The Power of Simple Things

Well it’s been ages since the last time I updated this blog. To be honest, I did want to write something, only if I haven’t been too busy sleeping. I am currently just doing research, reading literatures, and singing in the choir. Apparently doing research is really time-consuming, since I need to read many literatures and understand them before proceeding with the real research. Also the material I am currently studying is different with what I learned back in ITB, so I need to give more of my time to do the self-learning.

Speaking about reading literatures, I am currently self-learning one particular structure in mathematics called semisimple algebra. I won’t bother you with all the details, but it is an interesting object to study with, though. In short, an algebra A is a ring which is also a vector space (so we can multiply vector with vector as well as scalar with vector). Now consider a module U over A. Modules are basically just like vector space, where now the scalar set is no longer a field but a ring$. We call U as an A-module. Now U is called simple if its only submodules are the zero submodule and itself. We call U semisimple if U can be written as a direct sum of simple modules. Finally, we called an algebra A to be semisimple if every non-zero module over A is semisimple. What I am currently investigating is the special case when A is actually a group algebra \mathbb{C}G of \mathbb{C}-linear combination of elements in G, where G is any finite group.

Phew. That was long enough. Algebra is not really my forte, though. (I do miss integral and its friends!!) However, after spending so much time reading this particular thing, I admit I become more fascinated with algebra. To get the idea of this subject, I must have a strong understanding about group theory, but in a fancy level. At first I was quite skeptical about the idea of ‘fancy’ group theory. I didn’t have much experience dealing with groups. I did once learn it back then in my Algebra course in ITB, but what could be made complicated from just a set equipped with one single well-defined operation satisfying associativity, existence of the identity element, and existence of inverse elements?

Turns out I was wrong. Like, totally wrong. Up until now I am still amazed by how rich group theory is. I read so many new things I haven’t even heard before, like derived groups, composition series, chief series, solvability; not to mention things I have already heard but I have no idea what they are, e.g. Sylow theorems, general linear groups, group action, representation theory (which has a deep connection with semisimple algebras and module theory). That is just group theory. I haven’t talked about another more complex structures which I believe to possess a richer theory.

Before you stop reading because of what you just read, bear with me a little bit longer. What I really want to say from that rather abstract concepts which I believe not all of you would bother to understand is that we shouldn’t underestimate the power of simple things. Simple things can turn out to be not-so-simple and in fact they can probably involve a very serious and deeper understanding to master. Simple things are also the basic building blocks to grasp the more complicated ideas, so speaking that way, they are important.

Why do I use such a nonconcrete analogue just to explain that? Well probably because I do feel impressed with all this learning process and I don’t want to wait any longer to write something LOL. Hopefully it won’t take too long time for me to update this blog after this post. Anyway, thanks for reading!