# Hopf algebra of Chevalley group from the root system

Has anyone worked out a uniform way of constructing the Hopf algebra of a Chevalley group out of the root system (or, more precisely out of the root datum for reductive groups).

By "uniform", I mean a method that works for any type, not case by case.

If yes, can anyone point out a reference in the literature? Ideally, I'd like to see the construction over the integers, but I'd still be interested in constructions over a field.

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There is a uniform construction of the split semisimple Lie algebra attached to root system. The corresponding simply connected semisimple group is that attached to Tannakian category of representations of the Lie algebra. This works over any field. For reductive groups, you have to take a subcategory of representations of the Lie algebra. This all may work over Z, but I haven't seen it written out. – mephisto May 18 '11 at 14:52
You have 3 Hopf algebras monkeying around: the group algebra, the dual group algebra and a Z-form of the functions on the corresponding algebraic groups. There may be even more. Which one do you need? – Bugs Bunny May 18 '11 at 15:20
And all of them can be "unifirmly" constructed from the root system. You'd better say what you want to do with your Hopf algebra and how explicitly you want your construction to be... – Bugs Bunny May 18 '11 at 15:22
SGA 3 Exp. XXIII and XXV together give constructions of reductive groups from pinned reduced root data over any scheme, including $\mathbb{Z}$. Is that what you want? – S. Carnahan May 18 '11 at 15:42
Make that: any field F of characteristic zero. – mephisto May 18 '11 at 18:18

A no-nonsense construction, over $Z$, following work of Kostant and Chevalley, is given in Lusztig's paper "Twelve bridges from a reductive group to its Langlands dual". The heart of the construction of the Hopf algebra is in Section 5.

This is easy enough to find online, and according to Lusztig's webpage, it can also be found published in Contemp. Math. 478 (2009), 125-143.

Good luck!

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Lusztig's work improves (and in some ways corrects) what Chevalley did earlier while completing the program in Kostant's 1965 AMS Summer Institute paper. The full details are in a paper not linked from his home page: Study of a Z-form of the coordinate ring of a reductive group, J.Amer.Math.Soc. 22(2009), 739-769. – Jim Humphreys May 18 '11 at 18:09
Thanks for both of these references. At first sight, they seem to be what I am looking for, although I need to work through the notation and the notions to be really sure. – Baptiste Calmès May 18 '11 at 19:18
P.S. Preprint versions of the two papers by Lusztig mentioned here are on arXiv, while Chevalley's Bourbaki talk is available online at www.numdam.org. Also, Takeuchi's approach is treated in Jantzen's book Representations of Algebraic Groups, in II.1.14, though that may not be so relevant to your question – Jim Humphreys May 18 '11 at 20:32
By the way, the paper of Kostant himself is short and useful ("Groups over Z", in Algebraic groups and discontinuos subgroups, 1966). – Victor Petrov May 19 '11 at 18:08
I'm accepting this answer (especially the paper cited by Jim Humphrey in his comment), because it seems to reflect the best available in the literature in the spirit of what I was asking, but I wish the description was simpler: I do not understand very well the constants involved, they look very complicated to me, and I wish there was a more "invariant" description. It might be because I'm not very familiar with universal enveloping algebras, and their quantized versions. – Baptiste Calmès Jul 14 '11 at 13:26

It depends on what do you mean by "construct". There is a paper of Chevalley "Certain schemas de groupes semi-simples", where he gives a uniform construction, but it is not something you can easily put into your computer ;). Much more computational approach is in Takeuchi's "Generators and relations for hyperalgebras of reductive groups", where he constructs a Hopf algebra which is dual (in some certain sense) to the coordinate Hopf algebra of a reductive group.

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Thanks Victor, I'll have a look. – Baptiste Calmès May 18 '11 at 19:19