2 9 Fe b 20 04 QUASI SIMPLE LIE ALGEBRAS

We investigate a class of Lie algebras called quasi-simple Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac-Moody Lie algebras. A quasi-simple 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 more general class of quasi-simple 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 are able to show that the core modulo the center of a quasi-simple Lie algebra is a direct sum of cores modulo centers of some indecomposable quasi-simple 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 (QSLA for short) by proposing a system of fairly natural and not so much restrictive axioms. These Lie algebras are roughly characterized by a symmetric nondegenerate invariant bilinear form, a finite dimensional Cartan subalgebra, a discrete root system and the ad-nilpotency of the root spaces attached to non-isotropic roots. In [HK-T], the authors extract some basic properties of QSLAs from the axioms, but for the further study of such Lie algebras they assume the irre-ducibility of the corresponding root systems. Namely, a QSLA is called irre-ducible if the set of non-isotropic roots is indecomposable and isotropic roots are non-isolated (see Definition 1.11 for terminology). Following [AABGP], we call an irreducible QSLA an extended affine Lie algebra (EALA for short). We note that EALA have been under intensive investigation in recent years,