0000000001045696
AUTHOR
Mohamed Selmi
Separation of representations with quadratic overgroups
AbstractAny unitary irreducible representation π of a Lie group G defines a moment set Iπ, subset of the dual g⁎ of the Lie algebra of G. Unfortunately, Iπ does not characterize π. If G is exponential, there exists an overgroup G+ of G, built using real-analytic functions on g⁎, and extensions π+ of any generic representation π to G+ such that Iπ+ characterizes π.In this paper, we prove that, for many different classes of group G, G admits a quadratic overgroup: such an overgroup is built with the only use of linear and quadratic functions.
Erratum to “Separation of representations with quadratic overgroups” [Bull. Sci. Math. 135 (2) (2011) 141–165]
Abstract In the paper entitled “Separation of representations with quadratic overgroups”, we defined the notion of quadratic overgroups, and announced that the 6-dimensional nilpotent Lie algebra g 6 , 20 admits such a quadratic overgroup. There is a mistake in the proof. The present Erratum explains that the proposed overgroup is only weakly quadratic, and g 6 , 20 does not admit any natural quadratic overgroup.
Moment Map and Gelfand Transform for the Enveloping Algebra
International audience; Describing the Gelfand construction for the analytic states on an universal enveloping algebra, we characterize pure states and re-find the main result of a preceding work with L. Abdelmoula and J. Ludwig on the separation of unitary irreducible representations of a connected Lie group by their generalized moment sets.
Separation of unitary representations of connected Lie groups by their moment sets
AbstractWe show that every unitary representation π of a connected Lie group G is characterized up to quasi-equivalence by its complete moment set.Moreover, irreducible unitary representations π of G are characterized by their moment sets.
Séparation des orbites coadjointes d'un groupe exponentiel par leur enveloppe convexe
Resume Revenant sur la question de la separation des representations unitaires irreductibles d'un groupe de Lie exponentiel G par leur application moment, nous presentons ici une nouvelle solution : au lieu de prolonger l'application moment a l'algebre enveloppante de G , nous proposons de definir une application (non lineaire) Φ de g ∗ dans le dual g + ∗ de l'algebre de Lie d'un groupe resoluble G + , de prolonger les representations de G a G + de telle facon que les orbites coadjointes correspondantes de G + soient caracterisees par l'adherence de leur enveloppe convexe. Ceci nous permet de separer les representations irreductibles de G .