Preliminaries

I'm going to assume everyone here is familiar with linear algebra, at least so far as matrices being used for turning vectors into other vectors. I'll want to go a little bit beyond that, but more on that as I figure it out.
Also I hope everyone is comfortable with the complex numbers, because a lot of what I want to do doesn't work well if we don't have square roots of everything. Some things can be done with just the real numbers, some can't. Similarly, I'm going to be assuming characteristic 0 unless I explicitly say otherwise, because representation theory over fields of positive characteristic gets really hairy really quickly.
Finally, I won't be proving much. Some proofs are great, but some are just tedious, and I mostly want to build things and then show why I'm building them rather than checking fiddly details to make sure the constructions work.

While we're here, I'm going to mention something that I did on G+ but is a little bit nonstandard: I'm going to be using algebraic infinitesimals to motivate certain things about Lie algebras. Often mathematicians get a little leery of infinitesimals, and for good reason, but those reasons don't apply to what I'm doing so I'm going to do it. It makes things a little nicer in a lot of cases.

But that won't come up for a bit, as I want to get all of my tensor stuff here so I can refer to it.