Distribution algebra of a formal group

Definition

Let $k$ be a field and let $G=\operatorname{Spf}(A)$ be a finite-dimensional formally smooth affine formal group over $k$ whose pointed formal scheme is a formal disc. Write $\varepsilon:A\to k$ for the identity section and $\mathfrak m=\ker\varepsilon$ . The distribution algebra of $G$ is

\operatorname{Dist}(G) := \bigcup_{n\geq0}\operatorname{Dist}_{\leq n}(G), \qquad \operatorname{Dist}_{\leq n}(G) := \operatorname{Hom}_k(A/\mathfrak m^{n+1},k).

Its multiplication is convolution against the comultiplication of the coordinate Hopf algebra:

(u*v)(a)=(u\otimes v)\bigl(\Delta(a)\bigr).

Why convolution is defined

An element of $\operatorname{Dist}_{\leq n}(G)$ is a linear functional on $A$ that vanishes on $\mathfrak m^{n+1}$ ; it is a distribution of order at most $n$ supported at the identity. Continuity of $\Delta:A\to A\widehat\otimes A$ and the augmentation filtration imply

\operatorname{Dist}_{\leq r}(G)* \operatorname{Dist}_{\leq s}(G) \subseteq \operatorname{Dist}_{\leq r+s}(G).

Hence convolution is a well-defined filtered associative product, with unit $\varepsilon$ .

Dualizing multiplication on the finite quotients gives a cocommutative coalgebra structure on $\operatorname{Dist}(G)$ , and the antipode of $A$ dualizes to an antipode. Thus $\operatorname{Dist}(G)$ is a filtered cocommutative Hopf algebra. It can be noncommutative: multiplication in $\operatorname{Dist}(G)$ is dual to the possibly noncocommutative comultiplication of $A$ .

Primitive distributions and the Lie bracket

A distribution $D$ is primitive when

\Delta_{\operatorname{Dist}}(D) =D\otimes1+1\otimes D.

Equivalently, as a functional on $A$ ,

D(ab)=D(a)\varepsilon(b)+\varepsilon(a)D(b).

Such functionals are derivations at the identity and form the tangent space

\operatorname{Lie}(G) \cong(\mathfrak m/\mathfrak m^2)^\vee.

Their commutator for convolution is the tangent Lie bracket. The universal property therefore gives a filtered Hopf-algebra homomorphism

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

In characteristic zero, the canonical map displayed above is an isomorphism by the separate distribution–enveloping-algebra theorem. In positive characteristic, higher divided-power distributions need not be generated by primitive elements.

Terminology

This algebra is also called the hyperalgebra of $G$ . Its elements are algebraic finite-order functionals supported at the identity, not Schwartz distributions on a manifold and not all continuous linear functionals on $A$ .

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, “Distributions.”
Michiel Hazewinkel, Formal Groups and Applications, Pure and Applied Mathematics 78, Academic Press, 1978; AMS reprint, 2012. Publisher record. Relevant: the chapters on formal Lie theory and Cartier theory.
William C. Waterhouse, Introduction to Affine Group Schemes, Graduate Texts in Mathematics 66, Springer, 1979. Publisher record. Relevant: the treatment of group schemes, tangent spaces, and Hopf algebras.