More Lie algebra constructionsJuly 28, 2009

Posted by Akhil Mathew in algebra, 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 ${L}$. Given representations ${V_1, V_2}$, we clearly have a representation ${V_1 \oplus V_2}$; given a morphism of representations ${V_1 \rightarrow V_2}$, i.e. one which respects the action of ${L}$, the kernel and image are themselves representations.

Proposition 1 The category ${Rep(L)}$ of finite-dimensional representations of ${L}$ is an abelian category.

(more…)

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…)

Why simple modules are often finite-dimensional IIJuly 22, 2009

Posted by Akhil Mathew in algebra, representation theory.
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I had a post a few days back on why simple representations of algebras over a field ${k}$ 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: ${k}$ is our ground field, which we no longer assume algebraically closed (thanks to a comment in the previous post), ${A}$ is a ${k}$-algebra, ${Z}$ its center. We assume ${Z}$ is a finitely generated ring over ${k}$, so in particular Noetherian: each ideal of ${Z}$ is finitely generated.

Theorem 1 (Dixmier, Quillen) If ${A}$ is a finite ${Z}$-module, then any simple ${A}$-module is a finite-dimensional ${k}$-vector space.

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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.

Representations of sl2, Part IJuly 17, 2009

Posted by Akhil Mathew in algebra, representation theory.
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${\mathfrak{sl}_2}$ is a special Lie algebra, mentioned in my previous post briefly. It is the set of 2-by-2 matrices over ${\mathbb{C}}$ of trace zero, with the Lie bracket defined by:

$\displaystyle [A,B] = AB - BA.$

The representation theory of ${\mathfrak{sl}_2}$ is important for several reasons.

1. It’s elegant.
2. It introduces important ideas that generalize to the setting of semisimple Lie algebras.
3. Knowing the theory for ${\mathfrak{sl}_2}$ is useful in the proofs of the general theory, as it is often used as a tool there.

In this way, ${\mathfrak{sl}_2}$ is an ideal example. Thus, I am posting this partially to help myself learn about Lie algebras.

Lie algebras: fundamentalsJuly 16, 2009

Posted by Akhil Mathew in algebra, representation theory, Uncategorized.
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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.

Lie Algebras

Throughout, we work over ${{\mathbb C}}$, or even an algebraically closed field of characteristic zero.

Definition 1 A Lie algebra is a finite-dimensional vector space ${L}$ with a Lie bracket ${[\cdot, \cdot]: L \times L \rightarrow L}$ satisfying:

• The bracket ${[\cdot, \cdot]: L \times L \rightarrow L}$ is ${{\mathbb C}}$-bilinear in both variables.
• ${[A,B] = -[B,A]}$ for any ${A,B \in L}$.
• ${[A, [B,C]] + [B, [C,A]] + [C, [A,B]] = 0}$. 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 SwindleJuly 12, 2009

Posted by Akhil Mathew in algebra, representation theory.
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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.

Definition

Consider an abelian category ${\mathbf{A}}$. Then:

Definition 1 The Grothendieck group of ${\mathbf{A}}$ is the abelian group ${K(\mathbf{A})}$ defined via generators and relations as follows: ${K(\mathbf{A})}$ is generated by symbols ${[M]}$ for each ${M \in \mathbf{A}}$, and by relations ${[M] - [M'] - [M'']}$ for each exact sequence

$\displaystyle 0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0.\ \ \ \ \ (1)$

Note here that if ${M,N}$ are isomorphic, then ${[M] = [N]}$ in ${K(\mathbf{A})}$ by considering the exact sequence

$\displaystyle 0 \rightarrow M \rightarrow N \rightarrow 0 \rightarrow 0.$

The Grothendieck group has an important universal property: (more…)

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…)