Let be a field of characteristic zero. The intuition is that in this case, a Lie algebra is the same data as a “germ” of a Lie group, or of an algebraic group. This is made precise in the following:
Theorem 1 There is an equivalence of categories between:
- Cocommutative Hopf algebras over which are generated by a finite number of primitive elements.
- Finite-dimensional Lie algebras.
- Infinitesimal formal group schemes over (with finite-dimensional tangent space), i.e. those which are thickenings of one point.
- Formal group laws (in many variables).
1. Formal duality
Let’s start by explaining the functor between cocommutative Hopf algebras and formal group schemes. In general, there is an antiequivalence of categories between (cocommutative) -coalgebras and profinite (commutative) -algebras, given by the duality functor. In other words, one starts with the anti-equivalence of categories between finite-dimensional -coalgebras and finite-dimensional -algebras (given by vector space duality). This extends to an anti-equivalence
between the ind-objects of finite-dimensional coalgebras and the pro-objects of finite-dimensional algebras. The former can be identified with all coalgebras (every coalgebra can be written as a filtered colimit of finite-dimensional subcoalgebras) and the latter corresponds to profinite rings. This is analogous to the Pontryagin duality anti-equivalence between profinite commutative groups and torsion abelian groups (given by .
The upshot of all this is that, given a cocommutative -coalgebra (not necessarily finite-dimensional) with structure maps and , we can extract a formal scheme as
where is regarded as a topological ring. When regarded as a functor on finite-dimensional -algebras, we have
here continuous means with respect to the discrete topology on . The vector space maps (identified with elements of ) which correspond to morphisms of schemes are precisely the grouplike elements of the -coalgebra : that is, those elements satisfying (for the counit) and .
If is a Hopf algebra as well, then so is the dual , in a topological sense: that is, is a “profinite Hopf algebra” equipped with a continuous map (continuous tensor product), or in other words, becomes a formal group scheme. This is effectively because we can multiply grouplike elements, in the above terminology. So we get an equivalence between cocommutative -Hopf algebras and “formal group schemes” in the formal sense
2. Lie theory
So far, this is mostly formal duality nonsense; the real content of the result (and the place where characteristic zero will be used) is the identification with Lie algebras.
The equivalence between cocommutative Hopf algebras (with appropriate finiteness conditions) and finite-dimensional Lie algebras is given by
where denotes the enveloping algebra and the functor sending a Hopf algebra to its subspace of primitive elements (which acquires the structure of a Lie algebra). The functors and are adjoint, and it is classical that for any Lie algebra (over a field of characteristic zero).
Proposition 2 (Milnor-Moore) Let be a cocommutative Hopf algebra over generated by its primitive elements. Then the natural map is an isomorphism.
This is one of the theorems of Milnor-Moore.
By assumption it is surjective, so we just need to show that it is injective. To see this, we will work geometrically, in the language of formal group schemes. We assume that there are a finite number of primitive generators, without loss of generality.
Consider and , which are cocommutative coalgebras, as formal schemes using the previous section: that is, we take the s of the dual. There is a morphism of (topologized) duals
which induces a morphism of formal schemes
This is actually a morphism of formal group schemes, because we started with Hopf algebras. Our goal will be to see that it is an isomorphism.
In order to do this, we will start by showing that the above two functors are deformation functors in the sense of Schlessinger. Let us regard as functors on the category of finite-dimensional -algebras; recall that, to a finite-dimensional -algebra , we have
and similarly for . Since we are working with profinite -algebras, it suffices to show that the associated functors on are isomorphic.
- Each of them assign to . In fact, by the previous discussion, it suffices to show that there is only one grouplike element in and . This will follow from a lemma below.
- As the ‘s of rings, the functors are product-preserving (in fact, they are filtered colimits of ‘s, and filtered colimits commute with products).
- The functors are formally smooth. This will follow from a theorem of Cartier that group schemes are smooth in characteristic zero (which extends to formal group schemes).
The crucial point is that the map is an isomorphism on the tangent space. In fact, we can identify the tangent space of with grouplike elements of , or equivalently with primitive elements of (by a calculation: is grouplike if and only if is primitive). Since the map
induces an isomorphism on primitive elements, we find that the morphism of functors induces an isomorphism on the tangent space.
Combining these three observations, we now use the following consequence of Schlessinger’s theorem to finish the proof:
Lemma 3 Let be two formally smooth, product-preserving functors with . Suppose is a morphism inducing an isomorphism on the tangent spaces , and that the tangent spaces are finite-dimensional. Then is an isomorphism.
Proof: In fact, then Schlessinger’s criterion shows that are both prorepresentable by a power series ring where is the dimension of the tangent space. The map corresponds to a map
which induces an isomorphism on the tangent space, and which is consequently an isomorphism. (In general, an endomorphism of a complete local
The purpose of the next section is to check these conditions, the most interesting being smoothness.
3. Cartier’s theorem on group schemes
With this in mind, we’ll need to check the conditions for . As already discussed, they are product-preserving. To see that they assign to , we need to show:
Lemma 4 A primitively generated cocommutative Hopf algebra has no nontrivial grouplike elements.
Proof: Indeed, let be a set of primitive generators of ; then we can define a filtration on such that consists of elements which can be represented as sums of at most products of elements of . Then the coproduct sends
and a grouplike element (other than ) cannot satisfy this.
The last thing to check was formal smoothness. This will follow from the following result of Cartier:
Theorem 5 (Cartier) Let be a prorepresentable functor with , isomorphic to for a profinite -algebra . Suppose has characteristic zero. Then is isomorphic to a power series ring, and is in particular formally smooth.
In particular, (finite-dimensional) group schemes are smooth in characteristic zero. Proof: The strategy is to note that has a lot of derivations, and to use the derivations together with Taylor’s theorem to define an isomorphism with the power series ring. Namely, recall that
meaning -algebra homomorphisms which reduce to the identity mod . Such a derivation induces a tangent vector of given by sending a map to .
However, conversely, given a tangent vector (or a -derivation ), we can use the group structure on to define the map
giving a derivation of lifting . In particular, if is a basis for the tangent space of , then there are continuous derivations
lifting the derivations into . Define now the map
where means “reduce mod .” In other words, we expand in a “Taylor series” using the derivations . This is a ring-homomorphism, and one can check that it is an isomorphism as follows: is an isomorphism on the tangent space. By continuity, it must be surjective. One can define an inverse by picking generators in for . The composite maps either way induce the identity on the tangent space, and one concludes by a general lemma that an endomorphism of a complete, local noetherian -algebra inducing the identity on the tangent space is an isomorphism.
4. Finishing the proof
It remains to wrap up the proof of the various equivalences of categories. We saw that primitively and finitely generated, cocommutative Hopf algebras are equivalent (under the enveloping algebra and primitive elements) to finite-dimensional Lie algebras over . The proof, however, went through formal group schemes and it now essentially suffices to prove:
Theorem 6 A fiber product-preserving deformation functor (i.e., a formal group scheme over ) is uniquely determined by its Lie algebra. That is, there is an equivalence
where the finiteness condition on the deformation functors is on the Lie algebra.
If is a functor as above, the tangent space acquires the structure of a Lie algebra. Namely, we can interpret as the primitive elements in a certain (cocommutative) Hopf algebra, which is a Lie algebra. Proof: We know that is prorepresentable (in fact, prorepresentable by a power series ring by Cartier’s theorem), so that
for a cocommutative Hopf algebra over . The Lie algebra of is now the Lie algebra of primitive elements in , . There is a map of cocommutative Hopf algebras inducing an isomorphism on the primitive elements, which induces a map of formal group schemes
Since this is a morphism of formally smooth functors inducing an identity on the tangent space (i.e., on the primitive elements), it is an isomorphism. It follows that we can define the functor from formal group schemes to Lie algebras by taking the tangent space at the identity, and the functor is
We have just shown that the composite is an isomorphism, and the other composite is an isomorphism because any Lie algebra is recovered as the primitive elements in its enveloping algebra (in characteristic zero).
The last thing to note is that infinitesimal formal group schemes over are the same thing as formal group laws, because by smoothness they correspond to a power series ring. Note that in particular we can define a formal group law from a Lie algebra, in characteristic zero, defining an “infinitesimal Lie group;” this is given explicitly in the Baker-Campbell-Hausdorff formula.