$A_\infty$-Algebras from Lie Pairs
Mathieu Sti\'enon, Luca Vitagliano, Ping Xu

TL;DR
This paper constructs an $A_$-algebra structure on a space derived from a Lie pair, revealing a universal enveloping algebra for a related $L_$-algebroid, with implications for cohomology and algebraic structures.
Contribution
It introduces a canonical $A_$-algebra structure associated with Lie pairs, linking homotopy equivalences and dg Lie algebroids to universal enveloping algebras.
Findings
The space admits a unique $A_$-algebra structure.
The Chevalley-Eilenberg cohomology gains a canonical associative algebra structure.
The $A_$-algebra acts as the universal enveloping algebra of an $L_$-algebroid.
Abstract
Given an inclusion of Lie algebroids sharing the same base manifold , i.e. a Lie pair, we prove that the space , where , admits an -algebra structure, unique up to -isomorphisms. As a consequence, the Chevalley-Eilenberg cohomology admits a canonical associative algebra structure. This -algebra can be considered as the universal enveloping algebra of the -algebroid . Our construction is based on the homotopy equivalence of the -algebroid and the dg Lie algebroid corresponding to the comma double Lie algebroid of Jotz-Mackenzie.
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