Representations of Algebras. Tame and Wild Behavior

Description

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