There is a textbook theorem that the categories of unipotent algebraic groups and nilpotent finite-dimensional Lie algebras are equivalent in characteristic zero. Indeed, the exponential map is an algebraic isomorphism in this case and the group structure can be defined in terms of the Lie algebra structure and vice versa via the Campbell-Hausdorff series, which is finite due to nilpotency.

My problem is that I am unable to locate any textbook where this textbook theorem is stated. The books by Borel, Humphreys, Springer, Serre do not seem to mention this theorem.

The only reference I was able to locate is this original paper by Hochschild (which refers to his earlier papers), but he does it in a heavy Hopf-algebra language that is good, too, but still leaves one desiring to find also a simple textbook-style exposition. Later Hochschild wrote a book "Basic Theory of Algebraic Groups and Lie Algebras" on the subject, to which I have presently no access, but judging by Parshall's review, it is certainly not textbook-style.

Could anyone suggest a simple reference for this textbook theorem?


Demazure-Gabriel, Groupes algebriques, Tome I (published in 1970) is a more explicit source, if available. Chapitre IV treats "groupes affines, nilpotents, resolubles", while Chapitre V specializes to commutative affine groups. Typically they work over an (almost) arbitrary field $k$, but IV.2.4 is devoted to "groupes unipotents en caracteristique 0". This seems to be as full an account as you will find in a textbook; see especially Corollaire 4.5 for the category equivalence you want.

Later textbooks in English including Hochschild focus mainly on the structure/classification of reductive rather than arbitrary affine groups. Even this much of the story told without scheme language is fairly long for graduate courses. It's regrettable from the reference viewpoint that Demazure-Gabriel gave up after one volume.

  • $\begingroup$ This solves my problem. Thank you very much! $\endgroup$ – Leonid Positselski Mar 25 '10 at 10:06

I would say that the result is implicit in Serre's Lie Algebras and Lie Groups. In LG IV and V he proves the categorical equivalence between formal groups and Lie algebras over any field of characteristic 0. He also remarks (end of Section V.4) that when the Lie algebra is nilpotent, the formal group is just a polynomial, so there are no convergence issues. He (almost) surely doesn't state the result explicitly since he doesn't talk about algebraic groups per se, but I think that stringing together the big theorem and the comment is an acceptable, albeit not ideal, reference.

By coincidence, I have Hochschild's book checked out of the library, so I tried to look up the result in it. Not much luck -- indeed it is technical (even?) compared to most other books on linear algebraic groups, not so well indexed, and uses some nonstandard notation. (I have little doubt though that if I could read it from cover to cover my understanding of the subject would be greatly enriched.)

  • $\begingroup$ Thank you for your helpful answer. What you suggest is certainly one possible kind of reference. I would of course prefer a more explicit, detailed statement of the unipotent algebraic group case, but this is already something. $\endgroup$ – Leonid Positselski Jan 4 '10 at 22:52
  • $\begingroup$ No problem. It's a shame we just missed each other in grad school. Since you are only three years older than I, it is clearly your fault: you were too young when you got your PhD. $\endgroup$ – Pete L. Clark Jan 4 '10 at 23:08
  • $\begingroup$ Actually, we might have seen each other in September 1998. I left Harvard at the end of the month (my degree is November 98). $\endgroup$ – Leonid Positselski Jan 4 '10 at 23:19


To my surprise, I was unable to find such a statement in SGA3. There is the infinitesimal formal group equivalence that Pete mentioned in exp 7B corollary 3.3.2, but I came up empty looking in the places I expected (e.g., the part on unipotent groups). Exp 17 Lemma 3.9ter. only gives an equivalence of categories in the commutative case (where Baker-Campbell-Hausdorff collapses).

SGA3 references Seminaire Chevalley 1956/58 Exp 6 and 9 and Bourbaki's book but I don't have them handy. Have you looked there?

  • $\begingroup$ No, I haven't. But thank you, I will. $\endgroup$ – Leonid Positselski Jan 5 '10 at 1:17
  • $\begingroup$ Seminaire Chevalley 1956/58 is accessible from numdam.org/numdam-bin/browse?id=SCC_1956-1958__1_ ; I looked through Exp 6 and 9 and haven't found such a theorem there. Bourbaki's Groupes et Algebres de Lie, Chapitres 1-9 (which I have in a Russian translation) covers the Campbell-Haudorf series, but does not seem to discuss algebraic or unipotent groups at all. $\endgroup$ – Leonid Positselski Jan 6 '10 at 0:42

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