The previous version of this question was rather badly broken, and I hope this version makes some sense.

There have been a few questions on MathOverflow about how much representation-theoretic information is lost when passing from a Lie group to its Lie algebra, e.g., away from the semisimple case, Lie algebras have many more representations. In the algebraic setting, there is an intermediate construction between an algebraic group and its Lie algebra, given by the formal group. One completes the algebraic group along the identity to get a formal scheme equipped with a group law, and one can pass from there to the tangent space to get the Lie algebra. In characteristic zero, the tangent space functor is an equivalence of categories from formal groups to Lie algebras, but in positive characteristic, formal groups form an honest intermediate category since the tangent space can lose a lot of information. For example, there is only one isomorphism class of one-dimensional Lie algebra, but one-dimensional formal groups have a rich arithmetic theory, with a moduli space stratified by positive integer heights. The completions at the identity of the additive group and the multiplicative group have very distinct formal group structures, and one way to explain the lack of isomorphism is by the presence of denominators in the usual logarithm and exponential power series.

It seems to me that in positive characteristic, there could be an intermediate construction between formal groups and Lie algebras, given by passing to PD rings and replacing the coordinate ring of the formal group with the divided power envelope of the identity section. If I'm not mistaken, this construction yields a group object in PD formal schemes.

Here is a bit of explanation for the uninitiated (see Berthelot-Ogus for more): PD rings are triples $(A,I,\gamma)$, where $A$ is a commutative ring, $I$ is an ideal, and $\gamma = \{ \gamma_n: I \to A \}_{n \geq 0}$ is a system of divided power operations. I think they arose when Grothendieck tried to get De Rham cohomology to give the expected answers for proper varieties in characteristic $p$, since the naïve definition tended to yield infinite dimensional spaces. There is a forgetful functor $(A,I,\gamma) \mapsto (A,I)$ from PD rings to ring-ideal pairs, and it has a left adjoint, called the divided power envelope. In characteristic zero, $\gamma$ is canonically given as $\gamma_n(x) = x^n/n!$, so both functors are equivalences in that case. The notion of PD ring can be sheafified and localizations have canonical PD structures, so one has notions of PD scheme and PD formal scheme.

Question: Do PD formal groups contain any more information than the underlying Lie algebra?

I have a suspicion that the answer is "no" and the answer to the title question is "yes". Vague word-association suggests that the divided power structure is exactly what one needs to get a formal logarithm, but maybe there is a more fundamental obstruction.

I was originally motivated by the question of how Gelfand-Kazhdan formal geometry would differ in charateristic $p$ if I switched between ordinary and PD structures (cf. David Jordan's question). Unfortunately, I was laboring under some misconceptions about formal completions, and I'm still a bit confused about the precise structure of the automorphism group of the completion (PD or ordinary) of a smooth variety at a point in characteristic $p$.

The previous version of this question was rather badly broken, and I hope this version makes some sense.

There have been a few questions on MathOverflow about how much representation-theoretic information is lost when passing from a Lie group to its Lie algebra, e.g., away from the semisimple case, Lie algebras have many more representations. In the algebraic setting, there is an intermediate construction between an algebraic group and its Lie algebra, given by the formal group. One completes the algebraic group along the identity to get a formal scheme equipped with a group law, and one can pass from there to the tangent space to get the Lie algebra. In characteristic zero, the tangent space functor is an equivalence of categories from formal groups to Lie algebras, but in positive characteristic, formal groups form an honest intermediate category since the tangent space can lose a lot of information. For example, there is only one isomorphism class of one-dimensional Lie algebra, but one-dimensional formal groups have a rich arithmetic theory, with a moduli space stratified by positive integer heights. The completions at the identity of the additive group and the multiplicative group have very distinct formal group structures, and one way to explain the lack of isomorphism is by the presence of denominators in the usual logarithm and exponential power series.

It seems to me that in positive characteristic, there could be an intermediate construction between formal groups and Lie algebras, given by passing to PD rings and replacing the coordinate ring of the formal group with the divided power envelope of the identity section. If I'm not mistaken, this construction yields a group object in PD formal schemes.

Here is a bit of explanation for the uninitiated (see Berthelot-Ogus for more): PD rings are triples $(A,I,\gamma)$, where $A$ is a commutative ring, $I$ is an ideal, and $\gamma = \{ \gamma_n: I \to A \}_{n \geq 0}$ is a system of divided power operations. I think they arose when Grothendieck tried to get De Rham cohomology to give the expected answers for proper varieties in characteristic $p$, since the naïve definition tended to yield infinite dimensional spaces. There is a forgetful functor $(A,I,\gamma) \mapsto (A,I)$ from PD rings to ring-ideal pairs, and it has a left adjoint, called the divided power envelope. In characteristic zero, $\gamma$ is canonically given as $\gamma_n(x) = x^n/n!$, so both functors are equivalences in that case. The notion of PD ring can be sheafified and localizations have canonical PD structures, so one has notions of PD scheme and PD formal scheme.

Question: Do PD formal groups contain any more information than the underlying Lie algebra?

I have a suspicion that the answer is "no" and the answer to the title question is "yes". Vague word-association suggests that the divided power structure is exactly what one needs to get a formal logarithm, but maybe there is a more fundamental obstruction.

I was originally motivated by the question of how Gelfand-Kazhdan formal geometry would differ in charateristic $p$ if I switched between ordinary and PD structures (cf. David Jordan's question). Unfortunately, I was laboring under some misconceptions about formal completions, and I'm still a bit confused about the precise structure of the automorphism group of the completion (PD or ordinary) of a smooth variety at a point in characteristic $p$.

The following is my understanding, which may be broken:

If X is a smooth variety, then the formal completion of X at a closed point x (isomorphic to Spf of a Cohen ring) has a canonical formal group structure, with multiplication arising from the completion of $X \times X$ at (x,x). These formal groups encode some kind of information about infinitesimal translations near x. If I'm not mistaken, this formal group is always a product of formal additive groups in characteristic zero, because once we choose a generating set of the augmentation ideal, we can recursively construct a formal logarithm homomorphism. In characteristic p, the standard counterexamples are the multiplicative formal group and the formal group of an elliptic curve. Here, the logarithm would require denominators with factors of p. I think there is a fancier explanation involving p-divisible groups.

Instead of a formal group at a closed point, we can consider a more global object, given by completing $X \times X$ along the diagonal embedding of X. This has a natural formal groupoid structure, and taking a quotient of X by its action yields an object called the de Rham stack of X. As a functor its R-points are X(R/I), where I is the nilradical of R. There is an equivalence of categories between quasicoherent modules over the sheaf of ordinary differential operators on X and quasicoherent sheaves on the de Rham stack (which seems to mean a quasicoherent sheaf on X that is equivariant under the groupoid action).

When trying to make de Rham cohomology give the expected answers for proper varieties in characteristic p, one typically adds divided power (PD) structures (which are canonical in characteristic zero) to the defining ideals. In particular, one considers formal groups and groupoids given by PD completions instead of plain formal completions. There is a canonical map from the usual formal group/groupoid to the formal PD group/groupoid that is an isomorphism in characteristic zero. Sheaves that are equivariant under the PD groupoid action are called crystals (I'm not sure about this - the usual definition involves the crystalline site, but Berthelot's thesis seems to say the groupoid is universal). There is a theorem giving an equivalence of categories between crystals on X and quasicoherent sheaves on X with a flat connection.

My question is: Are these PD formal groups always isomorphic to a power of the formal (PD) additive group?

Vague word-association suggests that the divided power structure is exactly what one needs to get a logarithm, but maybe there is a more fundamental geometric obstruction.

(More motivation: I'm looking for a crystalline Gelfand-Kazhdan equivalence, like in David Jordan's question. Ideally, this would already exist in the literature. Is it a stupid idea?)

The previous version of this question was rather badly broken, and I hope this version makes some sense.

There have been a few questions on MathOverflow about how much representation-theoretic information is lost when passing from a Lie group to its Lie algebra, e.g., away from the semisimple case, Lie algebras have many more representations. In the algebraic setting, there is an intermediate construction between an algebraic group and its Lie algebra, given by the formal group. One completes the algebraic group along the identity to get a formal scheme equipped with a group law, and one can pass from there to the tangent space to get the Lie algebra. In characteristic zero, the tangent space functor is an equivalence of categories from formal groups to Lie algebras, but in positive characteristic, formal groups form an honest intermediate category since the tangent space can lose a lot of information. For example, there is only one isomorphism class of one-dimensional Lie algebra, but one-dimensional formal groups have a rich arithmetic theory, with a moduli space stratified by positive integer heights. The completions at the identity of the additive group and the multiplicative group have very distinct formal group structures, and one way to explain the lack of isomorphism is by the presence of denominators in the usual logarithm and exponential power series.

It seems to me that in positive characteristic, there could be an intermediate construction between formal groups and Lie algebras, given by passing to PD rings and replacing the coordinate ring of the formal group with the divided power envelope of the identity section. If I'm not mistaken, this construction yields a group object in PD formal schemes.

Here is a bit of explanation for the uninitiated (see Berthelot-Ogus for more): PD rings are triples $(A,I,\gamma)$, where $A$ is a commutative ring, $I$ is an ideal, and $\gamma = \{ \gamma_n: I \to A \}_{n \geq 0}$ is a system of divided power operations. I think they arose when Grothendieck tried to get De Rham cohomology to give the expected answers for proper varieties in characteristic $p$, since the naïve definition tended to yield infinite dimensional spaces. There is a forgetful functor $(A,I,\gamma) \mapsto (A,I)$ from PD rings to ring-ideal pairs, and it has a left adjoint, called the divided power envelope. In characteristic zero, $\gamma$ is canonically given as $\gamma_n(x) = x^n/n!$, so both functors are equivalences in that case. The notion of PD ring can be sheafified and localizations have canonical PD structures, so one has notions of PD scheme and PD formal scheme.

Question: Do PD formal groups contain any more information than the underlying Lie algebra?

I have a suspicion that the answer is "no" and the answer to the title question is "yes". Vague word-association suggests that the divided power structure is exactly what one needs to get a formal logarithm, but maybe there is a more fundamental obstruction.

I was originally motivated by the question of how Gelfand-Kazhdan formal geometry would differ in charateristic $p$ if I switched between ordinary and PD structures (cf. David Jordan's question). Unfortunately, I was laboring under some misconceptions about formal completions, and I'm still a bit confused about the precise structure of the automorphism group of the completion (PD or ordinary) of a smooth variety at a point in characteristic $p$.