Groups and quandles
Mohamad Maassarani
TL;DR
The paper investigates the relationship between a finite quandle $Q$ and its enveloping group $G(Q)$, including how the quandle structure embeds into ${\mathrm{Conj}}(G)$ and how representations behave. It develops the adjunction between the quandle and group categories via $G$ and ${\mathrm{Conj}}$, analyzes the image quandle $Q_{conj}$, and establishes that finite $Q$ yields a faithful linear representation into $GL_n(Z)$, making $G(Q)$ linear and residually finite. It computes key invariants: irreducible complex representations of $G(Q)$ have dimensions dividing $|Inn(Q)|$, the rational cohomology satisfies $H^*(G(Q),\mathbb{Q}) \cong \Lambda^*(\mathbb{Q}(Q/Inn(Q)))$, and the Malcev Lie algebra is abelian with gr$_1 G(Q) \cong \mathbb{Z}(Q/Inn(Q))$. The work further connects nilpotent and solvable quandle structures to central extensions and subquotients of nilpotent groups, providing a framework for classifying finite injective quandles via group-theoretic invariants.
Abstract
We are intereseted in quandles and their enveloping groups. Various results are proven. We show that a quandle $Q$ and its image in the enveloping group $G(Q)$ have isomorphic enveloping groups. The image quandle is injective. For $Q$ a finite quandle, we show that $G(Q)$ admits a faithfull representation $ρ: G(Q)\to GL_n(\mathbb{Z})$ for some $n$; an irreducible representation of $G(Q)$ over $\mathbb{C}$ is finite dimensional an its degree divides the order of the group $Inn(Q)$ of inner automorphism of $Q$. We determine the Malcev Lie algebra and the rational cohomology ring of $G(Q)$ for $Q$ finite. We prove that a finite injective quandle is a subquandle (for conjugacy) of a finite group. We also prove that the only finite subquandles (for conjugacy) of uniquely divisible groups are trivial quandles and that morphisms from quandles to nilpotent groups (for conjugacy) are constant on the indecomposable components. Implication of these results are considered.