A Survey of Racks and Quandles: Some Recent Developments
نویسندگان
چکیده
منابع مشابه
Extensions of Racks and Quandles
A rack is a set equipped with a bijective, self-right-distributive binary operation, and a quandle is a rack which satisfies an idempotency condition. In this paper, we introduce a new definition of modules over a rack or quandle, and show that this definition includes the one studied by Etingof and Graña [9] and the more general one given by Andruskiewitsch and Graña [1]. We further show that ...
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We revisit finite racks and quandles using a perspective based on permutations which can aid in the understanding of the structure. As a consequence we recover old results and prove new ones. We also present and analyze several examples.
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Given a rack Q and a ring A , one can construct a Yang-Baxter operator cQ : V ⊗ V → V ⊗ V on the free A-module V = AQ by setting cQ(x ⊗ y) = y ⊗ x y for all x, y ∈ Q . In answer to a question initiated by D.N.Yetter and P.J. Freyd, this article classifies formal deformations of cQ in the space of Yang-Baxter operators. For the trivial rack, where x = x for all x, y , one has, of course, the cla...
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A rack of order n is a binary operation B on a set X of cardinality n, such that right multiplication is an automorphism. More precisely, (X,B) is a rack provided that the map x 7→ x B y is a bijection for all y ∈ X, and (x B y) B z = (x B z) B (y B z) for all x, y, z ∈ X. The paper provides upper and lower bounds of the form 2cn 2 on the number of isomorphism classes of racks of order n. Simil...
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1. We consider differential topology to be the study of differentiable manifolds and differentiable maps. Then, naturally, manifolds are considered equivalent if they are diffeomorphic, i.e., there exists a differentiable map from one to the other with a differentiable inverse. For convenience, differentiable means C; in the problems we consider, C' would serve as well. The notions of different...
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ژورنال
عنوان ژورنال: Algebra Colloquium
سال: 2020
ISSN: 1005-3867,0219-1733
DOI: 10.1142/s1005386720000425