Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation
By Victoria Lebed, Leandro Vendramin. 2015
83 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/087ab3593c8ba0e39a8fb6aefc8726dd7692085c
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In Computational Theory and Mathematics
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Citations in the corpus
- On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation (2017, background): “Thus, to any solution one can associate its structure rack (X, r) [25,27,34], which is trivial if and only if the solution is involutive.”
- Problems on skew left braces (2018, background): “I do not discuss problems related to homology of solutions [49], semibraces [19] or trusses [12].”