In the last of the three sections covered in our Algebra course, we had Representation theory. We studied that representations correspond to modules over group algebras and that irreducible representations are simple modules. There is a classification of simple modules over an algebra, viz, they must occur as a matrix ring over a division algebra. This result generalizes to semisimple algebras and is a result due to Wedderburn. The proof is a culmination of beautiful ideas that involve rings, modules, algebras and their homomorphisms. One also encountered the Double Centralizer theorem, that asserts that that the centralizer of the centralizer of a module is itself, under certain conditions.

Since I couldn’t attend most classes, I took the trouble to type the notes, with annotations, improvised proofs and appendices. One can have a look at it in the notes page or click here.