论文标题
Atiyah课程和TODD类别的回调DG谎言代数与Lie对相关
Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs
论文作者
论文摘要
对于Lie代数$ L $和A Lie subalgebroid $ a $，即躺椅$（l，a）$，我们研究atiyah class和todd class and todd class the the the the the plackback dg（即差异分级）lie elgebroid $π^！ L $ l $沿束投影$π的$ $ l $：a [1] \至M $的移动矢量束$ a [1] $。应用同源扰动引理，我们提供了stiénon-vitagliano-xu的新结构，与Cochain Complex $ \ big（γ（π^！l），\ Mathit {q} \ big）$ $π^^！ l $ to chevalley-eilenberg复合物$（γ（λ^\ bullet a^\ vee \ otimes（l/a）），d^{\ mathrm {bott}}）$的d^{\ mathrm {bott}}）$。使用这种收缩，我们构造了两个同构：第一个标识了Cochain复合物$（γ（（（π^！l）^\ vee \ otimes \ Mathrm {end}（π^！l）（π^！l））），\ Mathit {q}）$与Chevalley-- Eilenerenberg Cohomology一起$ h^\ bullet _ {\ mathrm {ce}}}（a，（l/a）^\ vee \ otimes \ mathrm {end}（l/a））$是由bott代表产生的$ h^\ bullet _ {\ mathrm {ce}}}（a，λ（l/a）^\ vee）$。我们证明，这对同构鉴定了Atiyah类和DG的Todd类，属于代数$π^！ l $分别与Atiyah类和Lie Pair $（L，A）$的Todd类。
For a Lie algebroid $L$ and a Lie subalgebroid $A$, i.e. a Lie pair $(L,A)$, we study the Atiyah class and the Todd class of the pullback dg (i.e. differential graded) Lie algebroid $π^! L$ of $L$ along the bundle projection $π:A[1] \to M$ of the shifted vector bundle $A[1]$. Applying the homological perturbation lemma, we provide a new construction of Stiénon--Vitagliano--Xu's contraction relating the cochain complex $\big(Γ(π^! L),\mathit{Q}\big)$ of sections of $π^! L$ to the Chevalley--Eilenberg complex $(Γ(Λ^\bullet A^\vee\otimes(L/A)),d^{\mathrm{Bott}})$ of the Bott representation. Using this contraction, we construct two isomorphisms: the first identifies the cohomology of the cochain complex $(Γ((π^! L)^\vee\otimes\mathrm{End}(π^! L)),\mathit{Q})$ with the Chevalley--Eilenberg cohomology $H^\bullet_{\mathrm{CE}}(A,(L/A)^\vee\otimes\mathrm{End}(L/A))$ arising from the Bott representation, while the second identifies the cohomologies $H^\bullet(Γ(Λ(π^! L)^\vee),\mathit{Q})$ and $H^\bullet_{\mathrm{CE}}(A,Λ(L/A)^\vee)$. We prove that this pair of isomorphisms identifies the Atiyah class and the Todd class of the dg Lie algebroid $π^! L$ with the Atiyah class and the Todd class of the Lie pair $(L,A)$, respectively.
