While the common approach to algebraic groups is via representable functors, it seems that there is no such for differential algebraic groups (defined by differential polynomials). Neither the book by E. Kolchin, nor the texts by Ph. J. Cassidy contain anything like this — they work only with the groups of points over differential fields (and, naturally, don't say the words "group of points").

Concerning difference algebraic groups, i.e. defined by polynomials with some fixed endomorphism (also, I don't like the ambiguity with the notion of "difference algebraic equation"), there is no systematic treatment at all, although some of these groups are intensively studied (twisted groups of Lie type as an example).

So the question is: is there really no modern (scheme-like) exposition of the subject? If so, why?

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    I'm not close enough to this subject to offer a definite answer, but it's relevant to point to the lecture notes: Alexandru Buium, Differential algebraic groups of finite dimension. Lecture Notes in Mathematics, 1506. Springer-Verlag, Berlin, 1992. – Jim Humphreys Nov 20 '12 at 17:11
up vote 10 down vote accepted

A functorial-schematic approach to differential/difference algebraic groups is surely possible. Why this is not to be found in the literature is probably due to historic reasons. The major bulk of results in differential and difference algebra was obtained in a time where algebraic geometry in the style of Weil's "Foundations of algebraic geometry" was in vogue. In differential/difference algebra there never really emerged someone of the format of Grothendieck who was willing to rewrite all the foundations. (It is not only lack of motivation, there are also some mathematically highly non-trivial issues here.) Moreover, some people in the field seem to doubt the value of such an effort and model theorists, who always had a strong impact on the field, are also keeping the universal domain alive. So even nowadays, Kolchin's book remains the ultimate reference for differential algebra and differential algebraic geometry.

However, I guess if Kolchin had lived ten years later, he would have written his book in Grothendieckien style and we would not be discussing this topic here. It is also not the case that people in differential and difference algebra are ignorant of the scheme theoretic developements. For example, J. Kovacic has rewritten and clarified quite beautifully Kolchin's Galois theory of strongly normal differential field extensions. For difference equations, in my opinion, Hrushovskies proof of the twisted Lang-Weil estimates is also a manifest for the necessity and superiority of the scheme theoretic approach. Finally, this article http://arxiv.org/abs/1111.7285 contains a tannakian approach to differential/difference algebraic groups using scheme theoretic language.

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    Thanks for the answer, especially for the reference to the arXiv paper. Concerning the paper of Kovacic, I think it's better to mention his other work, titled "Differential schemes" (MR1921695). It turns out that there was quite a lot of work done on the subject by different people. I guess, I had to look first for plain differential schemes instead of differential group schemes. – Andrei Smolensky Feb 22 '13 at 6:08
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    Here arxiv.org/abs/1302.7198 is another paper that uses a representable functors approach to difference algebraic groups. – anonymous Mar 2 '13 at 9:33

For difference algebraic groups, I think the paper

Michael Wibmer: Affine difference algebraic groups

is what you asked for.

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