Infinitesimal deformations of Lie algebroid pairs

TL;DR

This paper develops a comprehensive deformation theory for Lie algebroid pairs, identifying governing $L_ abla$-algebras and connecting to complex and foliation deformations, advancing understanding of their infinitesimal deformations.

Contribution

It introduces a new $L_ abla$-algebra framework for infinitesimal deformations of Lie algebroid pairs, linking to extended deformation theory and classical geometric structures.

Findings

Identifies governing $L_ abla$-algebras for deformation functors.

Connects deformation theory of Lie algebroid pairs to complex structures.

Recovers deformation theories of complex structures and foliations.

Abstract

We study infinitesimal deformations of Lie algebroid pairs in the category of smooth manifolds enriched with a local Artinian algebra. Given a Lie algebroid pair $(L,A)$, i.e. a Lie algebroid $L$ together with a Lie subalgebroid $A$, we investigate isomorphism classes of infinitesimal deformations of $(L,A)$ modulo automorphisms from exponentials of derivations of $L$ and those from the exponentials of inner derivations of $L$, respectively. For the associated two deformation functors, we find the associated governing $L_\infty$-algebras in the sense of extended deformation theory. Furthermore, when $(L,A)$ is a matched Lie pair, i.e. the quotient $L/A$ is also a Lie subalgebroid of $L$, we investigate isomorphism classes of infinitesimal deformations modulo automorphisms from exponentials of derivations along the normal direction $L/A$. The extended deformation theory of the associated deformation functor recovers the formal deformation theory of complex structures and that of transversely holomorphic foliations.