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

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 $\mathrm{\Gamma }({\mathrm{\Lambda }}^{\bullet }{A}^{\vee }){\otimes }_R\frac{U(L)}{U(L)\cdot \mathrm{\Gamma }(A)}$, where $R=C^{\mathrm{\infty }}(M)$, admits an $A_{\mathrm{\infty }}$-algebra structure, unique up to $A_{\mathrm{\infty }}$-isomorphisms. As a consequence, the Chevalley-Eilenberg cohomology $H_{CE}^{\bullet }(A,\frac{U(L)}{U(L)\cdot \mathrm{\Gamma }(A)})$ admits a canonical associative algebra structure. This $A_{\mathrm{\infty }}$-algebra can be considered as the universal enveloping algebra of the $L_{\mathrm{\infty }}$-algebroid $A[1]{\times }_{M}L\mathrm{/}A$. Our construction is based on the homotopy equivalence of the $L_{\mathrm{\infty }}$-algebroid $A[1]{\times }_{M}L\mathrm{/}A$ and the dg Lie algebroid corresponding to the comma double Lie algebroid of Jotz-Mackenzie.