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Basic representation theory of algebras / Ibrahim Assem, Flávio U. Coelho.

By: Contributor(s): Material type: TextLanguage: English Series: Graduate texts in mathematics ; 283.Publication details: Cham : Springer, 2020.Description: 318 p. : illustrations (black and white) ; 24 cmISBN:
  • 9783030351182
  • 9783030991401
Subject(s): Additional physical formats: Print version:: Basic Representation Theory of AlgebrasDDC classification:
  • 512.2 23 ASS
Online resources:
Contents:
Intro -- Preface -- Contents -- I Modules, algebras and quivers -- I.1 Modules over finite dimensional algebras -- I.1.1 Algebras and modules -- I.1.2 The radical and indecomposability -- I.1.3 Idempotents, projectives and injectives -- I.1.4 The Grothendieck group and composition series -- Exercises for Section I.1 -- I.2 Quivers and algebras -- I.2.1 Path algebras and their quotients -- I.2.2 Quiver of a finite dimensional algebra -- I.2.3 Projective, injective and simple modules -- I.2.4 Nakayama algebras -- I.2.5 Hereditary algebras -- I.2.6 The Kronecker algebra.
Exercises for Section I.2 -- II The radical and almost split sequences -- II.1 The radical of a module category -- II.1.1 Categorical framework -- II.1.2 Defining the radical of ̀́39̀42̀"̇"613À̀45̀47̀""603AmodA -- II.1.3 Characterisations of the radical -- Exercises for Section II.1 -- II.2 Irreducible morphisms and almost split morphisms -- II.2.1 Irreducible morphisms -- II.2.2 Almost split and minimal morphisms -- II.2.3 Almost split sequences -- Exercises for Section II.2 -- II.3 The existence of almost split sequences -- II.3.1 The functor category ̀́39̀42̀"̇"613À̀45̀47̀""603AFunA.
II.3.2 Simple objects in ̀́39̀42̀"̇"613À̀45̀47̀""603AFunA -- II.3.3 Projective resolutions of simple functors -- Exercises for Section II.3 -- II.4 Factorising radical morphisms -- II.4.1 Higher powers of the radical -- II.4.2 Factorising radical morphisms -- II.4.3 Paths -- Exercises for Section II.4 -- III Constructing almost split sequences -- III.1 The Auslander-Reiten translations -- III.1.1 The stable categories -- III.1.2 Morphisms between projectives and injectives -- III.1.3 The Auslander-Reiten translations -- III.1.4 Properties of the Auslander-Reiten translations.
Exercises for Section III.1 -- III.2 The Auslander-Reiten formulae -- III.2.1 Preparatory lemmata -- III.2.2 Proof of the formulae -- III.2.3 Application to almost split sequences -- III.2.4 Starting to compute almost split sequences -- Exercises for Section III.2 -- III.3 Examples of constructions of almost split sequences -- III.3.1 The general case -- III.3.2 Projective-injective middle term -- III.3.3 Almost split sequences for Nakayama algebras -- III.3.4 Examples of almost split sequences over bound quiver algebras -- Exercises for Section III.3.
III.4 Almost split sequences over quotient algebras -- III.4.1 The change of rings functors -- III.4.2 The embedding of mod B inside mod A -- III.4.3 Split-by-nilpotent extensions -- Exercises for Section III.4 -- IV The Auslander-Reiten quiver of an algebra -- IV.1 The Auslander-Reiten quiver -- IV.1.1 The space of irreducible morphisms -- IV.1.2 Defining the Auslander-Reiten quiver -- IV.1.3 Examples and construction procedures -- IV.1.4 The combinatorial structure of the Auslander-Reiten quiver -- IV.1.5 The use of Auslander-Reiten quivers -- Exercises for Section IV.1.
Summary: This textbook introduces the representation theory of algebras by focusing on two of its most important aspects: the Auslander-Reiten theory and the study of the radical of a module category. It starts by introducing and describing several characterisations of the radical of a module category, then presents the central concepts of irreducible morphisms and almost split sequences, before providing the definition of the Auslander-Reiten quiver, which encodes much of the information on the module category. It then turns to the study of endomorphism algebras, leading on one hand to the definition of the Auslander algebra and on the other to tilting theory. The book ends with selected properties of representation-finite algebras, which are now the best understood class of algebras. Intended for graduate students in representation theory, this book is also of interest to any mathematician wanting to learn the fundamentals of this rapidly growing field. A graduate course in non-commutative or homological algebra, which is standard in most universities, is a prerequisite for readers of this book.
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General Books CUTN Central Library Sciences Non-fiction 512.2 ASS (Browse shelf(Opens below)) Available 54643

Description based upon print version of record.

IV.2 Postprojective and preinjective components.

Intro -- Preface -- Contents -- I Modules, algebras and quivers -- I.1 Modules over finite dimensional algebras -- I.1.1 Algebras and modules -- I.1.2 The radical and indecomposability -- I.1.3 Idempotents, projectives and injectives -- I.1.4 The Grothendieck group and composition series -- Exercises for Section I.1 -- I.2 Quivers and algebras -- I.2.1 Path algebras and their quotients -- I.2.2 Quiver of a finite dimensional algebra -- I.2.3 Projective, injective and simple modules -- I.2.4 Nakayama algebras -- I.2.5 Hereditary algebras -- I.2.6 The Kronecker algebra.

Exercises for Section I.2 -- II The radical and almost split sequences -- II.1 The radical of a module category -- II.1.1 Categorical framework -- II.1.2 Defining the radical of ̀́39̀42̀"̇"613À̀45̀47̀""603AmodA -- II.1.3 Characterisations of the radical -- Exercises for Section II.1 -- II.2 Irreducible morphisms and almost split morphisms -- II.2.1 Irreducible morphisms -- II.2.2 Almost split and minimal morphisms -- II.2.3 Almost split sequences -- Exercises for Section II.2 -- II.3 The existence of almost split sequences -- II.3.1 The functor category ̀́39̀42̀"̇"613À̀45̀47̀""603AFunA.

II.3.2 Simple objects in ̀́39̀42̀"̇"613À̀45̀47̀""603AFunA -- II.3.3 Projective resolutions of simple functors -- Exercises for Section II.3 -- II.4 Factorising radical morphisms -- II.4.1 Higher powers of the radical -- II.4.2 Factorising radical morphisms -- II.4.3 Paths -- Exercises for Section II.4 -- III Constructing almost split sequences -- III.1 The Auslander-Reiten translations -- III.1.1 The stable categories -- III.1.2 Morphisms between projectives and injectives -- III.1.3 The Auslander-Reiten translations -- III.1.4 Properties of the Auslander-Reiten translations.

Exercises for Section III.1 -- III.2 The Auslander-Reiten formulae -- III.2.1 Preparatory lemmata -- III.2.2 Proof of the formulae -- III.2.3 Application to almost split sequences -- III.2.4 Starting to compute almost split sequences -- Exercises for Section III.2 -- III.3 Examples of constructions of almost split sequences -- III.3.1 The general case -- III.3.2 Projective-injective middle term -- III.3.3 Almost split sequences for Nakayama algebras -- III.3.4 Examples of almost split sequences over bound quiver algebras -- Exercises for Section III.3.

III.4 Almost split sequences over quotient algebras -- III.4.1 The change of rings functors -- III.4.2 The embedding of mod B inside mod A -- III.4.3 Split-by-nilpotent extensions -- Exercises for Section III.4 -- IV The Auslander-Reiten quiver of an algebra -- IV.1 The Auslander-Reiten quiver -- IV.1.1 The space of irreducible morphisms -- IV.1.2 Defining the Auslander-Reiten quiver -- IV.1.3 Examples and construction procedures -- IV.1.4 The combinatorial structure of the Auslander-Reiten quiver -- IV.1.5 The use of Auslander-Reiten quivers -- Exercises for Section IV.1.

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This textbook introduces the representation theory of algebras by focusing on two of its most important aspects: the Auslander-Reiten theory and the study of the radical of a module category. It starts by introducing and describing several characterisations of the radical of a module category, then presents the central concepts of irreducible morphisms and almost split sequences, before providing the definition of the Auslander-Reiten quiver, which encodes much of the information on the module category. It then turns to the study of endomorphism algebras, leading on one hand to the definition of the Auslander algebra and on the other to tilting theory. The book ends with selected properties of representation-finite algebras, which are now the best understood class of algebras. Intended for graduate students in representation theory, this book is also of interest to any mathematician wanting to learn the fundamentals of this rapidly growing field. A graduate course in non-commutative or homological algebra, which is standard in most universities, is a prerequisite for readers of this book.

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