Some gim algebras and the associated indefinite Kac–Moody algebras

Let A[2]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])gim(A[2]) there are two indefinite Kac–Moody algebras g(A[2]˜) and g(A[2]˜τ), where A[2]˜ is the covering matrix of A[2]A[2] and A[2]˜τ is obtained from a diagram automorphism τ   of g(A[2]˜). We show that there is a bijection between non-isotropic positive imaginary roots of g(A[2]˜τ) and those of g(A[2]˜), which is applied to describe the 0-root space and the center of gim(A[2])gim(A[2]). Derivations of gim(A[2])gim(A[2]) which can be extended to derivations of g(A[2]˜) are determined.