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