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Über dieses Buch

This book is intended to serve as a textbook for a course in Representation Theory of Algebras at the beginning graduate level. The text has two parts. In Part I, the theory is studied in an elementary way using quivers and their representations. This is a very hands-on approach and requires only basic knowledge of linear algebra. The main tool for describing the representation theory of a finite-dimensional algebra is its Auslander-Reiten quiver, and the text introduces these quivers as early as possible. Part II then uses the language of algebras and modules to build on the material developed before. The equivalence of the two approaches is proved in the text. The last chapter gives a proof of Gabriel’s Theorem. The language of category theory is developed along the way as needed.

Inhaltsverzeichnis

Frontmatter

Quivers and Their Representations

Frontmatter

Chapter 1. Representations of Quivers

Abstract
In this chapter, we introduce the concept of quiver representations and their morphisms, discuss direct sums, kernels, and cokernels, and study short exact sequences of quiver representations. We also introduce some basic notions of category theory.
Ralf Schiffler

Chapter 2. Projective and Injective Representations

Abstract
Projective representations and injective representations are key concepts in representation theory. A representation P is called projective if the functor Hom(P, −) maps surjective morphisms to surjective morphisms. Dually a representation I is called injective if the functor Hom(−, I) maps injective morphisms to injective morphisms.
Ralf Schiffler

Chapter 3. Examples of Auslander–Reiten Quivers

Abstract
We have already pointed out in Sect.1.5 that Auslander–Reiten quivers provide a threefold information about the representation theory of the quiver, namely the indecomposable representations, the irreducible morphisms, and the almost split sequences—these in turn should be thought of the building blocks of arbitrary representations, morphisms, and short exact sequences, respectively.
Ralf Schiffler

Path Algebras

Frontmatter

Chapter 4. Algebras and Modules

Abstract
This chapter is an introduction to k-algebras and their modules, where k is an algebraically closed field. Since every algebra is a ring, we will often use certain notions from ring theory, like ideals and radicals. We introduce these notions in the first section. In the second and the third section we define k-algebras and their modules and present examples and basic properties. In the fourth section, we study the direct sum decomposition of a k-algebra (as a module over itself) determined by a choice of a complete set of primitive orthogonal idempotents \(e_{1},\ldots,e_{n}\). For the path algebra of a quiver Q, we are already familiar with this construction, namely the idempotent e i corresponds to the constant path at the vertex i in Q and the direct sum decomposition of the algebra corresponds to the direct sum of all indecomposable projective representations P(i). In the fifth section, we prove a useful criterion for the indecomposability of a module M. In fact, we show that M is indecomposable if and only if the algebra of all endomorphisms of M is a local algebra.
Ralf Schiffler

Chapter 5. Bound Quiver Algebras

Abstract
A bound quiver algebra is the quotient of a path algebra kQ by an ideal I which is required to satisfy a certain admissibility condition. Bound quiver algebras play a central role in representation theory, since, for any finite-dimensional algebra A over an algebraically closed field k, the category mod A is equivalent to the category mod kQI, for some bound quiver algebra kQI.
Ralf Schiffler

Chapter 6. New Algebras from Old

Abstract
In this chapter, we present several popular constructions for algebras, each one in a separate section. We introduce tilted algebras, trivial extensions, cluster-tilted algebras, triangular matrix algebras, and one-point extensions.
Ralf Schiffler

Chapter 7. Auslander–Reiten Theory

Abstract
Recall that the goal of representation theory is to classify the indecomposable modules and the morphisms between them. The Auslander–Reiten quiver is a first approximation of the module category. If the quiver is of finite representation type, then the Auslander–Reiten quiver gives a complete picture of the module category.
Ralf Schiffler

Chapter 8. Quadratic Forms and Gabriel’s Theorem

Abstract
The main goal of this chapter is to prove that the number of isoclasses of indecomposable representations of a connected quiver Q is finite if and only if Q is of Dynkin type \(\mathbb{A}, \mathbb{D}\) or \(\mathbb{E}\). The proof we are presenting uses the classification of positive definite integral quadratic forms associated to graphs and also a little algebraic geometry. For a different proof, using tilting theory, see [8, VII.5].
Ralf Schiffler

Backmatter

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