2012/06/01 by Rodrigo Vargas Le-Bert, Le-Bert, Rodrigo Vargas
Mathematics · #16W25 #17B15 #Advanced Topics in Algebra #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:16W25 #msc:17B15
paper · pdf · doi:10.48550/arxiv.1206.0210
12 pages. Author supported by Fondecyt Postdoctoral Grant N° 3110045
arxiv created 2012/06/01 · openalex publication_date 2012/06/01 · arxiv updated 2012/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a real Lie group with Lie algebra \mathfrak g. Given a unitary representation π of G, one obtains by differentiation a representation dπ of \mathfrak g by unbounded, skew-adjoint operators. Representations of \mathfrak g admitting such a description are called integrable, and they can be geometrically seen as the action of \mathfrak g by derivations on the algebra of representative functions g↦<ξ,π(g)η>, 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 \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.