Author Archives: limsup

The Group Algebra (III)

Posted on January 6, 2015 by limsup

As alluded to at the end of the previous article, we shall consider the case where K is algebraically closed, i.e. every polynomial with coefficients in K factors as a product of linear polynomials. E.g. K = C is a common choice. Having assumed … Continue reading →

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The Group Algebra (II)

Posted on January 5, 2015 by limsup

We continue our discussion of the group algebra. Constructing K[G]-modules Recall that such a module V is also called a representation of G over K, and corresponds to a group homomorphism (i) Given a K[G]-module V, a submodule W of V is precisely a vector subspace W such that g(W) ⊆ W for … Continue reading →

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The Group Algebra (I)

Posted on January 3, 2015 by limsup

[ Note: the contents of this article overlap with a previous series on character theory. ] Let K be a field and G a finite group. The group algebra K[G] is defined to be a vector space over K with basis , where “g” here is … Continue reading →

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Structure of Semisimple Rings

Posted on December 31, 2014 by limsup

It turns out there is a nice classification for semisimple rings. Theorem. Any semisimple ring R is a finite product: where each is a division ring and is the ring of n × n matrices with entries in D. Furthermore, the … Continue reading →

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Semisimple Rings and Modules

Posted on December 30, 2014 by limsup

After discussing simple modules, the next best thing is to look at semisimple modules, which are just direct sums of simple modules. Here’s a summary of the results we’ll prove: A module is semisimple iff it is a sum of simple … Continue reading →

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Simple Modules

Posted on December 29, 2014 by limsup

We briefly talked about modules over a (possibly non-commutative) ring R. An important aspect of modules is that unlike vector spaces, modules are usually not free, i.e. they don’t have a basis. For example, take the Z-module given by Z/2Z. [ Recall: a Z-module is … Continue reading →

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Coming up next…

Posted on December 29, 2014 by limsup

This blog has been dormant for a while, as I’ve been doing quite a bit of self-reading and ruminating over the stuffs I’ve read. I’d really like to post some of my thoughts, but there’s always the risk of misleading … Continue reading →

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From Euler Characteristics to Cohomology (II)

Posted on December 5, 2013 by limsup

Boundary Maps Here’s a brief recap of the previous article: we learnt that in refining a cell decomposition of an object M, we can, at each step, pick an i-dimensional cell and divide it in two. In this way, we … Continue reading →

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From Euler Characteristics to Cohomology (I)

Posted on December 2, 2013 by limsup

[ Warning: this is primarily an expository article, so the proofs are not airtight, but they should be sufficiently convincing. ] The five platonic solids were well-known among the ancient Greeks (V, E, F denote the number of vertices, edges and faces respectively): … Continue reading →

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Elementary Module Theory (IV): Linear Algebra

Posted on November 28, 2013 by limsup

Throughout this article, a general ring is denoted R while a division ring is denoted D. Dimension of a Vector Space First, let’s consider the dimension of a vector space V over D, denoted dim(V). If W is a subspace of V, we proved earlier that … Continue reading →

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