Exponential map and $L_\infty$ algebra associated to a Lie pair

TL;DR

This paper explores complex algebraic structures arising from Lie pairs of algebroids, revealing a canonical homotopy module structure on the quotient, enriching the understanding of Atiyah classes in this context.

Contribution

It introduces the Kapranov module structure on the quotient of a Lie pair, providing a new homotopy-rich algebraic framework linked to Atiyah classes.

Findings

Established a canonical homotopy module structure on L/A

Connected Atiyah classes to homotopy algebraic structures

Enhanced the algebraic understanding of Lie pairs and Atiyah classes

Abstract

In this note, we unveil homotopy-rich algebraic structures generated by the Atiyah classes relative to a Lie pair $(L,A)$ of algebroids. In particular, we prove that the quotient $L/A$ of such a pair admits an essentially canonical homotopy module structure over the Lie algebroid $A$, which we call Kapranov module.