Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation
- Autores
- Lebed, Victoria; Vendramin, Claudio Leandro
- Año de publicación
- 2017
- Idioma
- inglés
- Tipo de recurso
- artículo
- Estado
- versión publicada
- Descripción
- 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.
Fil: Lebed, Victoria. Trinity College Dublin; Irlanda
Fil: Vendramin, Claudio Leandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina - Materia
-
Birack
Braided Homology
Cubical Homology
Cycle Set
Extension
Quandle
Rack
Shelf
Yang–Baxter Equation - Nivel de accesibilidad
- acceso abierto
- Condiciones de uso
- https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
- Repositorio
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/55537
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Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equationLebed, VictoriaVendramin, Claudio LeandroBirackBraided HomologyCubical HomologyCycle SetExtensionQuandleRackShelfYang–Baxter Equationhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1This 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.Fil: Lebed, Victoria. Trinity College Dublin; IrlandaFil: Vendramin, Claudio Leandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaAcademic Press Inc Elsevier Science2017-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/55537Lebed, Victoria; Vendramin, Claudio Leandro; Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation; Academic Press Inc Elsevier Science; Advances in Mathematics; 304; 1-2017; 1219-12610001-8708CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1016/j.aim.2016.09.024info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0001870815302851info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:34:00Zoai:ri.conicet.gov.ar:11336/55537instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 09:34:00.604CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse |
dc.title.none.fl_str_mv |
Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation |
title |
Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation |
spellingShingle |
Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation Lebed, Victoria Birack Braided Homology Cubical Homology Cycle Set Extension Quandle Rack Shelf 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 |
dc.creator.none.fl_str_mv |
Lebed, Victoria Vendramin, Claudio Leandro |
author |
Lebed, Victoria |
author_facet |
Lebed, Victoria Vendramin, Claudio Leandro |
author_role |
author |
author2 |
Vendramin, Claudio Leandro |
author2_role |
author |
dc.subject.none.fl_str_mv |
Birack Braided Homology Cubical Homology Cycle Set Extension Quandle Rack Shelf Yang–Baxter Equation |
topic |
Birack Braided Homology Cubical Homology Cycle Set Extension Quandle Rack Shelf Yang–Baxter Equation |
purl_subject.fl_str_mv |
https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
dc.description.none.fl_txt_mv |
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. Fil: Lebed, Victoria. Trinity College Dublin; Irlanda Fil: Vendramin, Claudio Leandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina |
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. |
publishDate |
2017 |
dc.date.none.fl_str_mv |
2017-01 |
dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion http://purl.org/coar/resource_type/c_6501 info:ar-repo/semantics/articulo |
format |
article |
status_str |
publishedVersion |
dc.identifier.none.fl_str_mv |
http://hdl.handle.net/11336/55537 Lebed, Victoria; Vendramin, Claudio Leandro; Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation; Academic Press Inc Elsevier Science; Advances in Mathematics; 304; 1-2017; 1219-1261 0001-8708 CONICET Digital CONICET |
url |
http://hdl.handle.net/11336/55537 |
identifier_str_mv |
Lebed, Victoria; Vendramin, Claudio Leandro; Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation; Academic Press Inc Elsevier Science; Advances in Mathematics; 304; 1-2017; 1219-1261 0001-8708 CONICET Digital CONICET |
dc.language.none.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.aim.2016.09.024 info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0001870815302851 |
dc.rights.none.fl_str_mv |
info:eu-repo/semantics/openAccess https://creativecommons.org/licenses/by-nc-sa/2.5/ar/ |
eu_rights_str_mv |
openAccess |
rights_invalid_str_mv |
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/ |
dc.format.none.fl_str_mv |
application/pdf application/pdf |
dc.publisher.none.fl_str_mv |
Academic Press Inc Elsevier Science |
publisher.none.fl_str_mv |
Academic Press Inc Elsevier Science |
dc.source.none.fl_str_mv |
reponame:CONICET Digital (CONICET) instname:Consejo Nacional de Investigaciones Científicas y Técnicas |
reponame_str |
CONICET Digital (CONICET) |
collection |
CONICET Digital (CONICET) |
instname_str |
Consejo Nacional de Investigaciones Científicas y Técnicas |
repository.name.fl_str_mv |
CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas |
repository.mail.fl_str_mv |
dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar |
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1844613049565052928 |
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13.070432 |