On a Poisson-algebraic characterization of vector bundles
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866910362798915584 |
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| author | Mushengezi, Elie Zihindula |
| author_facet | Mushengezi, Elie Zihindula |
| contents | We prove that the $\mathbb{R}-$algebra $\mathcal{S}(\mathcal{P}(E,M)) $ of symbols of differential operators acting on the sections of the vector bundle $E\to M$ decompose into the sum \[ \mathcal{S}(\mathcal{P}(E,M))=\mathcal{J}(E)\oplus {\rm Pol}(T^*M) \] where $\mathcal{J}(E)$ is an ideal of $\mathcal{S}(\mathcal{P}(E,M))$ in which product of two elements is always zero. This induces that $\mathcal{S}(\mathcal{P}(E,M))$ cannot characterize $E \to M$ with its only structure of $\mathbb{R}-$ algebra. We prove that with its Poisson algebra structure, $\mathcal{S}(\mathcal{P}(E,M))$ characterizes the vector bundle $E\to M$ without the requirement to be considered as a ${\rm C}^\infty(M)-$module. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2008_13587 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On a Poisson-algebraic characterization of vector bundles Mushengezi, Elie Zihindula Differential Geometry We prove that the $\mathbb{R}-$algebra $\mathcal{S}(\mathcal{P}(E,M)) $ of symbols of differential operators acting on the sections of the vector bundle $E\to M$ decompose into the sum \[ \mathcal{S}(\mathcal{P}(E,M))=\mathcal{J}(E)\oplus {\rm Pol}(T^*M) \] where $\mathcal{J}(E)$ is an ideal of $\mathcal{S}(\mathcal{P}(E,M))$ in which product of two elements is always zero. This induces that $\mathcal{S}(\mathcal{P}(E,M))$ cannot characterize $E \to M$ with its only structure of $\mathbb{R}-$ algebra. We prove that with its Poisson algebra structure, $\mathcal{S}(\mathcal{P}(E,M))$ characterizes the vector bundle $E\to M$ without the requirement to be considered as a ${\rm C}^\infty(M)-$module. |
| title | On a Poisson-algebraic characterization of vector bundles |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2008.13587 |