Posts Tagged ‘math.RT’

An Almost-Proof of the Four Color Theorem

January 26, 2008

I recently gave an Olivetti (our graduate colloquium) on chord diagrams, effectively covering my first two posts on the subject. In preparation for the talk, I read a little bit more about some cool things one can do with them, and I finally got around to reading a paper of Bar-Natan on connections with the Four Color Theorem. I figured I should write a post on it before everyone completely forgot what I’ve said already; that said, this post should be readable even if you didn’t read my other posts on chord diagrams.

A Resolution of a Tensor Algebra

January 24, 2008

This post is about projective resolutions of algebras, thought of as a bimodules over themselves.  As long as $B$ is an associative, unital algebra (which it always will be in this post), there is a canonical projective resolution of $B$, called the bar resolution, which is sufficient for most purposes.  However, this resolution is of infinite length, and so it isn’t useful in bounding projective dimensions of modules.  For those purposes, it is natural to look for finite projective resolutions of $B$.

I came across such a problem in my research, and came up with limited and ultimately unhelpful results.  My interest was in the case that $B$ is a tensor algebra $T_AM$ of an algebra $A$ over a bimodule $M$.  Under what conditions on $A$ and $M$ would a nice, finite resolution of $T_AM$ exist?  My result is as follows:

Let $A$ be an algebra, $M$ be a bimodule over $A$, and let $\Omega^1A$ denote the kernel of the multiplication map $m: A\otimes A \rightarrow A$.  If $\Omega^1A$ is projective as an $A$ bimodule, and $Tor_1^A(M_A,_AM)=0$, then there is a projective resolution of $T_AM$ of length 3.

This is kinda neat, but its not super useful unless it can be used to produce projective resolutions of $T_AM$ modules.  Hence, the second result:

Let $A$, $M$, and $\Omega^1A$ be defined as above.  If $\Omega^1A$ is projective and $M$ is flat as a right $A$ module, then any left $T_AM$ module has a projective resolution of length 3.

Sadly, I wanted the assumptions of the former to prove the latter, which my techniques don’t.

Chord Diagrams and Lie Algebras

December 25, 2007

Merry Christmas!  In this post, we will build on some of the previous posts about chord diagrams.  In a bit of a tangent from previous thoughts, we will explore the relationship between chord diagrams and Lie algebras.  Explicitly, last time we came up with a relation, which we will henceforth call the IHX-relation: Remember that this was really another aspect of the 4T relation for regular chord diagrams.  We will see how this relationship is a pictorial representation of the Jacobi identity, which allows us to interpret generalized chord diagrams modulo the IHX relation as instructions on how to combine a large number of Lie brackets.

The first step is to introduce the Penrose’s tensor notation, which is a very natural tool for writing down instructions on manipulating tensor powers of vector spaces.  If we have a Lie algebra with an invariant inner product, we can turn a large class of graphs with extra data into such instructions.  This class of graphs includes chord diagrams, and we will see that in this framework, the IHX relation and the Jacobi relation are the same thing.

I should warn readers that there’s not much of a punch-line to this post.  Instead of being about building to some nifty conclusion, this is about a different perspective on chord diagrams and a series of nifty things you can do with them.  Also, this is a very long post that is a bit rambly.  I wouldn’t mind some constructive criticism on how some of this stuff could have been explained better.