Algebra VIII: Representations of finite-dimensional algebras. Transl. from the Russian
type: Item
instance of: scholarly article
title: Algebra VIII: Representations of finite-dimensional algebras. Transl. from the Russian
zbMATH Open document ID: 0839.16001
author: A. V. Rojter
author: P. Gabriel
published in: Encyclopaedia of Mathematical Sciences
publication date: 1993-01-23T00:00:00Z
review text: The main aim of this book is to present an overview of old and new results on representations of finite-dimensional algebras. Representation theory has developed much in the last twenty-five years and the lack of books containing the fundamentals of the theory has been a difficulty for those who want to start a research in this line. This book is very welcome because it presents some basic material and at the same time it presents some new insights of the theory. One point that should be mentioned is that the authors have preferred to use some new terminology instead of the one normally used in the literature so far. This may lead to some confusion for those beginners in the area. The contents of the book are: Section 1. Matrix problems. Here, the authors include a discussion of some classical matrix problems which are related to problems in representation theory of algebras. Section 2. Algebras, modules and categories. It contains some basic notions and establishes some terminology. Section 3. Radical, decomposition and aggregates. The objective of this section is to delimit a broad class of additive categories whose semigroups are free commutative. Section 4. Finitely spaced modules. The aim here is to present methods which reduce representations of algebras to matrix problems. Section 5. Finitely represented posets. Using an algorithm developed in the previous section, they characterize the posets which have finitely many isoclasses of indecomposable representations. Section 6. Roots. Here, the authors discuss the interplay of representation theory and integral quadratic forms. Section 7. Representation of quivers. This section is devoted to study further the relationship between representations of quivers and modules through a quadratic form. In particular, a proof of the classical Gabriel theorem is included. Section 8. Spectroids, quivers, coherence. Section 9. Almost split sequences. They discuss here the fundamental notion of almost split sequences, introduc…
Mathematics Subject Classification ID: 16-02
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E12
zbMATH DE Number: 107904
zbMATH Keywords: Algebra VIII
zbMATH Keywords: Finite-dimensional algebras
zbMATH Keywords: representations of finite-dimensional algebras
zbMATH Keywords: matrix problems
zbMATH Keywords: posets
zbMATH Keywords: indecomposable representations
zbMATH Keywords: integral quadratic forms
zbMATH Keywords: representations of quivers
zbMATH Keywords: almost split sequences
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: finite-dimensional representations
zbMATH Keywords: Dynkin quivers
reviewed by: Flávio U. Coelho
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 107904