Infinite Dimensional Lie Algebras of Type L

Abstract

In this paper, a class of infinite dimensional Lie algebras L(A, δ, α) over a field of characteristic 0 are studied. These Lie algebras, which we call here Lie algebras of type L, arose as one subclass in the recent classification of generalized Block algebras. We exhibit a large subclass of these algebras which are simple, as well as another subclass of these algebras which are never simple. For n > 1, simple Lie algebras of type L do not occur in any other known class of simple Lie algebras. In particular, for n > 1, these algebras have no toral elements. Simplicity in these algebras is equivalent to simplicity of an appropriate subalgebra. The notion of transitive ideal plays an important role in this theory.

Key Words:

• Cartan type Lie algebras
• Lie algebras of type L
• Simple Lie algebras
• Torus
• Transitive ideal

AMS Subject Classification:

• Primary: 17B40
• 17B65
• 17B66
• 17B68
• 17B70

Acknowledgments