Elements of the representation theory of associative algebras. Vol. 1: Techniques of representation theory.

*(English)*Zbl 1092.16001
London Mathematical Society Student Texts 65. Cambridge: Cambridge University Press (ISBN 0-521-58631-3/pbk; 0-521-58423-X/hbk). ix, 458 p. (2006).

The representation theory of associative algebras, especially of Artin algebras, has developed rapidly in the last 35 years. The present book is one of very few textbooks on this subject, and is concentrated on various important techniques of the modern representation theory. The main emphasis is made on the Auslander-Reiten theory and tilting theory.

The book consists of nine chapters and an appendix. The first three chapters are introductory and devoted to the general description of such notions as algebras, modules, representations, quivers. Section IV introduces the reader to almost-split sequences (including the functorial approach to them) and the Auslander-Reiten theory in general. More advanced questions related to this theory are treated in Section IX, where the reader will find a lot of information on directing modules and postprojective components. Section V is concentrated around various conditions on filtrations of modules, Nakayama algebras and representation finite group algebras. In Section VI the reader will find a rather detailed introduction to tilting theory. This theory led to the definition of what is now known as tilted algebras. Properties of such algebras are described in Section VIII. Section VII contains, in turn, material on representation-finite hereditary algebras, including the theory of integral quadratic forms, techniques of reflection functors and Gabriel’s theorem. Finally, in the Appendix the reader will find a collection of notation and terminology on categories, functors and homology, and also several basic facts from category theory and homological algebra.

This volume can be used both as a textbook for undergraduate and graduate courses and as an encyclopedia for researchers.

The book consists of nine chapters and an appendix. The first three chapters are introductory and devoted to the general description of such notions as algebras, modules, representations, quivers. Section IV introduces the reader to almost-split sequences (including the functorial approach to them) and the Auslander-Reiten theory in general. More advanced questions related to this theory are treated in Section IX, where the reader will find a lot of information on directing modules and postprojective components. Section V is concentrated around various conditions on filtrations of modules, Nakayama algebras and representation finite group algebras. In Section VI the reader will find a rather detailed introduction to tilting theory. This theory led to the definition of what is now known as tilted algebras. Properties of such algebras are described in Section VIII. Section VII contains, in turn, material on representation-finite hereditary algebras, including the theory of integral quadratic forms, techniques of reflection functors and Gabriel’s theorem. Finally, in the Appendix the reader will find a collection of notation and terminology on categories, functors and homology, and also several basic facts from category theory and homological algebra.

This volume can be used both as a textbook for undergraduate and graduate courses and as an encyclopedia for researchers.

Reviewer: Volodymyr Mazorchuk (Uppsala)

##### MSC:

16-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras |

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

16G10 | Representations of associative Artinian rings |

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

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

16B50 | Category-theoretic methods and results in associative algebras (except as in 16D90) |

16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |