Some automorphisms of generalized Kac-Moody algebras
Generalized Kac-Moody algebras are similar in many ways to ordinary Kac-Moody algebras, but allow for the possibility of imaginary simple roots. In this paper, the authors extend the results of \textit{J. Fuchs, B. Schellekens} and \textit{C. Schweigert} [Commun. Math. Phys. 180, No. 1, 39-97 (1996; Zbl 0863.17020)] to cover the case of symmetrizable generalized Kac-Moody algebras, and they greatly improve the theory behind some detailed results obtained by case analysis in the earlier paper. In more detail, given a generalized Kac-Moody algebra \(G\) with associated Cartan matrix \(A\) and index set \(I\), the authors consider a permutation of \(I\) fixing \(A\). This permutation induces an automorphism \(\omega\) of \(G\) preserving the usual triangular decomposition and in turn inducing a bijection on the dual of the Cartan subalgebra of \(G\). From \(\omega\), the authors obtain a linear map of \(G\)-modules, denoted \(\tau_{\omega}\), which in general is not a homomorphism, but a twining, or twisted homomorphism. However, the authors' main interest is in the situation where \(\tau_{\omega}\) is an isomorphism of (given) Verma modules and irreducible highest weight modules. In this case, they provide an explicit determination of the `twining' character in terms of the ordinary character of an associated Verma (or irreducible) module of a related Lie algebra called the orbit algebra.
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