Simultaneous deformations of a Lie algebroid and its Lie subalgebroid

Deformation problem is an interesting problem in mathematical physics. In this paper, we show that the deformations of a Lie algebroid are governed by a differential graded Lie algebra; and under certain regularity assumptions, an L∞-algebra can be constructed to govern the deformations of its Lie subalgebroid. Furthermore, by applying Y. Frégier and M. Zambon's result (0000, Thm. 3), these structures can be combined together to govern the simultaneous deformations. Applications of our results include deformations of a foliation, deformations of a Lie subalgebra, deformations of a complex structure, and deformations of a homomorphism of Lie algebroids.