Representation theory of finite-dimensional algebras
The representation theory of finite-dimensional algebras over fields is the systematic study of module categories of algebras or, in less mundane terminology, the theory of subalgebras of matrix algebras. While algebras have been studied for a long time and in many areas of mathematics, some of the concepts which are the foundational corner stones of a systematic theory are more recent. This includes, for instance the notion of a quiver of an algebra, which appears in work of Gabriel during the 1970s. It also includes the systematic use of categorical and homological methods. For the purpose of the present notes, we have chosen to make category theoretic comments from the very beginning, and collect in an appendix the relevant terminology as a reference. The same applies for the tensor product, which will be mentioned early on, with an appendix describing its construction and main properties in a systematic way.