Atiyah classes of Lie algebroid homotopy modules
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911910551617536 |
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| author | Batakidis, Panagiotis Lavau, Sylvain |
| author_facet | Batakidis, Panagiotis Lavau, Sylvain |
| contents | For a Lie algebroid pair $A\hookrightarrow L$ we study cocycles constructed from the extension to $L$ of the higher connection forms of a representation up to homotopy $E$ of the Lie algebroid $A$. We show that there exists a cohomology class with values in the endomorphism bundle of $E$ that is independent of the extension above and vanishes whenever a homotopy $A$-compatible extension exists. Whenever the representation up to homotopy $E$ is the resolution of a Lie algebroid representation $K$ of $A$, it is shown that there exists a quasi-isomorphism sending the new Atiyah class to the classical one, associated to extensions to $L$ of the Lie algebroid representation $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05204 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Atiyah classes of Lie algebroid homotopy modules Batakidis, Panagiotis Lavau, Sylvain Differential Geometry Mathematical Physics Algebraic Geometry For a Lie algebroid pair $A\hookrightarrow L$ we study cocycles constructed from the extension to $L$ of the higher connection forms of a representation up to homotopy $E$ of the Lie algebroid $A$. We show that there exists a cohomology class with values in the endomorphism bundle of $E$ that is independent of the extension above and vanishes whenever a homotopy $A$-compatible extension exists. Whenever the representation up to homotopy $E$ is the resolution of a Lie algebroid representation $K$ of $A$, it is shown that there exists a quasi-isomorphism sending the new Atiyah class to the classical one, associated to extensions to $L$ of the Lie algebroid representation $K$. |
| title | Atiyah classes of Lie algebroid homotopy modules |
| topic | Differential Geometry Mathematical Physics Algebraic Geometry |
| url | https://arxiv.org/abs/2406.05204 |