# The adjoint group of a Coxeter quandle

## TL;DR

This paper explicitly describes the adjoint group of a Coxeter quandle, revealing its structure as a central extension of the Coxeter group and its relation to Artin groups and root systems.

## Contribution

It provides a detailed description of the adjoint group of Coxeter quandles, including its construction, relation to Coxeter and Artin groups, and the connection to root systems.

## Key findings

- The adjoint group is an intermediate group between W and A_W.
- It fits into a central extension of W by a free abelian group.
- The root system is a rack and its adjoint group matches that of the quandle.

## Abstract

We give explicit descriptions of the adjoint group of the Coxeter quandle $Q_W$ associated with an arbitrary Coxeter group $W$. The adjoint group of $Q_W$ turns out to be an intermediate group between $W$ and the corresponding Artin group $A_W$, and fits into a central extension of $W$ by a finitely generated free abelian group. We construct $2$-cocycles of $W$ corresponding to the central extension. In addition, we prove that the commutator subgroup of the adjoint group of $Q_W$ is isomorphic to the commutator subgroup of $W$. Finally, the root system $\Phi_W$ associated with a Coxeter group $W$ turns out to be a rack. We prove that the adjoint group of $\Phi_W$ is isomorphic to the adjoint group of $Q_W$.