Representations of algebras : tame and wild behavior / / José Antonio de la Peña

• Record Nr.: UNINA9910620196603321
• Autore: de la Peña José Antonio
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 9783031122880; 9783031122873
• Descrizione fisica: 1 online resource (240 pages)
• Collana: Algebra and Applications ; ; v.30
• Disciplina: 512.2
• Soggetti: Representations of groups; Representacions de grups; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Intro -- 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.