More Lie algebra constructions July 28, 2009Posted by Akhil Mathew in algebra, representation theory.
Tags: bilinear forms, Lie algebras, Lie groups, representation theory
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The ultimate aim in the series on Lie algebras I am posting here is to cover the representation theory of semisimple Lie algebras. To get there, we first need to discuss some technical tools—for instance, invariant bilinear forms.
Generalities on representations
Fix a Lie algebra . Given representations , we clearly have a representation ; given a morphism of representations , i.e. one which respects the action of , the kernel and image are themselves representations.
Proposition 1 The category of finite-dimensional representations of is an abelian category.
Lie’s Theorem II July 27, 2009Posted by Akhil Mathew in algebra, representation theory.
Tags: eigenvalues, Lie algebras, Lie's theorem, linear algebra, representation theory, solvability
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Yesterday I was talking about Lie’s theorem for solvable Lie algebras. I went through most of the proof, but didn’t finish the last step. We had a solvable Lie algebra and an ideal such that was of codimension one.
There was a finite-dimensional representation of . For , we set
We assumed for some by the induction hypothesis. Then the following then completes the proof of Lie’s theorem, by the “fundamental calculation:”
Lemma 1 If , then .
Why simple modules are often finite-dimensional II July 22, 2009Posted by Akhil Mathew in algebra, representation theory.
Tags: algebra, finite-dimensional vector spaces, Nakayama's lemma, representation theory, simple modules
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I had a post a few days back on why simple representations of algebras over a field which are finitely generated over their centers are always finite-dimensional, where I covered some of the basic ideas, without actually finishing the proof; that is the purpose of this post.
So, let’s review the notation: is our ground field, which we no longer assume algebraically closed (thanks to a comment in the previous post), is a -algebra, its center. We assume is a finitely generated ring over , so in particular Noetherian: each ideal of is finitely generated.
Theorem 1 (Dixmier, Quillen) If is a finite -module, then any simple -module is a finite-dimensional -vector space.
Representations of sl2, Part II July 18, 2009Posted by Akhil Mathew in algebra, representation theory.
Tags: highest weights, Lie algebras, linear algebra, representation theory, sl2
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This post is the second in the series on and the third in the series on Lie algebras. I’m going to start where we left off yesterday on , and go straight from there to classification. Basically, it’s linear algebra.
We’ve covered all the preliminaries now and we can classify the -representations, the really interesting material here. By Weyl’s theorem, we can restrict ourselves to irreducible representations. Fix an irreducible .
So, we know that acts diagonalizably on , which means we can write
where for each , i.e. is the -eigenspace.
Representations of sl2, Part I July 17, 2009Posted by Akhil Mathew in algebra, representation theory.
Tags: algebra, Jordan decomposition, Lie algebras, representation theory, semisimplicity, sl2
is a special Lie algebra, mentioned in my previous post briefly. It is the set of 2-by-2 matrices over of trace zero, with the Lie bracket defined by:
The representation theory of is important for several reasons.
- It’s elegant.
- It introduces important ideas that generalize to the setting of semisimple Lie algebras.
- Knowing the theory for is useful in the proofs of the general theory, as it is often used as a tool there.
In this way, is an ideal example. Thus, I am posting this partially to help myself learn about Lie algebras.
Lie algebras: fundamentals July 16, 2009Posted by Akhil Mathew in algebra, representation theory, Uncategorized.
Tags: algebra, Lie algebras, representation theory
The following topic came up in the context of my project, which has been expanding to include new areas of mathematics that I did not initially anticipate. Consequently, I have had to learn about several new areas of mathematics; this is, of course, a common experience at RSI. For me, the representation theory of Lie algebras has been one of those areas, and I will post here about it to help myself understand it. Right here, I’ll aim to cover the groundwork necessary to get to the actual representation theory in a future post.
Throughout, we work over , or even an algebraically closed field of characteristic zero.
Definition 1 A Lie algebra is a finite-dimensional vector space with a Lie bracket satisfying:
- The bracket is -bilinear in both variables.
- for any .
- . This is the Jacobi identity.
To elucidate the meaning of the conditions, let’s look at a few examples. (more…)
Grothendieck Groups and the Eilenberg Swindle July 12, 2009Posted by Akhil Mathew in algebra, representation theory.
Tags: algebra, Eilenberg swindle, Grothendieck groups, k-theory, representation theory
The following topic came up in a discussion with my mentor recently. Since the material is somewhat general and well-known, but relevant to my project area, I decided to write this post partially to help myself understand it better.
Consider an abelian category . Then:
Note here that if are isomorphic, then in by considering the exact sequence
The Grothendieck group has an important universal property: (more…)
Basics of group representation theory July 10, 2009Posted by Akhil Mathew in algebra, representation theory.
Tags: algebra, groups, linear algebra, representation theory
Today, I want to talk a bit about group representation theory. Many of us (such as myself) are interested in representation theory in general and will likely talk more about it in the future, so it will be useful to summarize the essential ideas here to refer back. But the basics are well known and have been discussed at length on other blogs (see, e.g. here, which is discussing the subject right now), so I am merely going to summarize these facts without proofs. The interested reader can read these notes for full details. Then, I’ll mention a property to be used later on.
What is a group representation?
Start with a group . At least for now, we’re essentially going to be constructed with finite groups, but many of these constructions generalize. A representation of is essentially an action of on a finite-dimensional complex vector space .
Formally, we write:
Definition 1 A representation of the group is a finite-dimensional complex vector space and a group-homomorphism . In other words, it is a group homomorphism , where , and is the group of invertible -by- matrices.
An easy example is just the unit representation, sending each to the identity matrix. (more…)