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Lie’s Theorem II July 27, 2009

Posted by Akhil Mathew in algebra, representation theory.
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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 {L} and an ideal {I \subset L} such that {I} was of codimension one.

There was a finite-dimensional representation {V} of {L}. For {\lambda \in I^*}, we set

\displaystyle V_\lambda := \{ v \in V: Yv = \lambda(Y) v, \ \mathrm{all} \ Y \in I \}.

We assumed {V_\lambda \neq 0} for some {\lambda} by the induction hypothesis. Then the following then completes the proof of Lie’s theorem, by the “fundamental calculation:”

Lemma 1 If {V_\lambda \neq 0}, then {\lambda([L,I])=0}.

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Engel’s Theorem and Nilpotent Lie Algebras July 23, 2009

Posted by Akhil Mathew in algebra, representation theory.
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Now that I’ve discussed some of the basic definitions in the theory of Lie algebras, it’s time to look at specific subclasses: nilpotent, solvable, and eventually semisimple Lie algebras. Today, I want to focus on nilpotence and its applications.

Engel’s Theorem

To start with, choose a Lie algebra {L \subset \mathfrak{gl} (V)} for some finite-dimensional {k}-vector space {V}; recall that {\mathfrak{gl} (V)} is the Lie algebra of linear transformations {V \rightarrow V} with the bracket {[A,B] := AB - BA}. The previous definition was in terms of matrices, but here it is more natural to think in terms of linear transformations without initially fixing a basis.

Engel’s theorem is somewhat similar in its statement to the fact that commuting diagonalizable operators can be simultaneously diagonalized.

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Lie algebras II July 20, 2009

Posted by Akhil Mathew in algebra.
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I’m going to get back eventually to the story about finite-dimensional modules, but for now, Lie algebras are more immediate to my project, so I’ll talk about them here.

From an expository standpoint, jumping straight to {\mathfrak{sl}_2} basically right after defining Lie algebras was unsound. I am going to try to motivate them here and discuss some theorems, to lead into more of the general representation theory.

Derivations

So let’s consider a not-necessarily-associative algebra {A} over some field {F}. In other words, {A} is a {F}-vector space, and there is a {F}-bilinear map {A \times A \rightarrow A}, which sends say {(x,y) \rightarrow xy}, but it doesn’t have to either be commutative or associative (or unital). A Lie algebra with the Lie bracket would be one example.

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Representations of sl2, Part II July 18, 2009

Posted by Akhil Mathew in algebra, representation theory.
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This post is the second in the series on {\mathfrak{sl}_2} and the third in the series on Lie algebras. I’m going to start where we left off yesterday on {\mathfrak{sl}_2}, and go straight from there to classification.  Basically, it’s linear algebra.

Classification

We’ve covered all the preliminaries now and we can classify the {\mathfrak{sl}_2}-representations, the really interesting material here. By Weyl’s theorem, we can restrict ourselves to irreducible representations. Fix an irreducible {V}.

So, we know that {H} acts diagonalizably on {V}, which means we can write

\displaystyle V = \bigoplus_\lambda V_\lambda

where {Hv_\lambda = \lambda v_{\lambda}} for each {\lambda}, i.e. {V_\lambda} is the {H}-eigenspace.

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Basics of group representation theory July 10, 2009

Posted by Akhil Mathew in algebra, representation theory.
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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 {G}. At least for now, we’re essentially going to be constructed with finite groups, but many of these constructions generalize. A representation of {G} is essentially an action of {G} on a finite-dimensional complex vector space {V}.

Formally, we write:

Definition 1 A representation of the group {G} is a finite-dimensional complex vector space {V} and a group-homomorphism {G \rightarrow Aut(G)}. In other words, it is a group homomorphism {G \rightarrow GL_n(V)}, where {n = \dim \ V}, and {GL_n} is the group of invertible {n}-by-{n} matrices.

An easy example is just the unit representation, sending each {g \in G} to the identity matrix. (more…)