Formal and analytic deformations of the Witt algebra

For infinite dimensional Lie algebras g, the theory of formal deformations does not generally give decisive results about analytic deformations. In pa rticular, if g(t) is a family of Lie algebras depending holomorphically on t with g(o) = g, and if (g(t)) is formally equivalent to g[[t]], it does no t necessarily follow that g(t) is isomorphic to g for small \t\. However, o ne may ask if (g) over cap(t) is isomorphic to (g) over cap for small \t\, (g) over cap(t) and (g) over cap being suitable completions of g(t) and g r espectively. In this Letter, this question is discussed when g = W, the Wit t algebra.