Some automorphisms of Generalized Kac-Moody algebras

In this paper we consider some algebraic structures associated to a class of outer automorphisms of generalized Kac-Moody (GKM) algebras. These structures have recently been introduced in [2] for a smaller class of outer automorphisms in the case of ordinary Kac-Moody algebras with symmetrizable Cartan matrices. A GKM algebra G = G(A) is essentially described by its Cartan matrix, A = (aij)i,j∈I ; the index set I can be either a finite or a countably infinite set. For any permutation ω̇ of the set I which has finite order and leaves the Cartan matrix invariant, we find a family of outer automorphisms ω of the GKM algebra G(A) which preserve the Cartan decomposition. Such an outer automorphism gives rise to a linear bijection τω of G-modules, obeying the ω-twining property, i.e. if V is a G-module, then