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Abstract:

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.

Registro:

Documento: Artículo
Título:Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation
Autor:Lebed, V.; Vendramin, L.
Filiación:School of Mathematics, Trinity College Dublin, 2, Dublin, Ireland
Depto. de Matemática, FCEN, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, Buenos Aires, 1428, Argentina
Palabras clave:Birack; Braided homology; Cubical homology; Cycle set; Extension; Quandle; Rack; Shelf; Yang–Baxter equation
Año:2017
Volumen:304
Página de inicio:1219
Página de fin:1261
DOI: http://dx.doi.org/10.1016/j.aim.2016.09.024
Título revista:Advances in Mathematics
Título revista abreviado:Adv. Math.
ISSN:00018708
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v304_n_p1219_Lebed

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Citas:

---------- APA ----------
Lebed, V. & Vendramin, L. (2017) . Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation. Advances in Mathematics, 304, 1219-1261.
http://dx.doi.org/10.1016/j.aim.2016.09.024
---------- CHICAGO ----------
Lebed, V., Vendramin, L. "Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation" . Advances in Mathematics 304 (2017) : 1219-1261.
http://dx.doi.org/10.1016/j.aim.2016.09.024
---------- MLA ----------
Lebed, V., Vendramin, L. "Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation" . Advances in Mathematics, vol. 304, 2017, pp. 1219-1261.
http://dx.doi.org/10.1016/j.aim.2016.09.024
---------- VANCOUVER ----------
Lebed, V., Vendramin, L. Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation. Adv. Math. 2017;304:1219-1261.
http://dx.doi.org/10.1016/j.aim.2016.09.024