Hello everyone, today I would like to talk about Algebras which have a simple definition but are still very beautiful to me. This is possibly part of a series where I build up to Lie groups, or other exotic mathematical objects.

Definition: An algebra is a vector space V equipped with a bilinear product \(f : V \times V \rightarrow V\)

This in my opinion is a great example of mathematical intutions getting bogged up by definitions; The above definition just captures the sense of taking 2 numbers from some state space which we are concerned with, to output something in the same state space. The mathematical rigour just serves to ensure that we really get what we are trying to refer to.

This is a barebones structure that we can start with as a base for a lot of our mathematical constructions (such as associative algebras, lie groups, etc.). By making the product satisfy some properties (such as being unital, associative), We end up being able to say something about the elements of the state space. This seems like a very category theoretic way of looking at these things and reveals the beauty in generalization.