This book offers an original introduction to the representation theory of algebras, suitable for beginning researchers in algebra. It includes many results and techniques not usually covered in introductory books, some of which appear here for the first time in book form. The exposition employs methods from linear algebra (spectral methods and quadratic forms), as well as categorical and homological methods (module categories, Galois coverings, Hochschild cohomology) to present classical aspects of ring theory under new light. This includes topics such as rings with several objects, the Harada–Sai lemma, chain conditions, and Auslander–Reiten theory. Noteworthy and significant results covered in the book include the Brauer–Thrall conjectures, Drozd’s theorem, and criteria to distinguish tame from wild algebras. This text may serve as the basis for a second graduate course in algebra or as an introduction to research in the field of representation theory of algebras. The originality of the exposition and the wealth of topics covered also make it a valuable resource for more established researchers.
Author(s): José-Antonio de la Peña
Series: Algebra and Applications, 30
Publisher: Springer
Year: 2022
Language: English
Pages: 239
City: Cham
Preface
References
Contents
1 Introduction and First Examples
1.1 General Algebraic Problems
1.2 Some Terminology on Quadratic Forms
1.3 Geometrical and Linear Aspects of Unit Forms
1.4 Fundamental Examples
1.5 Some Categorical Notions
1.6 A Quick Overview on Algebraic Geometry
References
2 A Categorical Approach
2.1 Morphisms Between Indecomposable Modules
2.2 Harada-Sai Sequences
2.3 The Remak-Krull-Schmidt-Azumaya Decomposition
2.4 First Elements of Auslander-Reiten Theory
2.5 The Category of Additive Functors
2.6 First Brauer-Thrall Conjecture
References
3 Constructive Methods
3.1 The Lattice of Ideals
3.2 Other Brauer-Thrall Conjectures
3.3 The Post-Projective Components of a Triangular Algebra
3.4 A Generalization of Jacobi's Criterion
3.5 The Tits Quadratic Form
References
4 Spectral Methods in Representation Theory
4.1 Hereditary Algebras and the Coxeter Transformation
4.2 Coxeter Spectrum in the Study of Indecomposable Modules
4.3 The Canonical Representation of a Group of Symmetries
4.4 The Automorphism Group of a Graph
4.5 Canonical Algebras
4.6 Self-Injective Algebras
4.7 Further Spectral Properties
References
5 Group Actions on Algebras and Module Categories
5.1 The Group of Automorphisms of an Algebra
5.2 Constructions of Algebras Associated to Groups of Automorphisms (Coverings and Smash Products)
5.3 Coverings and the Representation Type of an Algebra
5.4 Balanced Functors
5.5 Galois Coverings of Algebras
5.6 Cycle-Finite Algebras
References
6 Reflections and Weyl Groups
6.1 Vinberg's Characterization of Dynkin Diagrams
6.2 M-Matrices and Positivity
6.3 Coxeter Matrices and Weyl Group
6.4 Very Sharp Reflections
6.5 On the Decomposition of the Coxeter Polynomial of an Algebra of Cyclotomic Type
References
7 Simply Connected Algebras
7.1 The Fundamental Group of a Triangular Algebra
7.2 A Separation Property
7.3 Strongly Simply Connected Algebras
7.4 Tame Quasi-Tilted Algebras
7.5 Weakly Separating Families of Coils
References
8 Degenerations of Algebras
8.1 Deformation Theory of Algebras: A Geometric Approach
8.2 Degenerations of Algebras: A Homological Interpretation
8.3 Tame and Wild Algebras: Definitions and Degeneration Property
8.4 The Tits Quadratic Form and the Degeneration of Algebras
References
9 Further Comments
9.1 More on Dichotomy Problems
9.2 Some Historical Notes
References
Index
