On the geometry underlying a real Lie algebra representation

TL;DR

This paper explores the geometric structures underlying representations of real Lie algebras, extending classical ideas to non-integrable cases using the coproduct of the universal enveloping algebra, and introduces a geometric framework for analyzing integrability.

Contribution

It generalizes the geometric interpretation of Lie algebra representations to non-integrable cases via the coproduct, establishing a new geometric approach to integrability questions.

Findings

Representations can be viewed as vector fields on homogeneous spaces.

The coproduct allows extension to non-integrable representations.

Provides a geometric framework for integrability analysis.

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.