On the integration of Manin pairs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910926927560704 |
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| author | Li-Bland, David Meinrenken, Eckhard |
| author_facet | Li-Bland, David Meinrenken, Eckhard |
| contents | It is a remarkable fact that the integrability of a Poisson manifold to a symplectic groupoid depends only on the integrability of its cotangent Lie algebroid $A$: The source-simply connected Lie groupoid $G\rightrightarrows M$ integrating $A$ automatically acquires a multiplicative symplectic 2-form. More generally, a similar result holds for the integration of Lie bialgebroids to Poisson groupoids, as well as in the `quasi' settings of Dirac structures and quasi-Lie bialgebroids. In this article, we will place these results into a general context of Manin pairs $(\mathbb{E},A)$, thereby obtaining a simple geometric approach to these integration results. We also clarify the case where the groupoid $G$ integrating $A$ is not source-simply connected. Furthermore, we obtain a description of Hamiltonian spaces for Poisson groupoids and quasi-symplectic groupoids within this formalism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17988 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the integration of Manin pairs Li-Bland, David Meinrenken, Eckhard Differential Geometry It is a remarkable fact that the integrability of a Poisson manifold to a symplectic groupoid depends only on the integrability of its cotangent Lie algebroid $A$: The source-simply connected Lie groupoid $G\rightrightarrows M$ integrating $A$ automatically acquires a multiplicative symplectic 2-form. More generally, a similar result holds for the integration of Lie bialgebroids to Poisson groupoids, as well as in the `quasi' settings of Dirac structures and quasi-Lie bialgebroids. In this article, we will place these results into a general context of Manin pairs $(\mathbb{E},A)$, thereby obtaining a simple geometric approach to these integration results. We also clarify the case where the groupoid $G$ integrating $A$ is not source-simply connected. Furthermore, we obtain a description of Hamiltonian spaces for Poisson groupoids and quasi-symplectic groupoids within this formalism. |
| title | On the integration of Manin pairs |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2411.17988 |