Modules and Representations

Abstract

Representation theory studies how algebras can act on vector spaces. The fundamental notion is that of a module, or equivalently, that of a representation. Perhaps the most elementary way to think of modules is to view them as generalizations of vector spaces, where the role of scalars is played by elements in an algebra, or more generally, in a ring. We introduce many examples, and study the fundamental notions of submodules, factor modules and module homomorphisms. For group algebras, and for path algebras of quivers we make the correspondence between modules and representations explicit.