An embedding of spherical quandles into Lie groups

TL;DR

This paper constructs smooth embeddings of spherical quandles into conjugation quandles of Lie groups like orthogonal, Spin, or Pin groups, and compares these with existing embeddings in dimensions 1 and 3.

Contribution

It introduces new smooth embeddings of spherical quandles into Lie group conjugation quandles and relates them to prior embeddings in specific low dimensions.

Findings

Embeddings into orthogonal, Spin, and Pin groups are constructed.

Comparison with Bergman and Akita's embeddings in dimensions 1 and 3.

Provides a framework for understanding spherical quandle embeddings 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 $1$ and $3$, we compare our embeddings with those due to Bergman and Akita.