology of left non-degenerate set-theoretic solutions to the yang–baxter equation
url http://hdl.handle.net/20.500.12110/paper_00018708_v304_n_p1219_Lebed
work_keys_str_mv AT lebedv homologyofleftnondegeneratesettheoreticsolutionstotheyangbaxterequation
AT vendraminl homologyofleftnondegeneratesettheoreticsolutionstotheyangbaxterequation
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