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2014 | Buch

Representation Theory

A Homological Algebra Point of View

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Introducing the representation theory of groups and finite dimensional algebras, first studying basic non-commutative ring theory, this book covers the necessary background on elementary homological algebra and representations of groups up to block theory. It further discusses vertices, defect groups, Green and Brauer correspondences and Clifford theory. Whenever possible the statements are presented in a general setting for more general algebras, such as symmetric finite dimensional algebras over a field.

Then, abelian and derived categories are introduced in detail and are used to explain stable module categories, as well as derived categories and their main invariants and links between them. Group theoretical applications of these theories are given – such as the structure of blocks of cyclic defect groups – whenever appropriate. Overall, many methods from the representation theory of algebras are introduced.

Representation Theory assumes only the most basic knowledge of linear algebra, groups, rings and fields and guides the reader in the use of categorical equivalences in the representation theory of groups and algebras. As the book is based on lectures, it will be accessible to any graduate student in algebra and can be used for self-study as well as for classroom use.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Rings, Algebras and Modules
Abstract
In this first chapter we provide the necessary facts in elementary module theory, we define the concept of a representation, and give elementary applications to representations of groups. We also provide a short introduction to the basic concepts leading to homological algebra, as far as it is necessary to understand the elementary modular representation theory of finite groups as it is developed in Chap. 2. We restrict ourselves to a selection of those properties that are going to be used in the sequel and avoid developing the theory in directions which are not explicitly used in later chapters. This way the book remains completely self-contained, without being encyclopedic, and the choice will also allows us to fix a coherent notation throughout.
Alexander Zimmermann
Chapter 2. Modular Representations of Finite Groups

We are now ready to apply the results from the previous chapter to group rings of finite groups. If the order of the group is invertible in the base field, Maschke’s Theorem 1.2.8 tells us that the group ring is semisimple, and semisimple rings are of less interest from the homological algebra point of view. Therefore, were are mostly interested in representations of finite groups \(G\) over fields \(k\) such that the characteristic of \(k\) divides the order of \(G\). In this chapter we will develop the classical part of the theory of these representations.

Alexander Zimmermann
Chapter 3. Abelian and Triangulated Categories

This chapter will provide a short introduction to the language of categories. We emphasize the concepts of abelian and triangulated categories and their immediate properties. We also provide a short introduction to spectral sequences.

Alexander Zimmermann
Chapter 4. Morita Theory
Abstract
We ask for practical conditions for two rings \(A\) and \(B\) to have equivalent module categories \(A\)-\(\textit{Mod}\simeq \) \(B\)-\(\textit{Mod}\). This question was completely solved in the 1950s by K. Morita and we present this result here. As application we give Puig theorem of nilpotent blocks, Gabriel’s theorem of the presentation of finite dimensional algebras by quiver and relations and a short introduction to Picard groups of algebras.
Alexander Zimmermann
Chapter 5. Stable Module Categories
Abstract
Morita equivalences provide a very strong relationship between two rings, and in particular their representation theory. However, one observes in examples similarities between module categories which are not given by a Morita equivalence. Nevertheless, a structural connection is reasonable. One of the possible connections is a stable equivalence. A stable equivalence is the weakest possible equivalence we study. We examine abstract equivalences as well as stable equivalences of Morita type. As application we give the structure theorem of blocks with cyclic defect group and Reiten’s theorem on the invariance of self-injectivity under stable equivalences.
Alexander Zimmermann
Chapter 6. Derived Equivalences

The derived category of an algebra was introduced in Chap. 3. In this Chap. 6 we shall study equivalences between the derived categories of two algebras. The main result is Rickard’s and Keller’s Morita theory for derived categories. We shall study many relations between this concept and stable equivalences. We also provide a long list of invariants under these equivalences and study singular categories as well as Picard groups for derived categories.

Alexander Zimmermann
Backmatter
Metadaten
Titel
Representation Theory
verfasst von
Alexander Zimmermann
Copyright-Jahr
2014
Electronic ISBN
978-3-319-07968-4
Print ISBN
978-3-319-07967-7
DOI
https://doi.org/10.1007/978-3-319-07968-4