Infinitesimal deformations of Lie algebroid pairs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914178842755072 |
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| author | Ni, Dadi Chen, Zhuo Hu, Chuangqiang Xiang, Maosong |
| author_facet | Ni, Dadi Chen, Zhuo Hu, Chuangqiang Xiang, Maosong |
| contents | We study infinitesimal deformations of Lie algebroid pairs in the category of smooth manifolds enriched with a local Artinian algebra. Given a Lie algebroid pair $(L,A)$, i.e. a Lie algebroid $L$ together with a Lie subalgebroid $A$, we investigate isomorphism classes of infinitesimal deformations of $(L,A)$ modulo automorphisms from exponentials of derivations of $L$ and those from the exponentials of inner derivations of $L$, respectively. For the associated two deformation functors, we find the associated governing $L_\infty$-algebras in the sense of extended deformation theory. Furthermore, when $(L,A)$ is a matched Lie pair, i.e. the quotient $L/A$ is also a Lie subalgebroid of $L$, we investigate isomorphism classes of infinitesimal deformations modulo automorphisms from exponentials of derivations along the normal direction $L/A$. The extended deformation theory of the associated deformation functor recovers the formal deformation theory of complex structures and that of transversely holomorphic foliations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03860 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinitesimal deformations of Lie algebroid pairs Ni, Dadi Chen, Zhuo Hu, Chuangqiang Xiang, Maosong Differential Geometry Mathematical Physics Quantum Algebra 13D10, 17B70, 16E45, 53C05, 53C12 We study infinitesimal deformations of Lie algebroid pairs in the category of smooth manifolds enriched with a local Artinian algebra. Given a Lie algebroid pair $(L,A)$, i.e. a Lie algebroid $L$ together with a Lie subalgebroid $A$, we investigate isomorphism classes of infinitesimal deformations of $(L,A)$ modulo automorphisms from exponentials of derivations of $L$ and those from the exponentials of inner derivations of $L$, respectively. For the associated two deformation functors, we find the associated governing $L_\infty$-algebras in the sense of extended deformation theory. Furthermore, when $(L,A)$ is a matched Lie pair, i.e. the quotient $L/A$ is also a Lie subalgebroid of $L$, we investigate isomorphism classes of infinitesimal deformations modulo automorphisms from exponentials of derivations along the normal direction $L/A$. The extended deformation theory of the associated deformation functor recovers the formal deformation theory of complex structures and that of transversely holomorphic foliations. |
| title | Infinitesimal deformations of Lie algebroid pairs |
| topic | Differential Geometry Mathematical Physics Quantum Algebra 13D10, 17B70, 16E45, 53C05, 53C12 |
| url | https://arxiv.org/abs/2512.03860 |