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**Algebra VIII: Representations of finite-dimensional algebras. Transl. from the Russian.**
*(English)*
Zbl 0839.16001

Encyclopaedia of Mathematical Sciences. 73. Berlin: Springer-Verlag. 177 p. (1992).

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, introduced by M. Auslander and I. Reiten. Such sequences are, roughly speaking, “minimal” non-split short exact sequences. Section 10. Postprojective components. It is devoted to the study of some components of the so-called representation quiver (also called Auslander-Reiten quiver because it is defined in terms of almost split sequences). Section 11. Representations of tame algebras. The aim of this section is to classify the finite-dimensional representations of the extended Dynkin quivers. They are tame because for any fixed dimension-function, the corresponding isoclasses either are finite or can be one-parametrized up to finitely many exceptions. Section 12. Derivations and tiltings. This section was written by B. Keller and intends to look at some invariants of two spectroids which have the same derived category. Section 13. Multiplicative basis. Section 14. Finitely represented algebras.

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, introduced by M. Auslander and I. Reiten. Such sequences are, roughly speaking, “minimal” non-split short exact sequences. Section 10. Postprojective components. It is devoted to the study of some components of the so-called representation quiver (also called Auslander-Reiten quiver because it is defined in terms of almost split sequences). Section 11. Representations of tame algebras. The aim of this section is to classify the finite-dimensional representations of the extended Dynkin quivers. They are tame because for any fixed dimension-function, the corresponding isoclasses either are finite or can be one-parametrized up to finitely many exceptions. Section 12. Derivations and tiltings. This section was written by B. Keller and intends to look at some invariants of two spectroids which have the same derived category. Section 13. Multiplicative basis. Section 14. Finitely represented algebras.

Reviewer: F.U.Coelho (São Paulo)

### MSC:

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16G20 | Representations of quivers and partially ordered sets |

16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |

16P10 | Finite rings and finite-dimensional associative algebras |

16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |

15A21 | Canonical forms, reductions, classification |

16G30 | Representations of orders, lattices, algebras over commutative rings |

11E04 | Quadratic forms over general fields |

11E12 | Quadratic forms over global rings and fields |