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For simplicity fix a base field $k$ of characteristic zero, and consider smooth affine algebraic $k$-groups. (It is understood that unipotent groups in positive characteristic are more complicated, as one might have interesting non-smooth ones.)

Question 1: forms of unipotent groups

If $k$ is algebraically closed, then it is clear that every connected unipotent $k$-group is a successive extension of $\mathbb{G}_\mathrm{a}$'s. Then what about the case $k$ not algebraically closed? Is there non-trivial $K$-forms of, say, the upper trangular unipotent $k$-group of $GL_n$, and of the unipotent radicals of Levi $k$-subgroups of $GL_n$, etc?

If $K$ is a finite Galois extension of $k$ of Galois group $\Gamma$, then a $K$-form of the split $k$-torus is the same as a $\Gamma$-module structure on the group of characters $\mathbb{Z}^d$. Is there analogous results for $K$-forms of unipotent $k$-groups?

For example, with $K$ a finite extension field of $k$. the scalar restriction $$Res_{K/k}\mathbb{G}_{\mathrm{m}}$$

is not split as $k$-torus, but $$Res_{K/k}\mathbb{G}_\mathrm{a}$$

splits into a direct sum of $\mathbb{G}_\mathrm{a}$, becasue $K$ is a finite vector space over $k$ viewed additively. It is from this example that I want to know if there are interesting examples of forms of unipotent groups.

Question 2: representations of unipotent groups

If one has the one dimensional unipotent group $U=\mathbb{G}_\mathrm{a}$, then an algebraic representation of $U$ on $V$ a finite dimensional $k$-vector space is the same as a unipotent operator on $V$. One can then extend this description naturally to obtain the Tannakian category of finite dimensional algebraic representations of $U$.

And what about general unipotent $k$-group $U$? By the theorem of Lie-Engel, we know that such a representation of $U$ on $V$ is upper-triangular: it stabilizes a full flag of $V$, and acts trivially on the successive quotients (because of unipotence). Is there more precise information one can find about these representations so as the determine the Tannakian category of representations of $U$?

Again, let $K$ be a finite Galois extension of $k$ with Galois group $\Gamma$, and $U$, $W$ two connected unipotent $k$-groups that are isomorphic over $K$. Then how can one distinghuish the representations of the two groups by some "action" of $\Gamma$ one the representations spaces, in the spirit one finds in representations of the $k$-torus $$Res_{K/k}\mathbb{G}_\mathrm{m}$$

thanks!

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In characteristic zero unipotent groups and nilpotent Lie algebras correspond to each other using the Campbell-Baker-Haussdorf formulas. –  Torsten Ekedahl Oct 4 '10 at 18:25
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1 Answer 1

up vote 2 down vote accepted

As Torsten points out, unipotent groups correspond naturally to nilpotent Lie algebras in characteristic 0. This is dealt with nicely on the scheme level, for example, in IV.2.4 of Demazure-Gabriel Groupes algebriques. They also treat in Chapter IV some questions about prime characteristic, which get quite tricky outside the commutative case. Both of your questions are more conveniently studied in the Lie algebra framework, I think, where standard Lie algebra methods for discussing forms in are available and where there is quite a bit of literature on structure, representations, and (in small dimensions) classification in characteristic 0. See for example Jacobson's 1962 book Lie Algebras.

Representation theory is potentially very complicated for nilpotent Lie algebras (say over the complex or real field), even in the finite dimensional situation: unlike the semisimple case, there is no nice general structure based on highest weights, etc. Dixmier and others have studied infinite dimensional representations extensively in connection with Lie groups. Classification of nilpotent Lie algebras is just about impossible in general, but up to dimension 7 or so there are lists. Anyway, there is a lot of literature out there. (Tori are on the other hand also studied a lot over fields of interest in number theory. They have at least the advantage of being commmutative.)

[ADDED] An older seminar write-up might be worth consulting, especially in prime characteristic, along with the relatively sparse literature published since then and best searched through MathSciNet: Unipotent Algebraic Groups by T. Kambayashi, M. Miyanishi, M. Takeuchi, Springer Lecture Notes in Math. 414 (1974). But as their treatment suggests, the main research challenges have occurred in prime characteristic. Over finite fields, there has been quite a bit of recent activity in studying the characters of finite unipotent groups related to the unipotent radical of a Borel subgroup.

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