论文标题
在通用$ l_ \ infty $ - 线性叶的algebroid上
On the universal $L_\infty$-algebroid of linear foliations
论文作者
论文摘要
我们计算了$ l_ \ infty $ -Algebroid结构，该结构是在某些类别的矢量空间$ v $上进行的分辨率分辨率，这是由$ \ mathfrak {gl}（v）$的某些Lie lie subalgebra引起的。这种$ l_ \ infty $ -Algebroid提供了单一叶子的不变，并且还提供了恒定的均替换奇异叶子。我们通过首先明确构建$ v $的自然线性作用而引起的奇异叶子的投射决议来做到这一点$ω$，然后计算$ l_ \ infty $ -AlgeBroid结构。然后，我们将这些构造概括为矢量束$ e $，在零节$ l $中，原点的角色现在占据。 然后，我们证明，可以直接从叶片中计算出任何单数叶子的构图分辨率的奇异点上的纤维，而无需投影分辨率。对于线性叶子，我们还提供了一种方法来计算各种纤维上的各向同性的作用，而无需投影分辨率。
We compute an $L_\infty$-algebroid structure on a projective resolution of some classes of singular foliations on a vector space $V$ induced by the linear action of some Lie subalgebra of $\mathfrak {gl}(V)$. This $L_\infty$-algebroid provides invariants of the singular foliations, and also provides a constant-rank replacement of the singular foliation. We do this by first explicitly constructing projective resolutions of the singular foliations induced by the natural linear actions of endomorphisms of $V$ preserving a subspace $W\subset V$, the Lie algebra of traceless endomorphisms, and the symplectic Lie algebra of endomorphisms of $V$ preserving a non-degenerate skew-symmetric bilinear form $ω$, and then computing the $L_\infty$-algebroid structure. We then generalize these constructions to a vector bundle $E$, where the role of the origin is now taken by the zero section $L$. We then show that the fibers over a singular point of a projective resolution of any singular foliation can be computed directly from the foliation, without needing the projective resolution. For linear foliations, we also provide a way to compute the action of the isotropy Lie algebra in the origin on these fibers, without needing the projective resolution.
