$A_{\infty }$-Algebras from Lie Pairs

Abstract

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 $\hat{I}“(\hat{I}{›}^{\bullet }{A}^{\vee }){\otimes }_R\frac{U(L)}{U(L)\cdot \hat{I}“(A)}$, where $R=C^{\infty }(M)$, admits an $A_{\infty }$-algebra structure, unique up to $A_{\infty }$-isomorphisms. As a consequence, the Chevalley-Eilenberg cohomology $H_{CE}^{\bullet }(A,\frac{U(L)}{U(L)\cdot \hat{I}“(A)})$ 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.

51 pages. v2: significantly revised, material added. Comments are welcome!