\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.
1
Institute of Mathematics "Simion Stoilow", Romanian Academy of Sciences, P. O. Box 1-764, Bucharest, Romania
Ingrid Beltita; Daniel Beltita. On Differentiability of Vectors in Lie Group Representations. Journal of Lie Theory, Tome 21 (2011) no. 4, pp. 771-785. http://geodesic.mathdoc.fr/item/JOLT_2011_21_4_a1/
@article{JOLT_2011_21_4_a1,
author = {Ingrid Beltita and Daniel Beltita},
title = {On {Differentiability} of {Vectors} in {Lie} {Group} {Representations}},
journal = {Journal of Lie Theory},
pages = {771--785},
year = {2011},
volume = {21},
number = {4},
url = {http://geodesic.mathdoc.fr/item/JOLT_2011_21_4_a1/}
}
TY - JOUR
AU - Ingrid Beltita
AU - Daniel Beltita
TI - On Differentiability of Vectors in Lie Group Representations
JO - Journal of Lie Theory
PY - 2011
SP - 771
EP - 785
VL - 21
IS - 4
UR - http://geodesic.mathdoc.fr/item/JOLT_2011_21_4_a1/
ID - JOLT_2011_21_4_a1
ER -
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%A Ingrid Beltita
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%T On Differentiability of Vectors in Lie Group Representations
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%P 771-785
%V 21
%N 4
%U http://geodesic.mathdoc.fr/item/JOLT_2011_21_4_a1/
%F JOLT_2011_21_4_a1
