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 $1$ and $3$, we compare our embeddings with those due to Bergman and Akita.

• The paper presents the complete solution for embedding spherical quandles into the conjugation quandle of specific Lie groups using smooth geometric methods.
• It introduces explicit constructions mapping S^n to Lie groups like O(2), Spin(n+1), or Pin⁺(n+1), leveraging symmetric space and covering space techniques.
• The work compares these geometric embeddings with earlier algebraic methods, highlighting differences in faithfulness and the unique group structures in low dimensions.

Embedding Spherical Quandles into Lie Groups

Introduction and Problem Statement

The paper "An embedding of spherical quandles into Lie groups" (2603.29479) addresses the longstanding problem of when a quandle, particularly a smooth and algebraically connected one, admits an injective smooth homomorphism into a conjugation quandle of a Lie group. The work gives a complete solution for the class of spherical quandles, which are smooth quandles associated to the spheres $S^n$ as Riemannian symmetric spaces. Previous progress focused mainly on discrete, algebraic, or combinatorial settings, while this work leverages smooth and geometric structures to obtain embedding results into classical compact Lie groups.

Main Results

Statement of Main Theorem

The paper proves that for every $n \geq 1$, the spherical quandle $S^n_{\mathbb{R}}$ can be smoothly, injectively embedded as a quandle into the conjugation quandle of an explicit Lie group $G_n$, with a closed-form description:

• $G_1 = O(2)$,
• $G_{n} = Spin(n+1)$ if $n$ is even and $n \geq 2$,
• $G_{n} = Pin^+(n+1)$ if $n$ is odd and $n \geq 1$0.

This embedding $n \geq 1$1 is both a smooth embedding of manifolds and a homomorphism of quandles into $n \geq 1$2, the quandle given by the conjugacy action in the group.

This result exploits the symmetric space structure of spheres and the Cartan-Dieudonné theorem to identify the relevant groups of symmetries and their double coverings, using the Pin and Spin groups as natural ambient Lie groups related to the inner automorphism structure of the spherical quandles.

Explicit Construction

For each $n \geq 1$3, a concrete construction of the embedding $n \geq 1$4 is provided. In low dimensions, the paper recovers and analyzes explicit embeddings:

• For $n \geq 1$5, an explicit diffeomorphism/quandle isomorphism $n \geq 1$6 matches the classical identification of $n \geq 1$7 and $n \geq 1$8 via rotation matrices.
• For $n \geq 1$9, the embedding $S^n_{\mathbb{R}}$0 factors through the identification of $S^n_{\mathbb{R}}$1 with the core quandle structure on $S^n_{\mathbb{R}}$2, embedded into $S^n_{\mathbb{R}}$3.

These constructions use smooth group actions, covering space theory, and the classification of transitive group actions on spheres to realize the quandle operations as conjugations in ambient Lie groups.

Comparison with Prior Embeddings

The work compares these geometric embeddings to prior algebraically-motivated embeddings, in particular:

• Bergman's embedding of core quandles into semidirect products involving $S^n_{\mathbb{R}}$4 and involutive automorphisms [Bergman2021core].
• Akita's embedding of generalized Alexander and twisted conjugation quandles, especially in the case of abelian groups [akita2022embedding].

For $S^n_{\mathbb{R}}$5 and $S^n_{\mathbb{R}}$6, explicit commutative diagrams are constructed showing the precise relationship and, in the abelian ($S^n_{\mathbb{R}}$7) case, the equivalence of the Suzuki embedding to these earlier constructions. Only in dimensions $S^n_{\mathbb{R}}$8 and $S^n_{\mathbb{R}}$9 do the spheres possess a Lie group structure compatible with the quandle operation, as dictated by the parallelizability of spheres.

Technical Highlights

Covering Space Methods and Lifting Group Actions

Key to the construction is a systematic procedure for lifting group actions through universal covering groups. Given the natural action of the orthogonal or special orthogonal group on the sphere or projective space, the embedding theorem is facilitated by passing to the Spin or Pin cover. The involution at the core of the quandle operation is connected to Cartan involutions and the structure of symmetric spaces.

Group schemes such as $G_n$0 and $G_n$1, together with coverings of conjugacy classes and stabilizers, are explicitly invoked. The construction leverages the connectedness and smooth structure of the quandles, crucially distinguishing the smooth context from the combinatorial one.

Faithful Versus Non-Faithful Quandles

The spherical quandles are notable for being non-faithful quandles: the canonical homomorphism to their inner automorphism group is not injective. However, the geometric and topological structure of the sphere, along with the properties of the associated Lie group coverings, enables a homomorphic embedding into a larger ambient group than the inner automorphism group. This circumvents obstacles encountered by algebraic methods, which depend on faithfulness for immediate embeddings into inner automorphism groups.

Broader Implications

Homogeneous Quandles and Symmetric Spaces

The approach yields an explicit geometric picture of quandles as arising naturally from symmetric spaces, broadening the connection between knot theory, symmetric space theory, and representation theory of Lie groups. Via this construction, the smooth category of quandles aligns with the classical theory of Lie groups and their homogeneous spaces.

The methods also serve as a foundation for embedding homogeneous quandles more generally; the first author has extended the results to cases such as real Grassmannians.

Limitations and Future Questions

The precise coincidence of Suzuki's embedding with Bergman's and Akita's constructions only occurs for $G_n$2 and $G_n$3, reflecting the parallelizability and group structure limitations on spheres; in higher dimensions these identifications fail due to the absence of a Lie group structure. The paper poses further questions about finding or classifying groups $G_n$4 where $G_n$5 is isomorphic as a quandle to a twisted conjugation quandle on $G_n$6 and about the influence of noncommutativity on their structures.

Further developments could extend embedding results to other classes of smooth, homogeneous, or symmetric quandles, refine the topological classification of quandles using Lie theory, and explore applications to invariants in geometric topology.

Conclusion

The paper provides a comprehensive construction of smooth, injective embeddings of spherical quandles into conjugation quandles of explicit Lie groups (orthogonal, Spin, Pin). This advances the understanding of the embedding problem for smooth, algebraically connected quandles and exposes deep connections among quandle cohomology, knot theory, and the theory of compact Lie groups. The geometric framework established here supports further investigations into the algebraic and topological classification of quandles and enriches the toolkit for studying objects at the intersection of geometry, algebra, and topology.