Hokkaido Mathematical Journal

Cohomological equations and invariant distributions on a compact Lie group

Aziz EL KACIMI ALAOUI and Hadda HMILI

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Abstract

This paper deals with two analytic questions on a connected compact Lie group G. i) Let aG and denote by γ the diffeomorphism of G given by γ (x) = ax (left translation by a). We give necessary and sufficient conditions for the existence of solutions of the cohomological equation f - f ∘ γ = g on the Fréchet space C (G) of complex C functions on G. ii) When G is the torus ${\Bbb T}^n$, we compute explicitly the distributions on ${\Bbb T}^n$ invariant by an affine automorphism γ, that is, γ (x) = A (x + a) with A ∈ GL(n, ℤ) and a ∈ ${\Bbb T}^n$. iii) We apply these results to describe the infinitesimal deformations of some Lie foliations.

Article information

Source
Hokkaido Math. J., Volume 43, Number 2 (2014), 151-173.

Dates
First available in Project Euclid: 1 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1404229920

Digital Object Identifier
doi:10.14492/hokmj/1404229920

Mathematical Reviews number (MathSciNet)
MR3229069

Zentralblatt MATH identifier
1294.22011

Subjects
Primary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32] 37A05: Measure-preserving transformations 37C10: Vector fields, flows, ordinary differential equations 58A30: Vector distributions (subbundles of the tangent bundles)

Keywords
Lie group cohomological equation distribution foliation deformation

Citation

EL KACIMI ALAOUI, Aziz; HMILI, Hadda. Cohomological equations and invariant distributions on a compact Lie group. Hokkaido Math. J. 43 (2014), no. 2, 151--173. doi:10.14492/hokmj/1404229920. https://projecteuclid.org/euclid.hokmj/1404229920


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