Stability of fixed points in Poisson geometry and higher Lie theory
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916656207364096 |
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| author | Singh, Karandeep J. |
| author_facet | Singh, Karandeep J. |
| contents | We provide a uniform approach to obtain sufficient criteria for a (higher order) fixed point of a given bracket structure on a manifold to be stable under deformations. Examples of bracket structures include Lie algebroids, Lie $n$-algebroids, singular foliations, Lie bialgebroids, Courant algebroids and Dirac structures in split Courant algebroids admitting a Dirac complement. We show that the stability problems are specific instances of the following problem: given a differential graded Lie algebra $\mathfrak g$, a differential graded Lie subalgebra $\mathfrak h$ of degreewise finite codimension in $\mathfrak g$ and a Maurer-Cartan element $Q\in \mathfrak h^1$, when are Maurer-Cartan elements near $Q$ in $\mathfrak g$ gauge equivalent to elements of $\mathfrak h^1$? We show that the vanishing of a finite-dimensional cohomology group associated to $\mathfrak g,\mathfrak h$ and $Q$ implies a positive answer to the question above, and therefore implies stability of fixed points of the geometric structures described above. In particular, we recover the stability results of Crainic-Fernandes for zero-dimensional leaves, as well as the stability results for higher order singularities of Dufour-Wade. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_16256 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Stability of fixed points in Poisson geometry and higher Lie theory Singh, Karandeep J. Differential Geometry Representation Theory Symplectic Geometry Primary: 17B70, 58H15. Secondary: 53D17, 58J10 We provide a uniform approach to obtain sufficient criteria for a (higher order) fixed point of a given bracket structure on a manifold to be stable under deformations. Examples of bracket structures include Lie algebroids, Lie $n$-algebroids, singular foliations, Lie bialgebroids, Courant algebroids and Dirac structures in split Courant algebroids admitting a Dirac complement. We show that the stability problems are specific instances of the following problem: given a differential graded Lie algebra $\mathfrak g$, a differential graded Lie subalgebra $\mathfrak h$ of degreewise finite codimension in $\mathfrak g$ and a Maurer-Cartan element $Q\in \mathfrak h^1$, when are Maurer-Cartan elements near $Q$ in $\mathfrak g$ gauge equivalent to elements of $\mathfrak h^1$? We show that the vanishing of a finite-dimensional cohomology group associated to $\mathfrak g,\mathfrak h$ and $Q$ implies a positive answer to the question above, and therefore implies stability of fixed points of the geometric structures described above. In particular, we recover the stability results of Crainic-Fernandes for zero-dimensional leaves, as well as the stability results for higher order singularities of Dufour-Wade. |
| title | Stability of fixed points in Poisson geometry and higher Lie theory |
| topic | Differential Geometry Representation Theory Symplectic Geometry Primary: 17B70, 58H15. Secondary: 53D17, 58J10 |
| url | https://arxiv.org/abs/2210.16256 |