Multiloop Lie algebras and the construction of extended affine Lie algebras

It is known that a multiloop Lie algebra, which is constructed using multiloop realization, can be a Lie Z(n)-torus if a given multiloop Lie algebra satisfies several conditions, and it is also known that a family of extended affine Lie algebras (EALAs) is obtained from a Lie Z(n)-torus. In many cases, however, even if a given multiloop Lie algebra does not satisfy these conditions, we can also construct a family of EALAs from it. In this paper, we study this construction, and prove that two families of EALAs constructed from two multiloop Lie algebras are coincide up to isomorphisms as EALAs if and only if two multiloop Lie algebras are "support-isomorphic". Also, we give a necessary and sufficient condition for two multiloop Lie algebras to be support-isomorphic. (C) 2010 Elsevier Inc. All rights reserved.