Some gim algebras and the associated indefinite Kac-Moody algebras

Let A([2]) be the 2-fold affinization of a Cartan matrix A of ADE type. Associated to the generalized intersection matrix algebra gim(A([2])) there are two indefinite Kac-Moody algebras g((A([2])) over tilde) and g((A([2])) over tilde (T)), where (A([2])) over tilde is the covering matrix of A([2]) and (A([2])) over tilde (T) is obtained from a diagram automorphism T of g((A([2])) over tilde). We show that there is a bijection between non-isotropic positive imaginary roots of g((A([2])) over tilde (T)) and those of g((A([2])) over tilde), which is applied to describe the 0-root space and the center of gim(A([2])). Derivations of gim(A([2])) which can be extended to derivations of g((A([2])) over tilde) are determined. (C) 2014 Elsevier Inc. All rights reserved.