On embeddings of homogeneous quandles

Abstract

In this paper, we study the embedding problem of homogeneous quandles. We give a necessary and sufficient condition under which a quandle homomorphism from the homogeneous quandle associated with a quandle triplet $(G,H,\sigma)$ into a conjugation quandle of a group is an embedding. This result provides a generalization of the embedding theorem of Dhanwani, Raundal and Singh for generalized Alexander quandles. As applications of the main theorem, we reinterpret Bergman's embedding of core quandles in the framework of homogeneous quandles, and construct explicit embeddings of several geometric examples, including unoriented and oriented Grassmann quandles and rotation quandles of $S^2$ arising from symmetric spaces.