## Lie’s Theorem IIJuly 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 ${\lambda([L,I])=0}$.

(more…)

## Engel’s Theorem and Nilpotent Lie AlgebrasJuly 23, 2009

Posted by Akhil Mathew in algebra, representation theory.
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1 comment so far

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.

## Lie algebras IIJuly 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.

## Representations of sl2, Part IIJuly 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.

## Basics of group representation theoryJuly 10, 2009

Posted by Akhil Mathew in algebra, representation theory.
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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 ${V}$.
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…)