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On embeddings of homogeneous quandles

Ayu Suzuki

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 $(G,H,σ)$ 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.

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 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 arising from symmetric spaces.
Paper Structure (13 sections, 15 theorems, 121 equations)

This paper contains 13 sections, 15 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation.
  4. Quandle
  5. Embedding of quandles
  6. Homogeneous quandles and quandle triplet
  7. Main statements and their proofs
  8. Main theorems
  9. Examples
  10. Bergman's embedding of the core quandle
  11. Embedding of the $\theta$--rotation quandle on the sphere $S^2$
  12. Embedding of the unoriented Grassmann quandle
  13. Embedding of the oriented Grassmann quandle

Key Result

Theorem 2.12

For any $n \in \mathbb{Z}_{>0}$, let $G_n$ be the Lie group defined by where $Spin(n+1)$ and $Pin^+(n+1)$ denote the non-trivial double coverings of $SO(n+1)$ and $O(n+1)$, respectively. Then exists an embedding of the spherical quandle

Theorems & Definitions (34)

  • Definition 2.1: Quandle
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5: $\theta$--rotation quandle on the sphere $S^2$
  • Example 2.6: Grassmann quandle
  • Example 2.7: Oriented Grassmann quandle
  • Definition 2.8: Quandle homomorphism
  • Definition 2.9
  • Definition 2.10: Embeddable quandle
  • ...and 24 more