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...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autor principal:	Vendramin, Leandro
Publicado:	2017
Materias:	Birack Braided homology Cubical homology Cycle set Extension Quandle Rack Shelf Yang–Baxter equation
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v304_n_p1219_Lebed http://hdl.handle.net/20.500.12110/paper_00018708_v304_n_p1219_Lebed
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_00018708_v304_n_p1219_Lebed
record_format	dspace
spelling	paper:paper_00018708_v304_n_p1219_Lebed2023-06-08T14:21:49Z Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation Vendramin, Leandro 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. 2017 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v304_n_p1219_Lebed 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 Vendramin, Leandro 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.
author	Vendramin, Leandro
author_facet	Vendramin, Leandro
author_sort	Vendramin, Leandro
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
publishDate	2017
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v304_n_p1219_Lebed http://hdl.handle.net/20.500.12110/paper_00018708_v304_n_p1219_Lebed
work_keys_str_mv	AT vendraminleandro homologyofleftnondegeneratesettheoreticsolutionstotheyangbaxterequation
_version_	1768545811331809280

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

Ejemplares similares