Appendix A Finite Dimensional Algebras

Abstract

This appendix reviews the necessary background material from the theory of finite dimensional algebras. Standard texts covering most of this subject matter are [CR88, Lam91, Ben98, ASS06]. Readers familiar with this material are urged to skim this chapter or skip it entirely. Very few proofs will be given here, as the results can be found in the references. Let us remark that, unlike the case of group algebras, monoid algebras are seldom semisimple, even over the complex numbers. This forces us to use more of the theory of finite dimensional algebras than would be encountered in a first course on the ordinary representation theory of finite groups. Some of the more elaborate tools, like projective covers, are not used except for in the final chapters of the text. Most of the text uses nothing beyond Wedderburn theory.