Infinitesimal deformations of Lie algebroid pairs

We study infinitesimal deformations of Lie algebroid pairs in the category of smooth manifolds enriched with a local Artinian K-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∞-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.