Math 583GC
Representations of algebras
S. Paul Smith
Spring 2004, Monday-Wednesday-Friday 2:30-3:20
This course concerns modules over finite dimensional algebras. Typical
such algebras are group algebras of finite groups and path algebras of
suitable quivers. A historically important question is to determine when
the algebra is of finite representation type, meaning that there are a
finite number of (finitely generated) indecomposable modules up to
isomorphism and every (finitely generated) module is isomorphic to a direct
sum of those. For example, among the path algebras this happens if and only
if the underlying quiver is a Dynkin diagram of type A, D, or E. Such
Dynkin diagrams turn up in many areas of mathematics: representation
theory, singularities of surfaces, semisimple Lie groups and algebras,
etc. This is a first indication of how the representation theory of
finite-dimensional algebras is connected to other areas of
mathematics. Beyond the finite representation type algebras, indecomposable
modules tend to occur in families, families that are parametrized by
interesting algebraic varieties. We will follow part of the book of
Auslander, Reiten, and Smalo.
Prerequisites: Math 504/5/6.