## 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.

## Why simple modules are often finite-dimensional, IJuly 19, 2009

Posted by Akhil Mathew in algebra.
Tags: , , , ,

Today I want to talk (partially) about a general fact, that first came up as a side remark in the context of my project, and which Dustin Clausen, David Speyer, and I worked out a few days ago.  It was a useful bit of algebra for me to think about.

Theorem 1 Let ${A}$ be an associative algebra with identity over an algebraically closed field ${k}$; suppose the center ${Z \subset A}$ is a finitely generated ring over ${k}$, and ${A}$ is a finitely generated ${Z}$-module. Then: all simple ${A}$-modules are finite-dimensional ${k}$-vector spaces.

We’ll get to this after discussing a few other facts about rings, interesting in their own right.

## Representations of sl2, Part IJuly 17, 2009

Posted by Akhil Mathew in algebra, representation theory.
Tags: , , , , ,

${\mathfrak{sl}_2}$ 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.
Tags: , , ,
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…)