2 Generalized Reductive Lie

We investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac-Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras have been under intensive investigation in recent years. They have also been called extended affine Lie algebras. The larger class of generalized reductive Lie algebras has not been so intensively investigated. We study them in this paper and note that one way they arise is as fixed point subalgebras of finite order automorphisms. We show that the core modulo the center of a generalized reductive Lie algebra is a direct sum of centerless Lie tori. Therefore one can use the results known about the classification of centerless Lie tori to classify the cores modulo centers of generalized reductive Lie algebras. 0. Introduction In 1990 Høegh-Krohn and B Torresani [HK-T] introduced a new interesting class of Lie algebras over field of complex numbers, called quasi simple Lie algebras by proposing a system of fairly natural and not very restrictive axioms. These Lie algebras are characterized by the existence of a symmetric nondegenerate invariant bilinear form, a finite dimensional Cartan subalgebra, a discrete root system which contains some nonisotropic roots, and the ad-nilpotency of the root spaces attached to non-isotropic roots. As it will appear from the sequel, these algebras are natural generalizations of reductive Lie algebras, and affine Kac–Moody Lie algebras. For this reason and other reasons indicated in the introduction of the paper [AABGP] we