Lecture notes on quiver representations

These are lecture notes that I have developed in the course of several years for the first year master course ”Quivers” at the University of Amsterdam. The goal of the master course is to give first-year master students a first taste of various core topics in algebra and geometry within the concrete setting of quiver representations. These topics include representation theory, category theory, homological algebra, Lie theory and algebraic geometry. In these lecture notes I assume the following foreknowledge: (1) Basics on linear algebra, rings and fields (typically the material covered in undergraduate courses on these topics). (2) Some basic facts on category theory. We will give precise references, mainly to the lecture notes [7] of the undergraduate course ”Modules and Categories” developed by Lenny Taelman. (3) In the last section of the lecture notes (Section 11), when discussing the proof of Gabriel’s Theorem, some results from algebraic geometry are needed, for which we refer to the classical text book [5]. The key ideas and steps in the proof of Gabriel’s Theorem will though be clear without foreknowledge of algebraic geometry. Initially the notes started as supplements to parts of the books by Schiffler [4] and Assem, Simson and Skowronski [1], but gradually grew out to become lecture notes covering the material of the whole master course. I tried to make the lecture notes as self-contained as possible, with references restricted mainly to the original sources [4, 1, 7] for the master course. If a proof is not given in detail, then a precise reference is given and these proofs can be understood without further additional foreknowledge unless explicitly stated otherwise. The main topics of the lecture notes are: (projective) quiver representations, Krull-Schmidt Theorem, (standard) projective resolutions, Jacobson’s Theorem for finite dimensional algebras, the presentation of basic finite dimensional algebras as bounded quiver algebras, induction and restriction, and Gabriel’s Theorem on the classification of quivers of finite representation type. These lecture notes are updated on a regular basis. Comments on the text are welcome.