论文标题
$ a_ \ infty $ - 代理来自谎言对
$A_\infty$-Algebras from Lie Pairs
论文作者
论文摘要
给定一个包含$ a \ hookrightArrow l lie代数共享相同的基本歧管$ m $，即一对谎言，我们证明了空间$γ（λ^\ bulet a^\ vee）\ otimes_ {r} $ r = c^\ infty（m）$，承认$ a_ \ infty $ -Algebra结构，唯一最多可达$ a_ \ infty $ iSomorphisms。结果，Chevalley-eilenberg共同体$ h^\ bullet_ {ce} \ big（a，\ frac {u（l）} {u（l）\cdotγ（a）} \ big）$录取了一种规范的algebra结构。这个$ a_ \ infty $ -Algebra可以被视为$ l_ \ infty $ -Algebroid $ a [1] \ times_m l/a $的通用包络代数。我们的构造基于$ l_ \ infty $ -Algebroid $ a [1] \ times_m l/a $的同型等效性，而DG则与Jotz-Mackenzie的Comma Double Algebroid相对应。
Given an inclusion $A\hookrightarrow L$ of Lie algebroids sharing the same base manifold $M$, i.e. a Lie pair, we prove that the space $Γ(Λ^\bullet A^\vee)\otimes_{R} \frac{U(L)}{U(L)\cdotΓ(A)}$, where $R=C^\infty(M)$, admits an $A_\infty$-algebra structure, unique up to $A_\infty$-isomorphisms. As a consequence, the Chevalley-Eilenberg cohomology $H^\bullet_{CE} \big( A, \frac{U(L)}{U(L)\cdotΓ(A)} \big)$ admits a canonical associative algebra structure. This $A_\infty$-algebra can be considered as the universal enveloping algebra of the $L_\infty$-algebroid $A[1]\times_M L/A$. Our construction is based on the homotopy equivalence of the $L_\infty$-algebroid $A[1]\times_M L/A$ and the dg Lie algebroid corresponding to the comma double Lie algebroid of Jotz-Mackenzie.
