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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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Why simple modules are often finite-dimensional, I July 19, 2009

Posted by Akhil Mathew in algebra.
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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.

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Representations of sl2, Part I July 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.

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Lie algebras: fundamentals July 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 Swindle July 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 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…)