Lie algebras of algebraic groups

Where can i find material about the definition of the exponential morphism from the Lie algebra of an algebraic affine group to the group?

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Have you tried Waterhouse's "Introduction to Affine Algebraic Groups" especially part 3? – stankewicz May 4 '11 at 19:21
What's up doc? Don't you know that $e^x$ is not usually a polynomial?? Thus, you have exponential maps only for nilpotent algebraic groups. In general, ther are some horrible formal guys living in some sort of hyperalgebra! Maybe, you start by explaining what you really want from your exponentials... – Bugs Bunny May 4 '11 at 20:42
Have you looked at the famous Steinberg notes on Chevalley groups? – isildur May 5 '11 at 0:23
As these comments illustrate, you need to provide more context for the question including restrictions on the ground field. – Jim Humphreys May 5 '11 at 12:44

2 Answers

In the classical theory of Lie groups and Lie algebras, the exponential map defined in terms of the usual power series is a standard tool for passing from the Lie algebra to the group. This makes sense for matrix groups over the real and complex fields because the series converges when evaluated at a square matrix, etc. When Chevalley set out to write a six-volume series of books on Lie-type groups over more general fields of characteristic 0, he adapted the classical theory to algebraic matrix groups by working with formal power series in one or more indeterminates, which allowed him to construct "generic points" in an algebraic group starting from its Lie algebra. This was in the spirit of algebraic geometry as studied at the time. Following his 1940s Princeton volume (in English) on Lie groups, he wrote only two more volumes (in French) before abandoning the project in favor of a study of algebraic groups over arbitrary algebraically closed fields based more on Borel's early work.

As far as I know, the only systematic attempt to use such formal exponential methods for the study of linear algebraic groups (in characteristic 0) was the second volume by Chevalley, Groupes algebriques (Hermann, Paris, 1951). His third volume was devoted essentially to Lie algebras. [Chevalley's characteristic 0 methods also appear in Chapter 4 of Hochschild's eclectic introductory text Basic Theory of Algebraic Groups and Lie Algebras (Springer, 1981). Hochschild presents an array of algebraic tools but these don't carry the theory very far in the direction of the structure and classification of semisimple groups in arbitrary characteristic.]

Over fields of characteristic $p>0$ it's also possible to make some use of the exponential power series, but only when applied to "nilpotent" elements of a restricted Lie algebra (also called Lie $p$-algebra) satisfying $x^{[p]}=0$. Such Lie algebras arise from linear algebraic groups, since the $p$-th power of a derivation is again a derivation. This technique continues to be useful in specialized questions involving nilpotent elements, although Chevalley's 1950s approach in characteristic 0 has been bypassed in favor of more powerful geometric techniques.

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Could be that Takeuchi's approach from "A hyperalgebraic proof of the isomorphism and isogeny theorems for reductive groups" is what you want. Steinberg in his lecture notes uses a slightly modified version of it (he doesn't mention any Hopf-theoretic stuff, though).

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