On universal enveloping algebras in a topological setting

Daniel Beltiţă¹ ; Mihai Nicolae²

¹Institute of Mathematics “Simion Stoilow” of the Romanian Academy P.O. Box 1-764 Bucureşti, Romania
²Petroleum-Gas University of Ploieşti Bd. Bucureşti no. 39 100680 Ploieşti, Romania and Institute of Mathematics “Simion Stoilow” of the Romanian Academy P.O. Box 1-764 Bucureşti, Romania

Studia Mathematica, Tome 230 (2015) no. 1, pp. 1-29

Résumé

We study some embeddings of suitably topologized spaces of vector-valued smooth functions on topological groups, where smoothness is defined via differentiability along continuous one-parameter subgroups. As an application, we investigate the canonical correspondences between the universal enveloping algebra, the invariant local operators, and the convolution algebra of distributions supported at the unit element of any finite-dimensional Lie group, when one passes from finite-dimensional Lie groups to pre-Lie groups. The latter class includes for instance all locally compact groups, Banach–Lie groups, additive groups underlying locally convex vector spaces, and also mapping groups consisting of rapidly decreasing Lie group-valued functions.

Détail

DOI : 10.4064/sm8003-12-2015

Keywords: study embeddings suitably topologized spaces vector valued smooth functions topological groups where smoothness defined via differentiability along continuous one parameter subgroups application investigate canonical correspondences between universal enveloping algebra invariant local operators convolution algebra distributions supported unit element finite dimensional lie group passes finite dimensional lie groups pre lie groups latter class includes instance locally compact groups banach lie groups additive groups underlying locally convex vector spaces mapping groups consisting rapidly decreasing lie group valued functions