# Complex root systems

This question is twofold.

1) What is the best reference on root systems?

2) Do complex root systems exist?

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Humphreys' "Lie Algebras" is a good reference for root systems. – Vinoth Dec 11 '10 at 2:56
Humphreys' book on finite reflection groups is better. – David Hill Dec 11 '10 at 7:36
oh, and yes to 2) – David Hill Dec 11 '10 at 7:37

To supplement what Pete says, I'd emphasize that "root systems" have been defined in a variety of ways for a variety of purposes related to Lie theory or to some type of "reflection" group. (For instance, Vinay Deodhar proposed a general notion in the setting of Coxeter groups, which I used in my 1990 book.) The traditional notion arose in the study of semisimple Lie groups and their complexified Lie algebras, where you have finite real reflection groups (Weyl groups = finite crystallographic Coxeter groups). Witt built an early bridge between Coxeter's viewpoint and this kind of Lie theory. Bourbaki's axiomatic treatment encodes all of this efficiently and also includes affine Weyl groups. Eventually many of these notions have been carried over to $p$-adic groups, Kac-Moody theory, etc.

Concerning "complex root systems", these arise in the separate but overlapping theory of finite complex (= unitary) reflection groups. After the classification by Shephard-Todd, Arjeh Cohen systematized somewhat the notion of root system in a 1976 paper "Finite complex reflection groups", freely available here. More recently Michel Broue and his collaborators have used complex reflection groups and associated diagrams (more elaborate versions of Dynkin diagrams) extensively in their study of representations of finite groups of Lie type. There is a useful 2009 Cambridge text by Lehrer and Taylor, Unitary Reflection Groups, with a suitable notion of root system defined at the end of the first chapter.

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Wow! Thanks for all that information Jim. – hypercube Dec 11 '10 at 16:58
+1.  – Pete L. Clark Dec 11 '10 at 17:23

For 1): Chapters 4-6 (this is a single bound text) of Bourbaki's Lie Groups and Lie Algebras is often said to be the most comprehensive introductory basic treatment on root systems. Much modern work in linear algebraic groups and related finite group theory makes reference to it.

(In fact I have heard it said that this is the high point of the entire Elements of Mathematics series, though this is obviously a matter of taste. For my part, I believe I currently like Commutative Algebra the best.)

For 2): apparently yes. See e.g.

www.math.uga.edu/~pete/Can06.pdf

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Your link is directing me to a UGA remote login page. – Ben Linowitz Dec 11 '10 at 3:52
@Ben: Ah, sorry about that. You should be able to get it now. – Pete L. Clark Dec 11 '10 at 4:04
Thanks for the quick reply. – hypercube Dec 11 '10 at 4:13
Dear Pete: Does (1) mean for learning, or as a general reference for research? Recommending Bourbaki to learn about root systems is like recommending EGA to learn algebraic geometry. For most people it is probably better to learn the basics from a more example-oriented and "concrete" source. The Bourbaki treatment is awesomely comprehensive (as EGA is for schemes), but books like Humphreys (on Lie algebras), Serre (on complex ss Lie alg.), and Springer (in chapter 8 of his book on algebraic groups) are more user-friendly ways to get a grip on things before diving into the ultimate resource. – BCnrd Dec 11 '10 at 6:24
@BCnrd: I said "most comprehensive", and I did not say anything about learning. The OP asked for a reference, so I decided that most comprehensive was one reasonable way to construe "best reference". I think giving other texts and explaining in what ways they may also be "best" would be a positive contribution. – Pete L. Clark Dec 11 '10 at 7:36