On Complex Lie Algebroids with Constant Real Rank
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915340032671744 |
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| author | Aguero, Dan |
| author_facet | Aguero, Dan |
| contents | We associate a real distribution to any complex Lie algebroid that we call distribution of real elements and a new invariant that we call real rank, given by the pointwise rank of this distribution. When the real rank is constant, we obtain a real Lie algebroid inside the original complex Lie algebroid. Under another regularity condition, we associate a complex Lie subalgebroid that we call the minimal complex subalgebroid. We also provide a local splitting for complex Lie algebroids with constant real rank. In the last part, we introduce the complex matched pair of skew-algebroids; these pairs produce complex Lie algebroid structures on the complexification of a vector bundle. We use this operation to characterize all the complex Lie algebroid structures on the complexification of real vector bundles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05274 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Complex Lie Algebroids with Constant Real Rank Aguero, Dan Symplectic Geometry Differential Geometry We associate a real distribution to any complex Lie algebroid that we call distribution of real elements and a new invariant that we call real rank, given by the pointwise rank of this distribution. When the real rank is constant, we obtain a real Lie algebroid inside the original complex Lie algebroid. Under another regularity condition, we associate a complex Lie subalgebroid that we call the minimal complex subalgebroid. We also provide a local splitting for complex Lie algebroids with constant real rank. In the last part, we introduce the complex matched pair of skew-algebroids; these pairs produce complex Lie algebroid structures on the complexification of a vector bundle. We use this operation to characterize all the complex Lie algebroid structures on the complexification of real vector bundles. |
| title | On Complex Lie Algebroids with Constant Real Rank |
| topic | Symplectic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2401.05274 |