The main object of study in this book is the finitely generated modules over artin algebras. A central role is played by the simple, projective and injective modules studied in the previous chapters. In this chapter we study some classes of algebras where the module categories have an alternative description which is sometimes easier to work with. The algebras we investigate are path algebras of quivers with or without relations, triangular matrix algebras, group algebras over a field and skew group algebras over artin algebras. These examples of algebras and their module categories are used to illustrate various concepts and results discussed in the first two chapters.

Quivers and their representations

In this section we introduce quivers and their representations over a field k. The notion of quiver and the associated path algebra come up naturally in the study of (not necessarily finite dimensional) tensor algebras of a bimodule over a semisimple k-algebra. The representations of a quiver with relations correspond to modules over a factor algebra of the associated path algebra. This way we get a concrete description of the modules in terms of vector spaces together with linear transformations. This is particularly effective in describing the simple, projective and injective modules. We show that any finite dimensional basic fe-algebra is given by a quiver with relations when k is algebraically closed.