The adjoint group of a Coxeter quandle

Toshiyuki Akita

doi:10.1215/21562261-2019-0061

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arXiv · 1702.07104

The adjoint group of a Coxeter quandle

Toshiyuki Akita

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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 $Q_W$.

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BibTeX RIS

Toshiyuki Akita. 2017-02-23. The adjoint group of a Coxeter quandle. https://doi.org/10.1215/21562261-2019-0061

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