Deformation theory and Lie algebras

2 Lie algebras and FMPs

Let \(\FMP \) be the category of formal moduli problems, and let \(\Lie \) be the category of Lie algebras over \(k\) (i.e dglas with quasi-isomorphisms inverted). The main result about formal moduli problems is is the following:

Theorem 2.1 (Lurie, Pridham). There is an equivalence \(\FMP \simeq \Lie \).

As currently stated, the equivalence is not explicit, but it can be made explicit in multiple ways, as will be explained. Roughly, we’d like to think of every formal moduli problem as being represented by a pro-Artinian spectral stack, and the above result says that every such stack can be recovered from the tangent complex of its loop space, which is a Lie algebra.

More precisely, we can first observe as in the previous section, that given a formal moduli problem \(F\), \(F(\Lambda [\ee _n])\) fit together to form the tangent complex \(T_F\). In fact, \(\Sigma ^{-1}T_F\) can be given the structure of a Lie algebra, and is the one corresponding to \(F\). We might expect this, since the construction \(F \mapsto T_F\) commutes with limits, so \(\Sigma ^{-1}T_F = T_{\Omega F}\), and \(\Omega F\) is a group in the category \(\FMP \), so it’s tangent space \(\Sigma ^{-1}T_F\) ought to be a Lie algebra.

Furthermore, it isn’t hard to see that if \(F \to F'\) is a map of formal moduli problems inducing an equivalence on tangent complexes, then it is an equivalence. This is because every object in \(\CAlg _{\art }\) is built out of \(k\) from finitely many square zero extensions, which are classified by a pullback square for which we can apply \((F2)\). This way we can inductively show that \(F \to F'\) is an equivalence for every \(R \in \CAlg _{\art }\). Thus it makes sense that \(T_F\), along with all the extra structure it carries, retains all the information of \(F\).

One way to construct the functor \(\Lie \to \FMP \), which we suggestively call \(\mg \mapsto B\mg \), is to use Koszul duality.

We have a functor

\[C^*(-):\Lie ^{op} \to \CAlg ^{\aug }\]

sending \(\mg \) to its Chevalley-Eilenberg cochains, which is canonically augmented because of the map \(\mg \to 0\). The inclusion \(\CAlg _{\art } \to \CAlg \) canonically lifts to \(\CAlg _{\art }\), since any augmentation of \(R \in \CAlg _{\art }\) must factor through the truncation \(\pi _0R\) which is Artinian local with residue field \(k\), so is uniquely augmented over \(k\).

\(C^*(-)\) has a left adjoint which we denote \(D:\CAlg ^{\aug } \to \Lie ^{op}\). This adjunction is fairly close to an equivalence: \(\Lie \) is equivalent via the bar construction (i.e Chevalley-Eilenberg chains) to coaugmented cocommutative coalgebras over \(k\), and \(D\) is the composite of this equivalence with the functor of taking \(k\)-linear duals.

Then given \(\mg \in \Lie \), the composite

(-tikz- diagram)

gives the formal moduli problem \(B\mg \).

A more explicit way to describe the formal moduli problem is via Maurer-Cartan elements. Given a dgla \(\mg \), we can define \(\MC _0(\mg )\) to be the set of elements \(x\in \mg _{-1}\) satisfying the Maurer-Cartan equation³:

\[dx + \frac 1 2[x,x]=0\]

As a warning \(\MC _0(\mg )\) is not a homotopy invariant of \(\mg \)! But we can make a homotopy invariant version of it, as we now explain.

Given a dgla \(\mg \) and a (possibly nonunital) cdga \(R\), \(\mg \otimes R\) inherits the structure of a dgla by viewing it as \(\mg \) base changed to a Lie algebra over \(R\). In otherwords, the Lie bracket is given by \([x\otimes a,y\otimes b] = (-1)^{|ay|}[x,y]\otimes ab\).

Let \(\Omega ^*(\Delta ^n)\) be the polynomial de Rham complex of the \(n\)-simplex: i.e it is the free cdga generated by \(t_0,\dots ,t_n\) in degree \(0\) with the equation \(\sum _0^nt_i =1\). \(\Omega ^*(\Delta ^n)\) fit together as \(n\) varies to form a simplicial cdga. Then we can define a simplicial set \(\MC _*(\mg )\) as having \(n\)-simplicies given by \(\MC _0(\mg \otimes \Omega ^*(\Delta ^n)\). This is a homotopy invariant functor of \(\mg \).

Now we can say what the formal moduli problem associated to a dgla \(\mg \) is: it assigns an artinian local cdga \(R\) with maximal ideal \(m\) to the simplicial set \(\MC _*(\mg \otimes m)\).

³ The name Mauer-Cartan element is related to the fact that on a Lie group, there is a canonically defined Maurer-Cartan form, which is a 1-form with values in the Lie algebra, and satisfies the Maurer-Cartan equation. In other words, the Maurer-Cartan form is a Maurer-Cartan element of the de Rham complex with coefficients in the Lie algebra. The Maurer-Cartan form is the form defining a principal connection on \(G\), thought of as a principal \(G\)-bundle over a point, and the Maurer-Cartan equation is the vanishing of the curvature of this connection.