On the geometry underlying a real Lie algebra representation

Abstract

Let $G$ be a real Lie group with Lie algebra $\mathfrak g$. Given a unitary representation $\pi$ of $G$, one obtains by differentiation a representation $d\pi$ of $\mathfrak g$ by unbounded, skew-adjoint operators. Representations of $\mathfrak g$ admitting such a description are called \emph{integrable,} and they can be geometrically seen as the action of $\mathfrak g$ by derivations on the algebra of representative functions $g\mapsto<\xi,\pi(g)\eta>$, which are naturally defined on the homogeneous space $M=G/\ker\pi$. 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 $\mathfrak g$ 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.