$A_\infty$-Algebras from Lie Pairs

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 $A\hookrightarrow L$ of Lie algebroids sharing the same base manifold $M$, i.e. a Lie pair, we prove that the space $\Gamma(\Lambda^\bullet A^\vee)\otimes_{R} \frac{U(L)}{U(L)\cdot\Gamma(A)}$, where $R=C^\infty(M)$, admits an $A_\infty$-algebra structure, unique up to $A_\infty$-isomorphisms. As a consequence, the Chevalley-Eilenberg cohomology $H^\bullet_{CE} \big( A, \frac{U(L)}{U(L)\cdot\Gamma(A)} \big)$ 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.