6 Generalized Kac - Moody Lie Algebras and Product Quivers

We construct the entire generalized Kac-Moody Lie algebra as a quotient of the positive part of another generalized Kac-Moody Lie algebra. The positive part of a generalized Kac-Moody Lie algebra can be constructed from representations of quivers using Ringel's Hall algebra construction. Thus we give a direct realization of the entire generalized Kac-Moody Lie algebra. This idea arises from the affine Lie algebra construction and evaluation maps. In [LL], we give a quantum version of this construction after analyzing Nakajima's quiver variety construction of integral highest weight representations of the quantized enveloping algebras in terms of the irreducible components of quiver varieties.