Classification of flat connected quandles

Abstract

Let A be an additive abelian group. Then the binary operation a∗b=2b−a gives a quandle structure on A, denoted by T(A), and called the Takasaki quandle of A. Viewing quandles as generalization of Riemannian symmetric spaces, Ishihara and Tamaru [Flat connected finite quandles, to appear in Proc. Amer. Math. Soc. (2016)] introduced flat quandles, and classified quandles which are finite, flat and connected. In this note, we classify all flat connected quandles. More precisely, we prove that a quandle X is flat and connected if and only if X≅T(A), where A is a 2-divisible group.