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 ] ˜ ) and g ( A [ 2 ] ˜ τ ) , where A [ 2 ] ˜ is the covering matrix of 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 ] ) . Derivations of gim ( A [ 2 ] ) which can be extended to derivations of g ( A [ 2 ] ˜ ) are determined.