Frobenius Algebras I: Basic Representation Theory (EMS Textbooks in Mathematics)

✍ Scribed by Andrzej Skowronski, Kunio Yamagata

• Publisher: European Mathematical Society/American Mathematical Society
• Year: 2011
• Tongue: English
• Leaves: 662
• Series: EMS Textbooks in Mathematics
• Category: Library

✦ Synopsis

This is the first of two volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras. The first part of the book serves as a general introduction to basic results and techniques of the modern representation theory of finite dimensional associative algebras over fields, including the Morita theory of equivalences and dualities and the Auslander-Reiten theory of irreducible morphisms and almost split sequences. The second part is devoted to fundamental classical and recent results concerning the Frobenius algebras and their module categories. Moreover, the prominent classes of Frobenius algebras, the Hecke algebras of Coxeter groups, and the finite dimensional Hopf algebras over fields are exhibited. This volume is self contained and the only prerequisite is a basic knowledge of linear algebra. It includes complete proofs of all results presented and provides a rich supply of examples and exercises. The text is primarily addressed to graduate students starting research in the representation theory of algebras as well as mathematicians working in other fields. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

✦ Table of Contents

Contents
Introduction
Chapter I. Algebras and modules
1 Algebras
2 Representations of algebras and modules
3 The Jacobson radical
4 The Krull–Schmidt theorem
5 Semisimple modules
6 Semisimple algebras
7 The Jordan–Hölder theorem
8 Projective and injective modules
9 Hereditary algebras
10 Nakayama algebras
11 The Grothendieck group and the Cartan matrix
12 Exercises
Chapter II. Morita theory
1 Categories and functors
2 Bimodules
3 Tensor products of modules
4 Adjunctions and natural isomorphisms
5 Progenerators
6 Morita equivalence
7 Morita–Azumaya duality
8 Exercises
Chapter III. Auslander–Reiten theory
1 The radical of a module category
2 The Harada–Sai lemma
3 The space of extensions
4 The Auslander–Reiten translations
5 The Nakayama functors
6 The Auslander–Reiten formulas
7 Irreducible and almost split homomorphisms
8 Almost split sequences
9 The Auslander–Reiten quiver
10 The Auslander theorem
11 The Bautista–Smalø theorem
12 Exercises
Chapter IV. Selfinjective algebras
1 The Frobenius theorem
2 The Brauer–Nesbitt–Nakayama theorems
3 Frobenius algebras
4 Symmetric algebras
5 Simple algebras
6 The Nakayama theorems
7 Non-Frobenius selfinjective algebras
8 The syzygy functors
9 The higher extension spaces
10 Periodic modules
11 Periodic algebras
12 The Green–Snashall–Solberg theorems
13 Dynkin and Euclidean graphs
14 Canonical mesh algebras of Dynkin type
15 The Riedtmann–Todorov theorem
16 Exercises
Chapter V. Hecke algebras
1 Finite reflection groups
2 Coxeter graphs
3 The Coxeter theorems
4 The Iwahori theorem
5 Hecke algebras
6 Exercises
Chapter VI. Hopf algebras
1 Coalgebras
2 Hopf algebras
3 The Larson–Sweedler theorems
4 The Radford theorem
5 The Fischman–Montgomery–Schneider formula
6 The module category
7 Exercises
Bibliography
Index