Deformation theory and differential graded Lie algebras

Question

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 m_A) modulo the gauge action of (L^0 tensor m_A), where m_A 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.

Answer 1

I hope to write more on this later, but for now let me make some general assertions: there are general theorems to this effect and give two references: arXiv:math/9812034, DG coalgebras as formal stacks, by Vladimir Hinich, and the survey article arXiv:math/0604504, Higher and derived stacks: a global overview, by Bertrand Toen (look at the very end to where Hinich's theorem and its generalizations are discussed).

The basic assertion if you'd like is the Koszul duality of the commutative and Lie operads in characteristic zero. In its simplest form it's a version of Lie's theorem: to any Lie algebra we can assign a formal group, and to every formal group we can assign a Lie algebra, and this gives an equivalence of categories. The general construction is the same: we replace Lie algebras by their homotopical analog, Loo algebras or dg Lie algebras (the two notions are equivalent --- both Lie algebras in a stable oo,1 category). We can associate to such an object the space of solutions of the Maurer-Cartan equations -- this is basically the classifying space of its formal group (ie formal group shifted by 1). Conversely from any formal derived stack we can calculate its shifted tangent complex (or perhaps better to say, the Lie algebra of its loop space). These are equivalences of oo-categories if you set everything up correctly. This is a form of Quillen's rational homotopy theory - we're passing from a simply connected space to the Lie algebra of its loop space (the Whitehead algebra of homotopy groups of X with a shift) and back.

So basically this "philosophy", with a modern understanding is just calculus or Lie theory: you can differentiate and exponentiate, and they are equivalences between commutative and Lie theories (note we're saying this geometrically, which means replacing commutative algebras by their opposite, ie appropriate spaces -- in this case formal stacks). Since any deformation/formal moduli problem, properly formulated, gives rise to a formal derived stack, it is gotten (again in characteristic zero) by exponentiating a Lie algebra.

Sorry to be so sketchy, might try to expand later, but look in Toen's article for more (though I think it's formulated there as an open question, and I think it's not so open anymore). Once you see things this way you can generalize them also in various ways -- for example, replacing commutative geometry by noncommutative geometry, you replace Lie algebras by associative algebras (see arXiv:math/0605095 by Lunts and Orlov for this philosophy) or pass to geometry over any operad with an augmentation and its dual...

Answer 2

Maybe http://arxiv.org/abs/0707.0889 could be of any help? It's general enough - representations of properads cover a huge variety of cases, from algebraic structures to formal differential geometric things.

Answer 3

I can offer an algebraic example generalising that of Hochschild cohomology. Let O → P be a morphism of operads, assume that O has O(1)=k (although augmented should be strong enough as well I think). Then we can form a cofibrant resolution, O' of O, this has the underlying structure of the free operad on a set of generators C, and this C has a cooperad structure.

We want to deform f. Well Hom(O',P) < Hom(FC,P) = Hom_S(C,P), where the first two hom's are in the category of operads and the final is in the category of collections. Recall that collections underlie operads, they can be given as functors from the category of finite sets and bijections into vector spaces.

But C is a cooperad and P is an operad, and this homset looks a lot like linear maps between them, so shouldn't we have a convolution operad structure. Well we do, but it's only a non-symmetric operad.

Non-symmetric operads have the natural structure of a pre-Lie algebra, the composition is defined by taking the sum of all possible ways of plugging one operation into another. If you haven't met pre-Lie algebras then don't worry as the anti-symmetrisation of the pre-Lie product is a Lie bracket. So our non-symmetric operad Hom(FC,P) naturally has the structure of a dg-Lie algebra.

The nature of the inclusion of Hom(O',P) into Hom(FC,P) should come as no surprise. They're precisely the Maurer-Cartan elements. So our morphism f corresponds to a MC element in a dg-Lie algebra.

Given a MC element in a graded-Lie algebra the deformations are MC elements in the dg-Lie algebra twisted by the original MC element.

Examples

1. Let P be an endomorphism operad ⊕Hom(A⊗...⊗A,A), then the theory above is the deformation theory for O algebras.
2. Let O be the associative operad, let P be the operad for associative algebras with an action of a comm. alg R by central elements and a Lie algebra g by derivations with some compatability conditions (in fact by a Lie-Rinehart algebra, or a Lie-algebroid). Then the deformation theory of the inclusion morphism is the study of deformations induced by Lie-algebroid actions. In fact the dg-Lie algebras involved are very often formal.

Answer 4

Perhaps notes of Kontsevich's lectures are helpfull.

Answer 5

I'm hesitant to put this up, since I haven't actually looked at the reference I'm about to suggest, but what about Illusie's Complexe Cotangents et Deformations I & II ? I've been under the impression that if I want to learn deformation theory, I should look there, though I have no clue if it contains the precise theorem you are looking for.

Answer 6

Some precise statements with proofs can be found in this paper of Manetti: http://arxiv.org/abs/math/9910071