The theory of finite-dimensional algebras is one of the most fundamental domains of modern algebra, applied in several parts of mathematics and in theoretical physics. This book, written by two of the leading researchers in the field and revised and augmented for the English edition, was translated from the Russian by a third leading specialist, who has contributed to it an appendix. The book presents both the basic classical theory and more recent results closely related to current research. The only prior knowledge assumed of the reader is linear algebra and, in places, a little group theory. Each chapter includes series of exercises. The book can be used as a textbook or as an up-to-date reference on the subject.
The theory of finite-dimensional algebras is one of the most fundamental domains of modern algebra, applied in several parts of mathematics and in theoretical physics. This book, written by two of the leading researchers in the field and revised and augmented for the English edition, was translated from the Russian by a third leading specialist, who has contributed to it an appendix. The book presents both the basic classical theory and more recent results closely related to current research. The only prior knowledge assumed of the reader is linear algebra and, in places, a little group theory. Each chapter includes series of exercises. The book can be used as a textbook or as an up-to-date reference on the subject.
Zusammenfassung
The theory of finite-dimensional algebras is one of the most fundamental domains of modern algebra, applied in several parts of mathematics and in theoretical physics. This book, written by two of the leading researchers in the field and revised and augmented for the English edition, was translated from the Russian by a third leading specialist, who has contributed to it an appendix. The book presents both the basic classical theory and more recent results closely related to current research. The only prior knowledge assumed of the reader is linear algebra and, in places, a little group theory. Each chapter includes series of exercises. The book can be used as a textbook or as an up-to-date reference on the subject.
Inhaltsverzeichnis
1. Introduction.- 1.1 Basic Concepts. Examples.- 1.2 Isomorphisms and Homomorphisms. Division Algebras.- 1.3 Representations and Modules.- 1.4 Submodules and Factor Modules. Ideals and Quotient Algebras.- 1.5 The Jordan-Hölder Theorem.- 1.6 Direct Sums.- 1.7 Endomorphisms. The Peirce Decomposition.- Exerises to Chapter 1.- 2. Semisimple Algebras.- 2.1 Schur's Lemma.- 2.2 Semisimple Modules and Algebras.- 2.3 Vector Spaces and Matrices.- 2.4 The Wedderburn-Artin Theorem.- 2.5 Uniqueness of the Decomposition.- 2.6 Representations of Semisimple Algebras.- Exercises to Chapter 2.- 3. The Radical.- 3.1 The Radical of a Module and of an Algebra.- 3.2 Lifting of Idempotents. Principal Modules.- 3.3 Projective Modules and Projective Covers.- 3.4 The Krull-Schmidt Theorem.- 3.5 The Radical of an Endomorphism Algebra.- 3.6 Diagram of an Algebra.- 3.7 Hereditary Algebras.- Exercises to Chapter 3.- 4. Central Simple Algebras.- 4.1 Bimodules.- 4.2 Tensor Products.- 4.3 Central Simple Algebras.- 4.4 Fundamental Theorems of the Theory of Division Algebras.- 4.5 Subfields of Division Algebras. Splitting Fields.- 4.6 Brauer Group. The Frobenius Theorem.- Exercises to Chapter 4.- 5. Galois Theory.- 5.1 Elements of Field Theory.- 5.2 Finite Fields. The Wedderburn Theorem.- 5.3 Separable Extensions.- 5.4 Normal Extensions. The Galois Group.- 5.5 The Fundamental Theorem of Galois Theory.- 5.6 Crossed Products.- Exercises to Chapter 5.- 6. Separable Algebras.- 6.1 Bimodules over Separable Algebras.- 6.2 The Wedderburn-Malcev Theorem.- 6.3 Trace, Norm, Discriminant.- Exercises to Chapter 6.- 7. Representations of Finite Groups.- 7.1 Maschke's Theorem.- 7.2 Number and Dimensions of Irreducible Representations.- 7.3 Characters.- 7.4 Algebraic Integers.- 7.5 Tensor Products ofRepresentations.- 7.6 Burnside's Theorem.- Exercises to Chapter 7.- 8. The Morita Theorem.- 8.1 Categories and Functors.- 8.2 Exact Sequences.- 8.3 Tensor Products.- 8.4 The Morita Theorem.- 8.5 Tensor Algebras and Hereditary Algebras.- Exercises to Chapter 8.- 9. Quasi-Frobenius Algebras.- 9.1 Duality. Injective Modules.- 9.2 Lemma on Separation.- 9.3 Quasi-Frobenius Algebras.- 9.4 Uniserial Algebras.- Exercises to Chapter 9.- 10. Serial Algebras.- 10.1 The Nakayama-Skornjakov Theorem.- 10.2 Right Serial Algebras.- 10.3 The Structure of Serial Algebras.- 10.4 Quasi-Frobenius and Hereditary Serial Algebras.- Exercises to Chapter 10.- 11. Elements of Homological Algebra.- 11.1 Complexes and Homology.- 11.2 Resolutions and Derived Functors.- 11.3 Ext and Tor. Extensions.- 11.4 Homological Dimensions.- 11.5 Duality.- 11.6 Almost Split Sequences.- 11.7 Auslander Algebras.- Exercises to Chapter 11.- References.- A.1 Preliminaries. Standard and Costandard Modules.- A.3 Basic Properties.- A.4 Canonical Constructions.- A.6 Final Remarks.- References to the Appendix.
