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2015 | Buch

Introduction to the Representation Theory of Algebras

verfasst von: Michael Barot

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book gives a general introduction to the theory of representations of algebras. It starts with examples of classification problems of matrices under linear transformations, explaining the three common setups: representation of quivers, modules over algebras and additive functors over certain categories. The main part is devoted to (i) module categories, presenting the unicity of the decomposition into indecomposable modules, the Auslander–Reiten theory and the technique of knitting; (ii) the use of combinatorial tools such as dimension vectors and integral quadratic forms; and (iii) deeper theorems such as Gabriel‘s Theorem, the trichotomy and the Theorem of Kac – all accompanied by further examples.
Each section includes exercises to facilitate understanding. By keeping the proofs as basic and comprehensible as possible and introducing the three languages at the beginning, this book is suitable for readers from the advanced undergraduate level onwards and enables them to consult related, specific research articles.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Matrix Problems

Abstract

In this chapter certain classification problems are solved. In each case an equivalence relation on the set of all matrices or the set of all matrices with a given subdivision is given and a “normal form” is sought: that is a family of matrices such that of each equivalence class one and only matrix, the representative, is chosen.

There is little theory within this chapter as the main goal is to establish the normal form of certain examples: the first ones are taken from linear algebra, then some more complicated are tackled. All of them are important in the theory and will be used frequently in the book for illustration. The phenomena which can be observed is that there are three distinct cases of normal forms: either it is a finite list, or it is an infinite but complete list, where for each size of matrices almost all members of the normal form are parametrized by one variable, or there exists one size of matrices for which there are representatives which are parametrized by two variables.

The technique applied to obtain the normal form is simple reduction using the allowed row and column transformations of the matrices. It is a powerful machinery and gives a fast start. But it also has its drawbacks: first it is quite easy to oversee errors and the technique delivers just a list without any internal structure.

Michael Barot

Chapter 2. Representations of Quivers

Abstract

In this chapter a new language is introduced to study the examples of matrix problems: that of representations of quivers. This approach leads naturally to a more sophisticated language known as categories and functors and large part of the chapter is devoted to the development of this new language. The benefit of it will be that the list of “normal forms” will be enhanced by some internal structure. At the end a the important example of a linear quiver is studied.

Michael Barot

Chapter 3. Algebras

Abstract

Representations of quivers is a modern language which now will be connected to the classic one of modules over an algebra. This translation will take up the whole chapter. With this three different languages are developed for the same thing, each of which with a distinctive flavour. All three languages have their own advantages and it is convenient to be able to switch freely between them as in the literature all three of them are used.

As we will see, it is quite easy to associate an algebra to a quiver, but the converse is rather more demanding and requires the development of certain concepts: some elements of an algebra called “idempotents” and others “radicals”. This are concepts, which also in the language of categories and functors play an important role.

Michael Barot

Chapter 4. Module Categories

Abstract

The focus is now changed towards modules. It is shown, that in our context the decomposition of a module into indecomposable summands is essentially unique. Several categorical notions for modules are developed: projective and injective, simple and semisimple modules are defined. In each case a full characterization of indecomposable modules with that particular property is achieved. This is a first step towards understanding the structure of module categories.

Michael Barot

Chapter 5. Elements of Homological Algebra

Abstract

Homological algebra is one of the most important tools if not the most important of all for the theory of representations of algebras. It gathers information of different points of views about its main subject, which is that of short exact sequences. First an equivalence relation is established on short exact sequences with fixed end terms. Then a left and a right multiplication with certain morphisms are defined and it is shown that this two multiplications are compatible. This enables to define a structure of vector space on the equivalence classes of short exact sequences with fixed end terms. Finally, for each module and each short exact sequence two so-called “long exact” sequences are associated. This makes it possible to compare the extension spaces for different end terms. Homological algebra prepares the basics for the second step for understanding the structure of module categories, which is called Auslander-Reiten theory and developed in the next chapter.

Michael Barot

Chapter 6. The Auslander-Reiten Theory

Abstract

In this chapter the insight into the structure of module category is deepened. One section is devoted to the example of the Kronecker algebra. A series of notions and results named after Auslander and Reiten is presented, all of them constitute the Auslander-Reiten theory: The “Auslander-Reiten translate”, which associates to every indecomposable non-projective module an indecomposable non-injective module and the the “Auslander Reiten sequences”, which are short exact sequences and in a certain sense minimal among the non-split sequences. The Theorem of Auslander-Reiten concerns the structure of module categories and yield a remarkable deep insight. Since it is valid in full generality it can be considered as the crown jewel of representation theory. It makes it possible to calculate combinatorially, via a process called “knitting” studied in the next chapter, important information about certain parts—and in some cases all of it—of the module category over some algebra.

Michael Barot

Chapter 7. Knitting

Abstract

The Auslander-Reiten Theorem can be translated into a combinatorial technique called knitting. It yields in many concrete cases large parts or all of the Auslander-Reiten quiver including the structure of the contained indecomposable modules. The development of the knitting technique takes up the main part of this chapter and will exhibited at concrete examples. It underlines the importance of combinatorial invariants, which will be studied with more detail in the next chapter. At the end an example will be discussed, which shows that the technique is not universal and exhibits how it may fail.

Michael Barot

Chapter 8. Combinatorial Invariants

Abstract

The reduction from modules to dimension vectors in the knitting can not only be seen as a loss of information but also as a gain in simplicity. Several useful tools are presented for working with dimension vectors: The first of them, called Grothendieck group, is based on homological algebra. The second linearizes the Auslander-Reiten translation as an action on the Grothendieck group and is called Coxeter transformation. The third is a quadratic form with integer coefficients. It plays a crucial role for characterizing the dimension vector of indecomposable modules. The combinatorial description of those dimension vectors is resumed in the definition of a root.

Michael Barot

Chapter 9. Indecomposables and Dimensions

Abstract

Some of the deepest results in the theory of representations of algebras are presented within this chapter. All of them deal with the representation type, in other words, with the amount of non-isomorphic indecomposable modules with a fixed dimension vector. Several results will only be indicated without proof.

First the two Brauer-Thrall conjectures are discussed, but only the first is proven. Both of them deal with the various possibilities of having finitely or infinitely many non-isomorphic representations of a fixed dimension. Next, Gabriel’s Theorem is proved. It yields a complete characterization of those algebras which among the hereditary algebras are of finite representation type. The role of the associated quadratic form will play a crucial role.

Then Kac’s Theorem will be indicated and the distinction between the tame and wild representation type is discussed. It is shown why in the wild case a classification of all indecomposable modules must seem a hopeless endeavour. At the end a geometric approach is presented to consider the classification problem afresh.

Michael Barot

Backmatter

Titel: Introduction to the Representation Theory of Algebras
verfasst von: Michael Barot
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-11475-0
Print ISBN: 978-3-319-11474-3
DOI: https://doi.org/10.1007/978-3-319-11475-0

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Matrix Problems

Chapter 2. Representations of Quivers

Chapter 3. Algebras

Chapter 4. Module Categories

Chapter 5. Elements of Homological Algebra

Chapter 6. The Auslander-Reiten Theory

Chapter 7. Knitting

Chapter 8. Combinatorial Invariants

Chapter 9. Indecomposables and Dimensions

Backmatter

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