:Deformation problem attracts a lot of interests in mathematical physics. In the this talk, we shall first consider the deformation problem of a Lie subalgebroid. Given a Lie subalgebroid $E$, an $L_\infty$-algebra associated with $E$ is construct, and under certain regularity assumptions, we prove that this $L_\infty$-algebra controls the deformations of $E$. We also obtain the result on simultaneous deformations of $E$ and the ambient Lie algebroid.

Moreover, similar results have been obtained for coisotropic submanifolds: every coisotropic submanifold $S$ of a Poisson manifold $(M,\pi)$ is attached with an $L_\infty$-algebra (A. S. Cattaneo & G. Felder), and when $\pi$ satisfies certain analyticity conditions this $L_\infty$-algebra controls the deformations of $S$ (F. Schatz and M. Zambon).

It is well-known that coisotropic submanifold and Lie subalgebroid are "equivalent" objects in geometry. From this viewpoint, we shall establish the "equivalence" of the two results on deformations described above, in the sense that we can recover the $L_\infty$-algebra and the regularity condition for one object from the other one.