Killing and the Coxeter transformation of Kac-Moody algebras
type: Item
instance of: scholarly article
title: Killing and the Coxeter transformation of Kac-Moody algebras
zbMATH Open document ID: 0679.17008
DOI: 10.1007/BF01393885
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143661
review text: Let A be an \(\ell \times \ell\) matrix with entries in an algebraically closed field with char \(K\neq 2\) and \(a_{ii}\neq 1\). Let V be an \(\ell\)- dimensional vector space with basis \(\{\alpha_ 1,...,\alpha_{\ell}\}\) and let \(S_ i\) be defined by \(S_ i(\alpha_ j)=\alpha_ j- a_{ij}\alpha_ i\). Let R be any product of distinct pseudoreflections \(R=S_{i_ 1},...,S_{i_ k}\). The author proves that if \(Rx=x\) then \(S_{i_ j}x=x\), \(1\leq j\leq k.\) Next, the author studies the Coxeter transformation \(R=S_{\ell}... S_ 1\). When A is the Cartan matrix of a finite-dimensional simple Lie algebra, it is known that R and \(R_{\sigma}\) are in the same conjugacy class for all permutations \(\sigma\) of 1,...,\(\ell\). This result is extended to the case when the graph of A is a tree. In the case when the graph is acyclic then the \(R_{\sigma}\) fall into \(\ell -1\) spectral classes. The paper is closely related to the paper of \textit{S. Berman}, \textit{Y. S. Lee} and \textit{R. V. Moody} (reviewed above) and in fact generalizes their results. The author makes frequent references to the work of Killing, ``in the hope of helping those of my readers who are not directly familiar with his work to gain some feeling of the extraordinary mathematical breakthrough he achieved''.
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 4112832
zbMATH Keywords: pseudoreflections
zbMATH Keywords: Coxeter transformation
zbMATH Keywords: Cartan matrix
zbMATH Keywords: Killing
author: A. John Coleman
MaRDI profile type: Publication
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OpenAlex ID: W2002678252
description: scientific article; zbMATH DE number 4112832