On Differentiability of Vectors in Lie Group Representations

I. Beltita ; D. Beltita

Journal of Lie theory, Tome 21 (2011) no. 4, pp. 771-785

Résumé

\def\g{{\frak g}} We address a linearity problem for differentiable vectors in representations of infinite-dimensional Lie groups on locally convex spaces, which is similar to the linearity problem for the directional derivatives of functions. In particular, we find conditions ensuring that if $\pi\colon G\to{\rm End}({\cal Y})$ is such a representation, and $y\in{\cal Y}$ is a vector such that ${\rm d}\pi(x)y$ makes sense for every $x$ in the Lie algebra $\g$ of $G$, then the mapping ${\rm d}\pi(\cdot)y\colon\g\to{\cal Y}$ is linear and continuous.

Détail

Classification : 22E65, 22E66, 22A10, 22A25
Mots-clés : Lie group, topological group, unitary representation, smooth vector