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.
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