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E-grāmata: Algebras and Representation Theory

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Algebras and Representation TheoryPalielināt
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This carefully written textbook provides an accessible introduction to the representation theory of algebras, including representations of quivers.

The book starts with basic topics on algebras and modules, covering fundamental results such as the Jordan-Hölder theorem on composition series, the Artin-Wedderburn theorem on the structure of semisimple algebras and the Krull-Schmidt theorem on indecomposable modules. The authors then go on to study representations of quivers in detail, leading to a complete proof of Gabriel's celebrated theorem characterizing the representation type of quivers in terms of Dynkin diagrams.





Requiring only introductory courses on linear algebra and groups, rings and fields, this textbook is aimed at undergraduate students. With numerous examples illustrating abstract concepts, and including more than 200 exercises (with solutions to about a third of them), the book provides an example-driven introduction suitable for self-study and use alongside lecture courses.

Recenzijas

The book under review is a text-book for higher undergraduate mathematics students or graduate students who have previous knowledge of results from linear algebra, and basic properties of rings and groups. It is also useful for non-experts (in representation theory of quivers), they may benefit from this book in several ways: by examining the numerous worked examples, or by working out the many exercises. (Bin Zhu, zbMATH 1429.16001, 2020)

1 Algebras
1(28)
1.1 Definition and Examples
1(8)
1.1.1 Division Algebras
4(2)
1.1.2 Group Algebras
6(1)
1.1.3 Path Algebras of Quivers
6(3)
1.2 Subalgebras, Ideals and Factor Algebras
9(4)
1.3 Algebra Homomorphisms
13(6)
1.4 Some Algebras of Small Dimensions
19(10)
2 Modules and Representations
29(32)
2.1 Definition and Examples
29(4)
2.2 Modules for Polynomial Algebras
33(2)
2.3 Submodules and Factor Modules
35(5)
2.4 Module Homomorphisms
40(7)
2.5 Representations of Algebras
47(14)
2.5.1 Representations of Groups vs. Modules for Group Algebras
51(2)
2.5.2 Representations of Quivers vs. Modules for Path Algebras
53(8)
3 Simple Modules and the Jordan--Holder Theorem
61(24)
3.1 Simple Modules
61(2)
3.2 Composition Series
63(6)
3.3 Modules of Finite Length
69(2)
3.4 Finding All Simple Modules
71(8)
3.4.1 Simple Modules for Factor Algebras of Polynomial Algebras
73(2)
3.4.2 Simple Modules for Path Algebras
75(2)
3.4.3 Simple Modules for Direct Products
77(2)
3.5 Schur's Lemma and Applications
79(6)
4 Semisimple Modules and Semisimple Algebras
85(18)
4.1 Semisimple Modules
85(6)
4.2 Semisimple Algebras
91(5)
4.3 The Jacobson Radical
96(7)
5 The Structure of Semisimple Algebras: The Artin--Wedderburn Theorem
103(14)
5.1 A Special Case
104(2)
5.2 Towards the Artin-Wedderburn Theorem
106(5)
5.3 The Artin-Wedderburn Theorem
111(6)
6 Semisimple Group Algebras and Maschke's Theorem
117(12)
6.1 Maschke's Theorem
117(3)
6.2 Some Consequences of Maschke's Theorem
120(2)
6.3 One-Dimensional Simple Modules and Commutator Groups
122(2)
6.4 Artin--Wedderburn Decomposition and Conjugacy Classes
124(5)
7 Indecomposable Modules
129(14)
7.1 Indecomposable Modules
129(4)
7.2 Fitting's Lemma and Local Algebras
133(4)
7.3 The Krull--Schmidt Theorem
137(6)
8 Representation Type
143(20)
8.1 Definition and Examples
143(6)
8.2 Representation Type for Group Algebras
149(14)
9 Representations of Quivers
163(22)
9.1 Definitions and Examples
163(6)
9.2 Representations of Subquivers
169(3)
9.3 Stretching Quivers and Representations
172(5)
9.4 Representation Type of Quivers
177(8)
10 Diagrams and Roots
185(18)
10.1 Dynkin Diagrams and Euclidean Diagrams
185(3)
10.2 The Bilinear Form and the Quadratic Form
188(9)
10.3 The Coxeter Transformation
197(6)
11 Gabriel's Theorem
203(36)
11.1 Reflecting Quivers and Representations
203(15)
11.1.1 The Reflection Σ+j at a Sink
206(4)
11.1.2 The Reflection Σ-j at a Source
210(5)
11.1.3 Compositions and Σ-j Σ+j Σ+j Σ-j
215(3)
11.2 Quivers of Infinite Representation Type
218(5)
11.3 Dimension Vectors and Reflections
223(4)
11.4 Finite Representation Type for Dynkin Quivers
227(12)
12 Proofs and Background
239(26)
12.1 Proofs on Reflections of Representations
239(7)
12.1.1 Invariance Under Direct Sums
240(3)
12.1.2 Compositions of Reflections
243(3)
12.2 All Euclidean Quivers Are of Infinite Representation Type
246(11)
12.2.1 Quivers of Type E6 Have Infinite Representation Type
247(3)
12.2.2 Quivers of Type E7 Have Infinite Representation Type
250(3)
12.2.3 Quivers of Type E8 Have Infinite Representation Type
253(4)
12.3 Root Systems
257(3)
12.4 Morita Equivalence
260(5)
A Induced Modules for Group Algebras 265(6)
B Solutions to Selected Exercises 271(26)
Index 297
Karin Erdmann's research focus lies on representation theory of finite groups, and finite-dimensional algebras. She has written many research articles, and is the author of a research monograph and a textbook.

Thorsten Holm is Professor of Mathematics at Leibniz Universität Hannover. His research interests include representation theory of finite groups and finite-dimensional algebras, and algebraic combinatorics.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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