$A_\infty$-Algebras from Lie Pairs
Mathieu Stiénon, Luca Vitagliano, Ping Xu
TL;DR
The paper constructs an $A_∞$-algebra structure on the dg module $ig(igwedge^ullet A^ ablaig) ensor_R U_{L/A}$ for any Lie pair $(L,A)$, using a dg Lie algebroid $reve{oldsymbol{pi}}L o A[1]$ and a contraction to transfer the dg associative structure to an $A_∞$-algebra. The construction relies on PBW-type isomorphisms, a carefully chosen contraction data $(P_0,I_0,H_0)$, and the homotopy transfer theorem for $A_∞$-algebras, ensuring the transferred structure is independent (up to isomorphism) of the choices of splitting and additional geometric data. Consequently, the Chevalley–Eilenberg cohomology $H^ullet_{CE}(A,U_{L/A})$ acquires a canonical associative algebra structure, and in special cases (e.g., foliations, complex manifolds, or matched pairs) this framework recovers classical structures or yields natural dg-algebra models for transverse or holomorphic differential operators. The results bridge the theory of $L_∞$-algebroids, double Lie algebroids, and deformation-quantization-type constructions, offering a canonical universal enveloping picture for the $L_∞$-algebroid associated to a Lie pair.
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 $Γ(Λ^\bullet A^\vee)\otimes_{R} \frac{U(L)}{U(L)\cdotΓ(A)}$, 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.