# Elementary reference for algebraic groups

I'm looking for a reference on algebraic groups which requires only knowledge of basic material on the theory of varieties which you could find in, for example, Basic Algebraic Geometry 1 by Shafarevich.

If possible, it would be nice to find such a book which also discusses representation theory, but that's not necessary.

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Milne has lectures notes which are probably excellent (I haven't looked at them). Whatever you read, the beef is str. theory of reductive alg. gps. In absence of schemes, some things in char. $> 0$ are a bit clunky in comparison with char. 0 because kernels can be non-smooth; e.g., ${\rm{SL}}_p \rightarrow {\rm{PGL}}_p$ in char. $p > 0$ with "kernel" $\mu_p$, akin to purely insep. isogeny of ell. curves. When you learn schemes and sheaves, some awkward things with quotients and non-smooth subgroups (and centralizers, and center, and so on) in char. $> 0$ will become more straightforward. –  BCnrd Sep 28 '10 at 2:04
I have to recuse myself from this question, having written an exposition of the Borel/Bass lecture notes in my carefree (and tenure-free) youth. All books mentioned here are useful, but for varied purposes and using geometry at different levels. One concrete early motivation for the algebraic group mixture of group theory and algebraic geometry is the Kolchin-Borel-Chevalley work showing the intrinsic nature of the multiplicative Jordan decomposition. This is elementary (albeit technical) but not conceptually obvious. Quotients and such get far more sophisticated. –  Jim Humphreys Sep 28 '10 at 17:43
@Jim: Could you please provide a reference? –  David Corwin Sep 29 '10 at 5:27
For discussion of Jordan decomposition in linear algebraic groups, look at the earlier MO post #30042 from June 30. –  Jim Humphreys Sep 29 '10 at 16:35

If you're interested in the theory of linear algebraic groups, Linear Algebraic Groups by Humphreys is a great book. The other two standard references are the books (with the same name) by Springer and Borel. All of the algebraic geometry you need to know is built from scratch in any of those books.

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Humphreys really fits the bill here. Especially since you are hoping it discusses rep theory. –  B. Bischof Sep 28 '10 at 4:50

My favourite: Waterhouse's Introduction to Affine Group Schemes! It is very friendly and clearly written and gives you the complete basic package on Affine Group Schemes in just 150 pages. With the firm grounding you get from this book you can gather whatever else you need to know (if anything) by skimming through other literature like the books by Humphreys, Springer, Demazure/Gabriel or articles.

For a very down to earth approach with lots of matrix examples, which give you a sense of reality and material for hands-on practice (e.g. while reading Waterhouse), check out Ulf Rehmann's five or six lectures here. Don't mind the unusual introduction - it's from a K-theory school...

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Waterhouse somehow managed to write an entire book on algebraic groups without discussing reductive or semisimple groups. I suppose that he was trying to make some kind of philosophical point, but the result is not helpful for most people who use algebraic groups. –  Nikita Sep 28 '10 at 19:50

The first book I read on algebraic groups was An Introduction to Algebraic Geometry and Algebraic Groups by Meinolf Geck. As I recall, the book includes a lot of examples about the classical matrix groups, and gives elementary accounts of such things like computing the tangent space at the identity to get the Lie algebra. Whatever algebraic geometry Geck needs (which isn't much) he develops from scratch in Chapter 1. This book is really aimed at advanced undergraduates, so is much more elementary than the previously suggested titles.

There's a little bit towards the end of Geck's book about (virtual) characters for finite groups of Lie type.

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If you are interested in algebraic groups over complex and real numbers only, try Onishchik and Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag 1990. This book contains also representation theory. (Then later you will have to learn the characteristic p case and algebraic groups over non-closed fields, say from Springer's book and Milne's lecture notes.)

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My favorite references are Springer's book and Milne's notes, both of which have already been mentioned. However, if you're encountering these things for the first time, I recommend reading "Lectures On Lie Groups And Lie Algebras" by Carter, Segal, and Macdonald. It is based on short lecture courses by the authors -- Carter discusses the basic of semisimple Lie algebras, Segal covers compact Lie groups, and Macdonald covers the basics of (linear) algebraic groups. It omits many proofs and most technicalities, but it gives a very nice, clutter-free overview of the subject.

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Thanks. The main reason I'm wondering is that I'm taking both algebraic geometry and representation theory of Lie groups/algebras (i.e. two courses) this semester, and in my algebraic geometry class, I might do a project which combines the two, and looks either at Lie groups, or at groups over more general fields, from a specifically algebro-geometric perspective, as opposed to the primarily analytic perspective one encounters in the general theory of Lie groups. –  David Corwin Sep 28 '10 at 6:12

If you have gone through Shafarevich, you don't really need it to be "elementary". Go for Jantzen's Representations of Algebraic Groups. Part 1 is groups and no representation theory and Part 2 has as much representation theory as you may ever need...

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Although RAGS is an amazing, thorough book, I think it's only worth diving into if you're interested in the positive-characteristic story. If you're interested in the characteristic 0 story, RAGS is overwhelming (indeed, I first tried to learn the characteristic 0 story out of RAGS, and had to give up and come back again later when I had a better feeling for the topic). –  Chuck Hague Sep 28 '10 at 12:50
Maybe, but most weights-roots business should not even be studied in the context of algebraic groups. Any book on Lie algebras is preferable. Then the first true "Algebraic Group Representation Theory" fact is BWB and RAGS has a decent expo of it (if I remember correctly - I won't go into library to check)... –  Bugs Bunny Sep 28 '10 at 13:49
Just note that by Shafarevich I said "Basic Algebraic Geometry 1," which has no schemes. –  David Corwin Sep 29 '10 at 21:27
RAGS will teach you schemes then... –  Bugs Bunny Sep 30 '10 at 10:14

The following is an emended excerpt from my answer to a related question1 about books about Lie groups for someone with algebraic geometry background. I might add that Procesi's book ideally fits your goals, since you are also interested in representation theory.

For someone with algebraic geometry background, I would heartily recommend Procesi's Lie groups: An approach through invariants and representations. It is masterfully written, with a lot of explicit results, and covers a lot more ground than Fulton and Harris. If you like "theory through exercises" approach then Vinberg and Onishchik, Lie groups and algebraic groups is very good (the Russian title included the word "seminar" that disappeared in translation).

If you aren't put off by a bit archaic notation and language, vol 2 of Chevalley's Lie groups is still good.

1That question is exactly one year old and, according to Anton's MO birthday post on meta, was the second "real" question asked on Mathoverflow.

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