Formal and Analytic Deformations of the Witt Algebra

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For infinite dimensional Lie algebras ℊ, the theory of formal deformations does not generally give decisive results about analytic deformations. In particular, if ℊt is a family of Lie algebras depending holomorphically on t with ℊ0 = ℊ, and if (ℊt) is formally equivalent to ℊ[[t]], it does not necessarily follow that ℊt is isomorphic to ℊ for small ∣t∣. However, one may ask if Symbolt is isomorphic to Symbol for small ∣t∣,Symbolt and Symbol being suitable completions of ℊt and ℊ respectively. In this Letter, this question is discussed when ℊ = 𝒲, the Witt algebra.

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