Diffeomorphism Groups, Quantization, and Su(∞)

A method of relating infinite dimensional Lie algebras to basis dependent sequences of finite dimensional Lie algebras is introduced. Diff A S 2 and Diff A T 2 are shown to be N → ∞ limits of SU(N), with a limiting procedure, and " SU (∞)'s", that differ from the usual Kac Moody ones. After a detailed presentation of these two 'principal' examples, the general construction is outlined, which seems to work for the Poisson algebra of more or less any homogenous symplectic manifold.