Abelian quandles and quandles with abelian structure group

Victoria Lebed (1) , Arnaud Mortier (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

Sets with a self-distributive operation (in the sense of (a ⊳ b) ⊳ c = (a ⊳ c) ⊳ (b ⊳ 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 ⊳ b) ⊳ c = (a ⊳ c) ⊳ 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.

Domains: Mathematics [math]; Group Theory [math.GR]; Mathematics [math]; K-Theory and Homology [math.KT]

Victoria Lebed, Arnaud Mortier. Abelian quandles and quandles with abelian structure group. 2019. ⟨hal-02266880v3⟩