Classical Poisson algebra of a vector bundle : Lie-algebraic characterization
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866913261232848896 |
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| author | Lecomte, P. B. A Mushengezi, Elie Zihindula |
| author_facet | Lecomte, P. B. A Mushengezi, Elie Zihindula |
| contents | We prove that the Lie algebra $\mathcal{S}(\mathcal{P}(E,M))$ of symbols of linear operators acting on smooth sections of a vector bundle $E\to M,$ characterizes it. To obtain this, we assume that $\mathcal{S}(\mathcal{P}(E,M))$ is seen as ${\rm C}^\infty(M)-$module and that the vector bundle is of rank $n>1.$ We improve this result for the Lie algebra $\mathcal{S}^1(\mathcal{P}(E,M))$ of symbols of first-order linear operators. We obtain a Lie algebraic characterization of vector bundles with $\mathcal{S}^1(\mathcal{P}(E,M))$ without the hypothesis of being seen as a ${\rm C}^\infty(M)-$module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_13495 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Classical Poisson algebra of a vector bundle : Lie-algebraic characterization Lecomte, P. B. A Mushengezi, Elie Zihindula Differential Geometry We prove that the Lie algebra $\mathcal{S}(\mathcal{P}(E,M))$ of symbols of linear operators acting on smooth sections of a vector bundle $E\to M,$ characterizes it. To obtain this, we assume that $\mathcal{S}(\mathcal{P}(E,M))$ is seen as ${\rm C}^\infty(M)-$module and that the vector bundle is of rank $n>1.$ We improve this result for the Lie algebra $\mathcal{S}^1(\mathcal{P}(E,M))$ of symbols of first-order linear operators. We obtain a Lie algebraic characterization of vector bundles with $\mathcal{S}^1(\mathcal{P}(E,M))$ without the hypothesis of being seen as a ${\rm C}^\infty(M)-$module. |
| title | Classical Poisson algebra of a vector bundle : Lie-algebraic characterization |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2008.13495 |