Invariant Forms on Lie Algebra Extensions

Abstract

For a perfect ideal 𝔥 of the Lie algebra 𝔤 with a 𝔤-invariant symmetric bilinear form ⟨·, ·⟩, we consider the continuous cohomology class in defined by the 2-cocycles of the form ⟨[X, ·], ·⟩ on 𝔥, X ∈ 𝔤. We determine the obstruction for extending this class to 𝔤. Invariant symmetric bilinear forms on corresponding Abelian extensions of 𝔤 by (𝔤/𝔥)* are constructed. The result is applied to central extensions of the Lie algebra of symplectic vector fields and of the Lie algebra of divergence free vector fields.

Key Words:

• Crossed module
• Lie algebra extension
• Symplectic and Hamiltonian vector field

2000 Mathematics Subject Classification:

• 17B66
• 17B56

ACKNOWLEDGMENT

The author thanks Karl-Hermann Neeb for helpful comments. Supported by Centre Bernoulli, Lausanne and by the FWF Projekt P 17108-N04.

Notes

Communicated by A. Elduque.