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Given a field K of char. zero the theorem of Milnor Moore states that taking the enveloping hopf algebra defines an embedding $\mathcal{U} $ from Lie algebras over K into hopf algebras over K. Taking the enveloping hopf algebra is left adjoint to taking the primitive elements $\mathcal{P}$ and so the Milnor-Moore theorem says that for any Lie algebra $L$ over K the unit $L \to \mathcal{P}\mathcal{U}(L)$ is an equivalence.

If a field K is of positive char., the theorem of Milnor-Moore remains valid if one replaces Lie algebras by restricted Lie algebras over K, and the enveloping hopf algebra by the enveloping restricted hopf algebra:

Taking the enveloping restricted hopf algebra defines an embedding $\mathcal{U}_\mathrm{res} $ from restricted Lie algebras over K into hopf algebras over K. $\mathcal{U}_\mathrm{res} $ is left adjoint to taking the primitive elements $\mathcal{P}$ (that carry a canonical restricted Lie algebra structure). So similarly the Milnor-Moore theorem says that for any restricted Lie algebra $L$ over K the unit $L \to \mathcal{P}\mathcal{U}_\mathrm{res}(L)$ is an equivalence.

By a theorem of Fresse equipping a Lie algebra over K with the structure of a restricted Lie algebra over K is the same as equipping it with divided powers.

So one could interprete the Milnor-Moore theorem as saying that there is an embedding from Lie algebras with divided powers over K into hopf algebras over K left adjoint to taking primitive elements (with their canonical structure of a Lie algebra with divided powers).

Divided power Lie algebras make sense over any commutative ring R and one can ask if there is an embedding from Lie algebras with divided powers over R into hopf algebras over R left adjoint to taking primitive elements. Especially I would like to know this for R the integers. Is this true and does anyone know a reference?

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