Lie algebra of homogeneous operators of a vector bundle

We prove that for a vector bundle \( E \to M\), the Lie algebra \(\mathcal{D}_{\mathcal{E}}(E)\) generated by all differential operators on \(E\) which are eigenvectors of \(L_{\mathcal{E}},\) the Lie derivative in the direction of the Euler vector field of \(E,\) and the Lie algebra \(\mathcal{D}_G...

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Veröffentlicht in:arXiv.org 2020-08
Hauptverfasser: Lecomte, P B A, Elie Zihindula Mushengezi
Format: Artikel
Sprache:eng
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Zusammenfassung:We prove that for a vector bundle \( E \to M\), the Lie algebra \(\mathcal{D}_{\mathcal{E}}(E)\) generated by all differential operators on \(E\) which are eigenvectors of \(L_{\mathcal{E}},\) the Lie derivative in the direction of the Euler vector field of \(E,\) and the Lie algebra \(\mathcal{D}_G(E)\) obtained by Grothendieck construction over the \(\mathbb{R}-\)algebra \(\mathcal{A}(E):= {\rm Pol}(E)\) of fiberwise polynomial functions, coincide up an isomorphism. This allows us to compute all the derivations of the \(\mathbb{R}-\)algebra \(\mathcal{A}(E)\) and to obtain an explicit description of the Lie algebra of zero-weight derivations of \(\mathcal{A}(E).\)
ISSN:2331-8422