Let G be a real Lie group with Lie algebra g. Given a unitary
representation π of G, one obtains by differentiation a representation
dπ of g by unbounded, skew-adjoint operators. Representations
of g admitting such a description are called \emph{integrable,} and
they can be geometrically seen as the action of g by derivations on
the algebra of representative functions g↦, which are
naturally defined on the homogeneous space M=G/kerπ. In other words,
integrable representations of a real Lie algebra can always be seen as
realizations of that algebra by vector fields on a homogeneous manifold. Here
we show how to use the coproduct of the universal enveloping algebra of
g to generalize this to representations which are not necessarily
integrable. The geometry now playing the role of M is a locally homogeneous
space. This provides the basis for a geometric approach to integrability
questions regarding Lie algebra representations.Comment: 12 pages. Author supported by Fondecyt Postdoctoral Grant N{\deg}
311004