Construction of a local Lie group from its Lie algebra

Abstract

In "Symmetry Groups and Their Applications", by W. Miller [1, p.152-206], Miller discusses local Lie group theory and certain resulting applications in special function theory, in the course of this discussion Miller considers local Lie transformation groups and Lie derivatives. Miller is able to prove that any Lie algebra of differential operators is the set of Lie derivatives for some local Lie transformation group (G, Q), where G = (V.?) is the underlying local Lie group and Q is the action. Miller's proof shows that the action Q can be found by solving a system of ordinary differential equations. His proof does not explicitly give the underlying local Lie group G. It only shows that such an underlying local Lie group exists. We show that if you restrict the Lie algebras of differential operators to ones with a basis of the form (see document).... Thus we have found a method of constructing a local Lie group from its Lie algebra when the Lie algebra is realized as differential operators having the above form. The fact that any Lie algebra of differential operators is the set of Lie derivatives for some local Lie transformation group is important in applying local Lie theory to special function theory. By means of local Lie groups that are not sets of matrices, we verify known addition formulas for polynomials of binomial type, 2Fo hypergeometric series, Eulerian polynomials and Hermite polynomials. Although our results can be derived by various special function techniques, our examples are interesting in that they show that the various addition formulas can all be obtained by using the same local Lie group theory.