In this work, we develop a homological theory for the graded Lie algebras, which gives new information on the structure of the Lorentzian Kac- Moody Lie algebras. The technique of the Hochschild-Serre spectral sequences offers a uniform method of studying the higher level root multiplicities and the principally specialized affine characters of Lorentzian Kac-Moody Lie algebras. In the past 20 years, the theory of Kac-Moody Lie algebras has developed rapidly and with great success. Surprising connections to areas such as combi- natorics, modular forms, and mathematical physics have shown Kac-Moody Lie algebras to be of uncommon interest. The discovery of the Macdonald identi- ties gave rise to an intensive study of the class of affine Kac-Moody Lie algebras and their representations (Mcd). The structure of such Lie algebras and their connections with other branches of mathematics and mathematical physics have been well-established and are being extensively investigated. The next natural step after the affine case is that of the hyperbolic Kac-Moody Lie algebras. One of the most ambitious goals of current research activity in infinite dimensional Lie algebras may be to construct geometric realizations of the hyperbolic Kac-Moody Lie algebras. Once that is accomplished, we will have a much deeper understanding of the structure of Kac-Moody Lie algebras and their connections with number theory. Unfortunately, many basic questions regarding the hyperbolic case are still unresolved. For example, the behavior of the root multiplicities is not well-understood. Feingold-Frenkel (F-F) and Kac- Moody-Wakimoto (K-M-W) made some progress in this area. They computed the level 2 root multiplicities for the hyperbolic Kac-Moody Lie algebras HA^ and HE% . Other important works on the hyperbolic Kac-Moody Lie algebras include (Fe2, Fr, LM, and M2). Recently, V. Kac has informed the author that he also discovered a level 3 root multiplicity formula for HA^ (unpublished). In this work, we develop a homological theory for the graded Lie algebras. Combining with the representation theory of affine Kac-Moody Lie algebras, we obtain new information on the structure of the Lorentzian Kac-Moody Lie algebras; i.e., Kac-Moody Lie algebras whose Carian matrix has a Lorentzian
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