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Lie algebras II
*July 20, 2009*

*Posted by Akhil Mathew in algebra.*

Tags: algebra, general theory, Lie algebras, linear algebra, quotients

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Tags: algebra, general theory, Lie algebras, linear algebra, quotients

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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 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 over some field . In other words, is a -vector space, and there is a -bilinear map , which sends say , 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.*

Tags: algebra, Hilbert basis theorem, Nullstellensatz, polynomials, simple modules

3 comments

Tags: algebra, Hilbert basis theorem, Nullstellensatz, polynomials, simple modules

3 comments

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 1Let be an associative algebra with identity over an algebraically closed field ; suppose the center is a finitely generated ring over , and is a finitely generated -module.Then: all simple -modules are finite-dimensional -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.*

Tags: algebra, Jordan decomposition, Lie algebras, representation theory, semisimplicity, sl2

5 comments

Tags: algebra, Jordan decomposition, Lie algebras, representation theory, semisimplicity, sl2

5 comments

is a special Lie algebra, mentioned in my previous post briefly. It is the set of 2-by-2 matrices over of trace zero, with the Lie bracket defined by:

The representation theory of is important for several reasons.

- It’s elegant.
- It introduces important ideas that generalize to the setting of semisimple Lie algebras.
- Knowing the theory for is useful in the proofs of the general theory, as it is often used as a tool there.

In this way, 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.*

Tags: algebra, Lie algebras, representation theory

13 comments

Tags: algebra, Lie algebras, representation theory

13 comments

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 , or even an algebraically closed field of characteristic zero.

Definition 1ALie algebrais a finite-dimensional vector space with aLie bracketsatisfying:

The bracket is -bilinear in both variables.for any .. This is theJacobi identity.

To elucidate the meaning of the conditions, let’s look at a few examples. (more…)

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Grothendieck Groups and the Eilenberg Swindle
*July 12, 2009*

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

Tags: algebra, Eilenberg swindle, Grothendieck groups, k-theory, representation theory

7 comments

Tags: algebra, Eilenberg swindle, Grothendieck groups, k-theory, representation theory

7 comments

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 . Then:

Definition 1TheGrothendieck groupof is the abelian group defined via generators and relations as follows: is generated by symbols for each , and by relations for each exact sequence

Note here that if are isomorphic, then in by considering the exact sequence

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

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Basics of group representation theory
*July 10, 2009*

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

Tags: algebra, groups, linear algebra, representation theory

6 comments

Tags: algebra, groups, linear algebra, representation theory

6 comments

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 . At least for now, we’re essentially going to be constructed with finite groups, but many of these constructions generalize. A **representation** of is essentially an action of on a finite-dimensional complex vector space .

Formally, we write:

Definition 1A representation of the group is a finite-dimensional complex vector space and a group-homomorphism . In other words, it is a group homomorphism , where , and is the group of invertible -by- matrices.

An easy example is just the unit representation, sending each to the identity matrix. (more…)