Weyl groups of some Kac-Moody algebras
The authors investigate the structure of the Weyl groups of a class of Kac-Moody algebras. Let \(A = (a_{ij})^n_{i, j = 0}\) be the indecomposable generalized Cartan matrix. The authors define a special vertex to be an index \(i\) such that \(a_{ij} a_{ji} \neq 1, 2, 3\) for any \(j \neq i\). For \(n > 2\), algebras with special vertices in their Dynkin diagrams are all indefinite. For convenience, they suppose 0 is a special vertex. In this case, the Weyl group \(W\) decomposes into a semi-direct product of Coxeter groups \(W = W_0 \ltimes \overline W\), where \(W_0 = \langle r_1, r_2, \dots, r_n \rangle\) is the Weyl group on the remaining vertices and \(\overline W = \langle r_\alpha |\alpha \in W_0 \alpha \rangle\). Theorem 2 proves that \(\overline W\) is a Coxeter group. The proofs involve technical arguments on reduced expressions of certain elements of the Weyl group.
scientific article; zbMATH DE number 840165