Infinitesimal deformations of Lie algebroid pairs

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.