Abelian quandles and quandles with abelian structure group

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:225

Article

Lebed, Victoria¹  Mortier, Arnaud¹ 

[1] Univ Caen Normandie, LMNO, BP 5186, F-14032 Caen, France

关键词: Quandle;    Structure group;    Yang-Baxter equation;    Rack homology;   

DOI  :  10.1016/j.jpaa.2020.106474

来源: Elsevier

【 摘 要 】

Sets with a self-distributive operation (in the sense of (a (sic) b) (sic) c = (a (sic) c) (sic) (b (sic) c)), in particular quandles, appear in knot and braid theories, Hopf algebra classification, the study of the Yang-Baxter equation, and other areas. An important invariant of quandles is their structure group. The structure group of a finite quandle is known to be either boring (free abelian), or interesting (non-abelian with torsion). In this paper we explicitly describe all finite quandles with abelian structure group. To achieve this, we show that such quandles are abelian (i.e., satisfy (a (sic) b) (sic) c = (a (sic) c) (sic) b); present the structure group of any abelian quandle as a central extension of a free abelian group by an explicit finite abelian group; and determine when the latter is trivial. In the second part of the paper, we relate the structure group of any quandle to its 2nd homology group H-2. We use this to prove that the H-2 of a finite quandle with abelian structure group is torsion-free, but general abelian quandles may exhibit torsion. Torsion in H-2 is important for constructing knot invariants and pointed Hopf algebras. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free