论文标题
泊松几何学和更高谎言理论中固定点的稳定性
Stability of fixed points in Poisson geometry and higher Lie theory
论文作者
论文摘要
我们提供了一种统一的方法，以获得在歧管上给定的支架结构的（高阶）固定点的足够标准，以在变形下稳定。括号结构的例子包括谎言代数，谎言$ n $ - 代数，奇异叶子，谎言双子，双ggebroids，courant algebroids和dirac结构，以分裂的courant courant代数为代数。我们表明稳定性问题是以下问题的特定实例：给定差异分级为代数$ \ mathfrak g $，一个差异分级为subergebra $ \ mathfrak $ \ mathfrak h $ in $ \ m m athfrak g $中的有限代码，mathfrak g $和maurer-cartan元素$ g $量规等于$ \ mathfrak h^1 $的元素？我们表明，与$ \ mathfrak g，\ mathfrak h $和$ q $相关的有限维共同体学组消失了，这意味着对上述问题有积极的答案，因此暗示上述几何结构的固定点的稳定性。特别是，我们恢复了零维叶子的Crainic-Fernandes的稳定性结果，以及Dufour-Wade高阶奇点的稳定性结果。
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.
