On the set-theoretical Yang-Baxter equation

By Jiang-Hua Lu, Min Yan, Yongchang Zhu. 2000

281 citations recorded by the source corpus.

We propose a general way of constructing set-theoretical solutions of the YangBaxter equation. We study the properties of the construction. We also show that our construction includes the earlier ones given by Weinstein-Xu and Etingof-SchedlerSoloviev.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8bcd4a9abdfa0f29a9d6cc97257ce0f148d4293f

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In Algebra and Number Theory

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Yang–Baxter deformations of quandles and racks (2004, background): “]. Can a similar interpretation be established for Yang-Baxter deformations of Qin general? Question 41 What can be said about deformations of set-theoretic YangBaxter operators in general? Following [8, 12, 26], consider a set Qequipped with a bijective map c: Q×Q→Q×Q satisfying (c×I)(I ×c)(c×I) = ”
Multipermutation Solutions of the Yang–Baxter Equation (2009, methodology): “s and we refer to the text in [GIM08] and references therein. In particular, this notion was used by Lu, Yan and Zhu to study the set-theoretic solution of YBE and the associated ‘braided group’, see [LYZ] and the excellent review [Ta]. The notion of a matched pair of monoids , is developed in [GIM0”
Involutive Yang-Baxter groups (2008, background): “, r2 is the identity map on X2) have received recently a lot of attention by Etingof, Schedler and Soloviev [5], Gateva-Ivanova and Van den Bergh [6, 8], Lu, Yan and Zhu [13], Rump [16, 17], Jespers and Okninski [10, 11] and others.”
On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation (2017, background): “6 from [Sol00] and Proposition 6 from [LYZ00].”
Skew left braces of nilpotent type (2018, background): “hat στ = τσ and r(x,y) = (σ(y),τ(x)), then the pair (X,r) is a non-degenerate set-theoretic solution of the Yang–Baxter equation. The ﬁrstpaperswherenon-degenerateset-theoreticsolutionswerestudied are[11, 15, 21, 31]. It turns out that set-theoretic solutions have connections for example with groups”
Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks (2016, methodology): “e, and because some of the best results about classiﬁcation of skew left braces, our ultimate goal, are done in this setting. 2 Skew left braces: Initial concepts and results Motivated by the work of [39, 33] (which we will recall in the next section), Guarnieri and Vendramin introduced in [28] the ”
Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity (2010, background): “ 44] and have been under intensive study during the last decade. There are numerous works on set-theoretic solutions and related structures, of which a relevant selection for the interested reader is [44, 25, 39, 12, 29, 18, 17, 40, 45, 21, 22, 23]. 6 TATIANA GATEVA-IVANOVA Denition 2.5. Let (X;r) b”
Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation (2023, background): “Moreover, Lu-Yan-Zhu’s construction of the braided groups is functorial, i.e. there is a structure group functor F : NBS → BG given by F(X,R) = (GX, R̄) for all (X,R) ∈ NBS. Theorem 3.19.”
Geometry of the Yang-Baxter Maps: pencils of conics and quadrirational mappings (2003, background): “set-theoretical solutions of the Yang–Baxter equation; a number of examples for the case when X is a ﬁnite set was found in [9], and classiﬁcation results for this case were achieved in [7], see also [11]. It turned out that the following non-degeneracy notion is natural and useful for such maps. Le”