Distribution algebra and universal enveloping algebra

Statement

Let $k$ be a field of characteristic zero and let $G$ be a finite-dimensional formally smooth affine formal group over $k$ , with tangent Lie algebra $\mathfrak g=\operatorname{Lie}(G)$ . Inclusion of the primitive distributions extends uniquely to a canonical isomorphism of filtered Hopf algebras

U(\mathfrak g) \xrightarrow{\ \sim\ } \operatorname{Dist}(G),

where $U(\mathfrak g)$ is the universal enveloping algebra and $\operatorname{Dist}(G)$ is the distribution algebra at the identity.

Associated-graded proof

Formal smoothness identifies the associated graded coalgebra and algebra of finite-order distributions with

\operatorname{gr}\operatorname{Dist}(G) \cong \operatorname{Sym}_k(\mathfrak g).

The Poincaré–Birkhoff–Witt theorem gives the same associated graded algebra for the standard filtration on $U(\mathfrak g)$ . The canonical map is the identity in filtration degree one and induces the symmetric-algebra isomorphism on associated gradeds, so it is an isomorphism.

Characteristic dependence

Characteristic zero is essential. In characteristic $p>0$ , divided-power distributions of higher order generally are not generated by primitive order-one distributions, and

U(\operatorname{Lie}(G))\longrightarrow\operatorname{Dist}(G)

need not be surjective. This is one manifestation of the failure of tangent classification in positive characteristic.

References

Jens Carsten Jantzen, Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs 107, American Mathematical Society, 2003. Publisher record. Relevant: Part I, §7.
Michiel Hazewinkel, Formal Groups and Applications, Pure and Applied Mathematics 78, Academic Press, 1978; AMS reprint, 2012. Publisher record. Relevant: formal Lie theory.