Representations of semidirect products

So, the blog stats show that semisimple Lie algebras haven’t exactly been popular.  Traffic has actually been unusually high, but people have been reading about the heat equation or Ricci curvature rather than Verma modules.  Which is interesting, since I thought there was a dearth of analysts in the mathosphere.  At MathOverflow, for instance, there have been a few complaints that everyone there is an algebraic geometer.     Anyway, there wasn’t going to be that much more I would say about semisimple Lie algebras in the near future, so for the next few weeks I plan random and totally disconnected posts at varying levels (but loosely related to algebra or algebraic geometry, in general).

I learned a while back that there is a classification of the simple modules over the semidirect product between a group and an commutative algebra which works the same way as the (more specific case) between a group and an abelian group.  The result for abelian groups rather than commutative rings appears in a lot of places, e.g. Serre’s Linear Representations of Finite Groups or Pavel Etingof’s notes.  I couldn’t find a source for the more general result though.  I wanted to work that out here, though I got a bit confused near the end, at which point I’ll toss out a bleg.

Let be a finite group acting on a finite-dimensional commutative algebra over an algebraically closed field of characteristic prime to the order of . Then, the irreducible representations of correspond to maximal ideals in , or equivalently (by Hilbert’s Nullstellensatz!) homomorphisms , called characters. In other words, acts on a 1-dimensional space via the character .

 

Why simple modules are often finite-dimensional, I

Today I want to talk (partially) about a general fact, that first came up as a side remark in the context of my project, and which Dustin Clausen, David Speyer, and I worked out a few days ago.  It was a useful bit of algebra for me to think about.

Theorem 1 Let be an associative algebra with identity over an algebraically closed field ; suppose the center is a finitely generated ring over , and is a finitely generated -module. Then: all simple -modules are finite-dimensional -vector spaces.

We’ll get to this after discussing a few other facts about rings, interesting in their own right.