There is supposed to be a philosophy that, at least over a field of characteristic zero, every "deformation problem" is somehow "governed" or "controlled" by a differential graded Lie algebra. See for example http://arxiv.org/abs/math/0507284
I've seen this idea attributed to big names like Quillen, Drinfeld, and Deligne -- so it must be true, right? ;-)
An example of this philosophy is the deformation theory of a compact complex manifold: It is "controlled" by the Kodaira-Spencer dg Lie algebra: holomorphic vector fields tensor Dolbeault complex, with differential induced by del-bar on the Dolbeault complex, and Lie bracket induced by Lie bracket on the vector fields (I think also take wedge product on the Dolbeault side).
I seem to recall that there is a general theorem which justifies this philosophy, but I don't remember the details, or where I heard about it. The statement of the theorem should be something like:
Let k be a field of characteristic zero. Given a functor F: (Local Artin k-algebras) -> (Sets) satisfying some natural conditions that a "deformation functor" should satisfy, then there exists a dg Lie algebra L such that F is isomorphic to the deformation functor of L, which is the functor that takes an algebra A and returns the set of Maurer-Cartan solutions (dx + [x,x] = 0) in (L^1 tensor mA) modulo the gauge action of (L^0 tensor mA), where mA denotes the maximal ideal of A.
Furthermore, I think such an L should be unique up to quasi-isomorphism.
Does anyone know a reference for something along these lines?
Any other nice examples of cases where this philosophy holds would also be appreciated.