Killing and the Coxeter transformation of Kac-Moody algebras
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''.
scientific article; zbMATH DE number 4112832
scientific article