Tangent Lie algebra of a formal group

Core idea

Let $\mathcal G$ be a finite-dimensional formal group over a field $k$ , with identity $e$ . Its tangent Lie algebra is

\operatorname{Lie}(\mathcal G):=T_e\mathcal G

with the bracket obtained by extending tangent vectors uniquely to left-invariant formal vector fields and taking their commutator.

Invariant derivations

In affine coordinates, formal vector fields are continuous derivations of the complete coordinate ring. Multiplication on $\mathcal G$ defines the left-translation condition. Evaluation at $e$ identifies left-invariant derivations with $T_e\mathcal G$ , and the commutator of derivations preserves left invariance. Transporting that commutator back to the tangent space gives the bilinear alternating bracket satisfying the Jacobi identity.

Formula from a formal group law

Choose coordinates and write the corresponding formal group law as

F(X,Y)=X+Y+B(X,Y)+O(3),

where $B$ is the homogeneous bilinear part. Then

[u,v]=B(u,v)-B(v,u).

The antisymmetric part of the quadratic multiplication therefore contains the first nonabelian information. In BCH coordinates, $B(u,v)=\tfrac12[u,v]$ , recovering the original bracket.

Functoriality

For a formal group homomorphism $\phi:\mathcal G\to\mathcal H$ , the differential at the identity

d\phi_e:T_e\mathcal G\longrightarrow T_e\mathcal H

preserves brackets. Hence

\operatorname{Lie}:\mathbf{FGrp}^{\mathrm{fd}}_k \longrightarrow\mathbf{LieAlg}^{\mathrm{fd}}_k

is a functor.

In characteristic zero this functor is an equivalence on the formal-disc category. In positive characteristic it remains useful but is not a complete invariant.

References

A. Fröhlich, Formal Groups, Lecture Notes in Mathematics 74, Springer, 1968. Publisher record. Relevant: Chapter 2, Lie theory.
Jean-Pierre Serre, Lie Algebras and Lie Groups, second edition, Springer, 1992. Publisher record. Relevant: Lie functors and invariant vector fields.