Invariant Hochschild Cohomology of Smooth Functions
Journal of Lie Theory, Tome 31 (2021) no. 2, pp. 557-574
Voir la notice de l'article provenant de la source Heldermann Verlag
Given an action of a Lie group on a smooth manifold, we discuss the induced action on the Hochschild cohomology of smooth functions, and notions of invariance on this space. Depending on whether one considers invariance of cochains or invariance of cohomology classes, two different spaces of invariants arise. We perform a general comparison of these notions, give an interpretation of the lower orders of the invariant cohomology spaces and conclude as our main result that for proper group actions both spaces are isomorphic. As a corollary and a geometric interpretation, an invariant version of the Hochschild-Kostant-Rosenberg theorem is given, identifying the cohomology of invariant cochains with invariant multivector fields. Using this theorem, we shortly discuss the invariant Hochschild cohomology in the case of homogeneous spaces.
Classification :
13D03, 16W22
Mots-clés : Differential geometry, homological algebra, deformation quantization
Mots-clés : Differential geometry, homological algebra, deformation quantization
Affiliations des auteurs :
Lukas Miaskiwskyi 1
Lukas Miaskiwskyi. Invariant Hochschild Cohomology of Smooth Functions. Journal of Lie Theory, Tome 31 (2021) no. 2, pp. 557-574. http://geodesic.mathdoc.fr/item/JOLT_2021_31_2_a13/
@article{JOLT_2021_31_2_a13,
author = {Lukas Miaskiwskyi},
title = {Invariant {Hochschild} {Cohomology} of {Smooth} {Functions}},
journal = {Journal of Lie Theory},
pages = {557--574},
year = {2021},
volume = {31},
number = {2},
url = {http://geodesic.mathdoc.fr/item/JOLT_2021_31_2_a13/}
}