Irreducible Representations of Infinite-Dimensional Transformation Groups and Lie Algebras, I

This paper contains some general results on irreducibility and inequivalence of representations of certain kinds of infinite dimensional Lie algebras, related to transformation groups. The main abstract theorem is a generalization of a classical result of Burnside. Applications are given, especially to the study of the Dirac quantization problem of representing the Lie algebra of functions on a symplectic manifold.