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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2007.14649 |
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Table of Contents:
- We prove a Lie-algebraic characterization of vector bundle for the Lie algebra $\mathcal{D}(E,M),$ seen as ${\rm C}^\infty(M)-$module, of all linear operators acting on sections of a vector bundle $E\to M$. We obtain similar result for its Lie subalgebra $\mathcal{D}^1(E,M)$ of all linear first-order differential operators. Thanks to a well-chosen filtration, $\mathcal{D}(E,M)$ becomes $\mathcal{P}(E,M)$ and we prove that $\mathcal{P}^1(E,M)$ characterizes the vector bundle without the hypothesis of being seen as ${\rm C}^\infty(M)-$module. 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. We obtain a similar result with the Lie algebra $\mathcal{S}^1(\mathcal{P}(E,M))$ of symbols of first-order linear operators without the hypothesis of being seen as a ${\rm C}^\infty(M)-$module.