On embeddings of homogeneous quandles

TL;DR

This paper investigates the conditions under which homomorphisms from homogeneous quandles can be embedded into conjugation quandles, generalizing existing theorems and providing explicit embeddings for various geometric quandles.

Contribution

It establishes a necessary and sufficient condition for embeddings of homogeneous quandles into conjugation quandles, extending previous embedding theorems and applying them to geometric examples.

Findings

Derived a general embedding criterion for homogeneous quandles

Reinterpreted Bergman's embedding within the homogeneous quandle framework

Constructed explicit embeddings for geometric quandles such as Grassmann and rotation 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.