On symmetries of singular foliations
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866910164074889216 |
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| author | Louis, Ruben |
| author_facet | Louis, Ruben |
| contents | This paper shows that a weak symmetry action of a Lie algebra $\mathfrak{g}$ on a singular foliation $\mathcal F$ induces a unique up to homotopy Lie$\infty$-morphism from $\mathfrak{g}$ to the DGLA of vector fields on a universal Lie $\infty$-algebroid of $\mathcal F$. Such a Lie $\infty$-morphismwas studied by R. Mehta and M. Zambon as $L_\infty$-algebra action. We deduce from this general result several geometrical consequences. For instance, we give an example of a Lie algebra action on an affine sub-variety which cannot be extended on the ambient space. Last, we introduce the notion of bi-submersion towers over a singular foliation and lift symmetries to those. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_01585 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On symmetries of singular foliations Louis, Ruben Differential Geometry This paper shows that a weak symmetry action of a Lie algebra $\mathfrak{g}$ on a singular foliation $\mathcal F$ induces a unique up to homotopy Lie$\infty$-morphism from $\mathfrak{g}$ to the DGLA of vector fields on a universal Lie $\infty$-algebroid of $\mathcal F$. Such a Lie $\infty$-morphismwas studied by R. Mehta and M. Zambon as $L_\infty$-algebra action. We deduce from this general result several geometrical consequences. For instance, we give an example of a Lie algebra action on an affine sub-variety which cannot be extended on the ambient space. Last, we introduce the notion of bi-submersion towers over a singular foliation and lift symmetries to those. |
| title | On symmetries of singular foliations |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2203.01585 |