Journal of Lie Theory

no. 4
On Differentiability of Vectors in Lie Group Representations
Journal of Lie Theory, Volume 21 (2011) no. 4, pp. 771-785
\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.
DOI: 10.5802/jolt.648
Classification: 22E65, 22E66, 22A10, 22A25
Keywords: Lie group, topological group, unitary representation, smooth vector 
@article{JOLT_2011_21_4_a1,
     author = {I. Beltita and D. Beltita},
     title = {On {Differentiability} of {Vectors} in  {Lie} {Group} {Representations}},
     journal = {Journal of Lie Theory},
     pages = {771--785},
     year = {2011},
     volume = {21},
     number = {4},
     doi = {10.5802/jolt.648},
     zbl = {1232.22014},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.648/}
}
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I. Beltita; D. Beltita. On Differentiability of Vectors in  Lie Group Representations. Journal of Lie Theory, Volume 21 (2011) no. 4, pp. 771-785. doi: 10.5802/jolt.648

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