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An embedding of spherical quandles into Lie groups

Ayu Suzuki, Kentaro Yonemura

Abstract

We construct smooth embeddings of spherical quandles into conjugation quandles of Lie groups, where the ambient Lie groups can be taken to be orthogonal, Spin, or Pin groups. Moreover, in dimensions $1$ and $3$, we compare our embeddings with those due to Bergman and Akita.

An embedding of spherical quandles into Lie groups

Abstract

We construct smooth embeddings of spherical quandles into conjugation quandles of Lie groups, where the ambient Lie groups can be taken to be orthogonal, Spin, or Pin groups. Moreover, in dimensions and , we compare our embeddings with those due to Bergman and Akita.

Paper Structure

This paper contains 18 sections, 24 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quandle basics
  3. Action of Lie groups on smooth manifolds
  4. Smooth actions of Lie groups and the Lie-Palais theorem
  5. Lifting Lie group actions
  6. The relationship between the lifted Lie group actions and coverings
  7. Embedding spherical quandles in Lie groups
  8. Preliminaries
  9. In the case $n=1$
  10. Lifting orthogonal group actions on spheres
  11. Embedding quandles defined on projective space into Lie groups
  12. A covering space defined over a conjugacy class of a Lie group
  13. The proof of main theorem in the case $n\geq 2$
  14. Bergman's embedding and Akita's embedding
  15. Construction of embeddings
  16. ...and 3 more sections

Key Result

Theorem 1.3

For any positive integer $n$, there is a Lie group $G_{n}$ and a smooth embedding $\iota_{n}:S^{n}\to G_{n}$ such that $\iota_n$ is an injective quandle homomorphism from the spherical quandle $S^n_{\mathbb{R}}$ to the conjugation quandle $\operatorname{Conj}G_n$. Moreover, $G_n$ may be chosen as fo

Figures (1)

  • Figure 1: Geometric interpretation of the axioms in Definition \ref{['def_quandle']}.

Theorems & Definitions (40)

  • Theorem 1.3
  • Definition 2.1: Joyce1982Matveev1982
  • Proposition 2.2: Nosaka2017
  • Proposition 2.3
  • Theorem 3.1: Lie-Palais theorem; Palais1957
  • Proposition 3.2
  • Proposition 3.3: LiftingActionMontaldiOrtega2009
  • Proposition 3.4
  • Remark 4.1
  • Proposition 4.2
  • ...and 30 more