Infinitesimal ideal systems and the Atiyah class

Atiyah代数胚的Lie结构中编码的几何结构

Geometric structures encoded in the Lie structure of an Atiyah algebroid

本文从李代数空间方法的角度研究了Atiyah代数胚，即主丛的无穷小对象.首先证明了如果两个Atiyah代数胚的光滑截面上的李代数同构，则相应的基流形必然同构。进一步，我们给出了与半单结构群的主丛相关联的Atiyah代数胚的截面李代数的同构的两个刻画。例如，我们证明了在半单的情况下，李代数的部分是同构的当且仅当相应的李代数胚，或者，以及，当且仅当集成主丛是局部同构的。最后，我们应用这些结果来描述的同构的情况下，约简结构群-令人惊讶的是，他们不再是由向量丛同构，并涉及分歧的基础流形。

Atiyah classes of strongly homotopy Lie pairs

The subject of this paper is strongly homotopy (SH) Lie algebras, also known as $L_\infty$-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an (SH) Lie algebra $A$ when it is extended to $L$. In fact, given such an SH Lie pair $(L, A)$, and any $A$-module $E$, there associates a canonical cohomology class, the Atiyah class $[α^E]$, which generalizes earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class $[α^{L/A}]$ induces a graded Lie algebra structure on $\operatorname{H}^\bullet_{\mathrm{CE}}(A,L/A[-2])$, and th…

Obstructions to representations up to homotopy and ideals

This paper considers the Pontryagin characters of graded vector bundles of finite rank, in the cohomology vector spaces of a Lie algebroid over the same base. These Pontryagin characters vanish if the graded vector bundle carries a representation up to homotopy of the Lie algebroid. As a consequence, this gives a strong obstruction to the existence of a representation up to homotopy on a graded vector bundle of finite rank. In particular, if a graded vector bundle $E[0]\oplus F[1]\to M$ carries a $2$-term representation up to homotopy of a Lie algebroid $A\to M$, then all the (classical) $A$-P…

关于与 spray 相关的李代数

On Lie algebras associated with a spray

本文研究黎曼流形上无穷小等距李代数的结构，证明其至多含两个交换理想，分别源于典范联络的 Nijenhuis 张量水平零化空间及与度量无关的常向量场。