Generalized Kac–Moody Lie Algebras and Product Quivers

Abstract

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 [16], 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.

Key Words:

• Generalized Kac–Moody algebras
• Product quivers
• Ringel–Hall algebras

2000 Mathematics Subject Classification:

• 17B67
• 16G20

ACKNOWLEDGMENTS

The authors thank Jie Xiao and Bangming Deng for stimulating discussions on the questions of realizing the whole Lie algebras. In particular, Bangming Deng pointed out that there is no need to introduce the new variables h _i for the semisimple part. We thank the referee for valuable comments and suggestions.

Notes

Communicated by D. Nakano.