The adjoint group of a Coxeter quandle

Abstract

We give explicit descriptions of the adjoint group $Ad\left(Q_W\right)$ of the Coxeter quandle $Q_W$ associated with an arbitrary Coxeter group $W$. The adjoint group $Ad\left(Q_W\right)$ turns out to be an intermediate group between $W$ and the corresponding Artin group $A_W$, and it 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 $Ad\left(Q_W\right)$ 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 $Ad\left({\Phi }_W\right)$ of ${\Phi }_W$ is isomorphic to the adjoint group of $Q_W$.