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I am currently doing a course on Lie groups, Lie Algebras and Representation theory based on Brian Hall's book of the same name. We should cover upto chapter 4/5 in this book by the end of the semester. Now for this course, our lecturer has suggested that we come up with a final project in the form of an approximately 15-page essay on any topic that we like related to Lie algebras. The difficulty of course is in choosing such a topic, perhaps those more experienced/familiar with the literature can help in suggesting one. So far, the following three suggestions have come up:

  1. A final project related to the differential geometry side of things, i.e. matrix Lie groups as manifolds, flows, vector fields,etc.

  2. A final project related to Algebraic Topology, e.g. perhaps classifying higher homotopy groups of the classical groups $(\textrm{SO}(n),\textrm{O}(n),\textrm{GL}_n, \textrm{Sp}_n$ etc).

  3. A final project related to Algebraic Groups, suggestions for a final topic have been for example "What is a Reductive Group".

The list above is (possibly) non-exhaustive. As far as Algebraic groups go, I have had a look at the books by Humphreys, Borel and Tom Springer as well as the notes of James Milne. At this moment, Springer's book looks the most accessible with just 20 pages or so of algebraic geometry in the beginning.

My question is: What would be a good topic to look at combining Lie algebras and Algebraic Groups? Also can anyone suggest any good books/course notes/ material that I can look at apart from what I listed above?

Thanks.

Edit: I would add that this question may also be for suggestions on further topics in Lie Theory.

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For what it is worth, I think that the way the question is asked (especially the original title) may qualify it as "too localized". However, this question, and the answers, give an idea of where one can go after a course on Basic Lie theory. And as such, I don't think it is too localized. –  M Turgeon Sep 4 '12 at 15:37
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@MTurgeon I agree. I have edited my question, as well as the title. –  fpqc Sep 4 '12 at 15:38
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This question is very welcome on this site: the relationship between Lie groups and algebraic groups has attracted the best 20th century mathematicians: Borel, Chevalley,Godement,... It should be of interest to many users here and the precise reason why BenjaLim asked it is irrelevant : the answers and bibliography will be of general interest. Whoever is not interested should move on, just as I did when someone recently asked about the meaning of the symbol "$\cdots$" (three dots) in algebra: I wouldn't dream of downvoting that question or voting to close it (and nobody else did either). –  Georges Elencwajg Sep 4 '12 at 16:35
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Dear @t.b., I am very happy about this endorsement, especially coming from someone I have had many opportunities to appreciate. –  Georges Elencwajg Sep 4 '12 at 16:41
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Dear BenjaLim, for your project you could take the theme of quotients of affine algebraic varieties by linearly reductive groups. This is non-trivial even for finite groups but yet rather elementary. Somehow it is not mentioned in the basic algebraic geometry books. As a reference you could consult Hans-peter Kraft's homepage . The relevant chapter is here . –  Georges Elencwajg Sep 5 '12 at 11:14
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up vote 6 down vote accepted

I am more algebraically inclined, and so this will reflect in my suggestions.

  1. Lie algebras appeared in the context of Lie groups, and as such, there were first defined over $\mathbb R$ and $\mathbb C$. Of course, we can define them over any field. Hence, you could investigate what happens when the field is not algebraically closed, or what happens when the characteristic is not zero.
  2. Depending on your knowledge of representation theory, you can look at the classification of finite-dimensional representations of semisimple Lie algebras over $\mathbb C$. At some point, you encounter Verma modules, which in general are not finite-dimensional. This leads to the notion of the Category $\mathcal O$ of a semisimple Lie algebra. (On this topic, there is a book by James Humphreys.)
  3. As you mentioned above, Algebraic groups are a natural place to look after you have studied Lie groups. However, your knowledge of classical algebraic geometry may be an obstacle to appreciating and understanding this topic. In the case you know close to nothing on this topic, I recommend Springer's book over the others.
  4. Continuing on the last point, here is an idea of a possibly interesting topic for your essay: there are strong connections between Lie groups and Algebraic groups in the structure and the ideas. What can we say about algebraic groups when we work over an algebraically closed field of characteristic zero? (This is Chapter 5 in Humphreys' book.)

Added: In my opinion, there are two ways to efficiently learn about algebraic groups:

  1. Probably the most obvious: take a course on the topic. However, I know that this can be complicated (for example, in your situation, you have to learn about it more quickly, but also, such a course is not offered everywhere). However, the lecturer will be able to give insight into the theory and possibly applications, which are very valuable.

  2. Once you know the basic terminology (say the first four chapters of Humphreys'), pick a random chapter in the book and start reading. Or read papers/books where algebraic groups are being used and see how the structure theory is used in concrete applications (personally, this was achieved through reading papers on $p$-adic groups, and later, when learning about automorphic forms and groups). Since you have taken a course on Lie groups, the structure theory should not surprise you (except maybe the fact that the focus is shifted from semisimple groups to reductive groups). The difficult part is proving the theorems we want, and this requires a good knowledge of algebraic geometry.

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Thanks Max for your suggestion. In any case if this question gets closed, I am sure your answer will be helpful not just to me but many other future learners of Lie theory. –  fpqc Sep 4 '12 at 15:23
    
There is this book by Malle and Testerman, Linear Algebraic Groups and Finite groups of Lie type, what do you think of it? –  fpqc Sep 8 '12 at 0:20
    
@BenjaLim I have just read the preface and the beginning of the first chapter, and maybe it is not the best place to learn about algebraic groups, if you don't already know algebraic geometry. But if you're ready to take it as a black box, you can learn how it is connected to finite groups of Lie-type. –  M Turgeon Sep 8 '12 at 2:59
    
Thanks for having a look. I guess I'll stick with Springer then. Now I have been advised by my lecturer that even though there may be a lot of things that I don't understand, I should just push on. For example, he has set it out that I should try to understand why if I have an algebraic group acting on a variety, the orbit of a point is open in its closure. Should I learn to take things as black boxes? –  fpqc Sep 8 '12 at 3:11
    
@BenjaLim I have added a few suggestions regarding your last comment. –  M Turgeon Sep 10 '12 at 13:48
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What about the combinatorics of Weyl groups? I don't know how much that is covered in your course, but this is a rich topic with direct connections to Lie Theory. Bjorner and Brenti's Combinatorics of Coxeter Groups and Humphreys' Reflection Groups and Coxeter Groups are two very readable texts which give a fairly comprehensive introduction to the subject. The other standard reference is Bourbaki's Lie Groups and Lie Algebras: Chapters 4-6, though this is a bit more daunting to someone first learning the subject.

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Thanks Michael for your answer, I will look in the topics above! –  fpqc Sep 4 '12 at 15:32
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I second Humphreys' book. It is a very readable account of an extremely fascinating topic. –  M Turgeon Sep 4 '12 at 15:34
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