# 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.