On universal enveloping algebras in a topological setting
Volume 230 / 2015
Abstract
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.
