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December 2020 The adjoint group of a Coxeter quandle
Toshiyuki Akita
Kyoto J. Math. 60(4): 1245-1260 (December 2020). DOI: 10.1215/21562261-2019-0061

Abstract

We give explicit descriptions of the adjoint group Ad ( Q W ) of the Coxeter quandle Q W associated with an arbitrary Coxeter group W . The adjoint group Ad ( Q W ) 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 ( Q W ) is isomorphic to the commutator subgroup of W . Finally, the root system Φ W associated with a Coxeter group W turns out to be a rack. We prove that the adjoint group Ad ( Φ W ) of Φ W is isomorphic to the adjoint group of Q W .

Citation

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Toshiyuki Akita. "The adjoint group of a Coxeter quandle." Kyoto J. Math. 60 (4) 1245 - 1260, December 2020. https://doi.org/10.1215/21562261-2019-0061

Information

Received: 4 September 2018; Revised: 19 September 2018; Accepted: 21 September 2018; Published: December 2020
First available in Project Euclid: 29 September 2020

MathSciNet: MR4175808
Digital Object Identifier: 10.1215/21562261-2019-0061

Subjects:
Primary: 20F55
Secondary: 08A05 , 19C09 , 20F36

Keywords: Artin group , Coxeter group , quandle , rack , root system

Rights: Copyright © 2020 Kyoto University

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Vol.60 • No. 4 • December 2020
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