Author | Drozd, Yurij A. author |
---|---|
Title | Finite Dimensional Algebras [electronic resource] / by Yurij A. Drozd, Vladimir V. Kirichenko |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1994 |
Connect to | http://dx.doi.org/10.1007/978-3-642-76244-4 |
Descript | XIII, 249p. online resource |
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-Hรถlder Theorem -- 1.6 Direct Sums -- 1.7 Endomorphisms. The Peirce Decomposition -- Exerises to Chapter 1 -- 2. Semisimple Algebras -- 2.1 Schurโs 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 Maschkeโs Theorem -- 7.2 Number and Dimensions of Irreducible Representations -- 7.3 Characters -- 7.4 Algebraic Integers -- 7.5 Tensor Products of Representations -- 7.6 Burnsideโs 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