##
**Finite dimensional algebras. With an appendix by Vlastimil Dlab. Transl. from the Russian by Vlastimil Dlab.**
*(English)*
Zbl 0816.16001

Berlin: Springer-Verlag. xiii, 249 p. (1994).

In spite of the fact that the theory of finite dimensional algebras is one of the oldest branches of modern algebra, there are only few modern textbooks and monographs about this. Therefore the present textbook is a welcome addition to the expository literature. A Russian edition appeared in 1980 [Zbl 0469.16001], the present English edition contains additionally a chapter on homological algebra and an Appendix, written by V. Dlab, concerning quasi-hereditary algebras. The book presents both the basic classical theory and more recent results closely related to current research.

Chapter I contains the basic concepts and elementary properties of modules including the Jordan-HĂ¶lder Theorem.

In Chapter II the theory of semisimple algebras is presented, in particular the Wedderburn-Artin Theorem is proved and the characterization of modules over semisimple algebras is given.

Chapter III deals with the radical of modules and algebras. Primary algebras are characterized and basic algebras are studied. Moreover, the concept of the diagram (quiver) of an algebra, which is widely used in representation theory of algebras, is explained. The chapter ends with the classification of hereditary algebras (over an algebraically closed field).

Chapter IV is devoted to central simple algebras. Here the techniques of tensor products and bimodules and fundamental facts of the theory of division algebras can be found, the concept of the Brauer group is discussed and the Frobenius Theorem on finite dimensional algebras over \(\mathbb{R}\) is stated.

Chapter V begins with the study of field theory with emphasis on finite fields, including the Wedderburn Theorem on finite skew-fields. Then, using the machinery of bimodules and tensor products, the foundations of Galois theory are developed. As an application of Galois theory to central simple algebras the construction of crossed products is given.

In Chapter VI some characterizations of separable algebras and the Wedderburn-Malcev Theorem can be found.

Based on the theory of semisimple algebras, in Chapter VII the basic results of classical group theory are developed up to the integral theorems and the Burnside Theorem on the solvability of a group of order \(p^ a q^ b\).

Chapter VIII deals with the Morita Theorem on equivalences of module categories. The necessary background from category theory and the techniques of tensor products and exact sequences are presented. Furthermore in this chapter the concept of the tensor algebra of a bimodule as a generalization of the path algebra is explained.

In Chapters IX and X the classes of quasi-Frobenius algebras, uniserial algebras and generalized uniserial (serial) algebras are studied. These algebras are the best understood non-semisimple ones, however some of the results presented here until now have been available only in journal articles.

Chapter XI gives a short introduction to the methods of homological algebra. Apart from the fundamental concepts the topics of homological dimension, almost split sequences and Auslander algebras are briefly examined.

In the Appendix V. Dlab reviews the basic results on quasi-hereditary algebras. These algebras play an important role in the concept of Kazhdan-Lusztig theory as developed by Cline, Parshall and Scott and in the study of the Bernstein-Gelfand-Gelfand category \(\mathcal O\).

The book is very well written. Each chapter contains many interesting exercises covering a wide spectrum of difficulty. The prerequisites are only linear algebra and, in places, a little group theory. In particular no knowledge of any preliminary information about the theory of rings and modules is assumed. Thus the textbook is accessible to a wide audience starting from students of the second and third year. Moreover, it provides a good approach to modern representation theory for the reader. On the other hand the book serves also as an excellent reference for algebraists and scientists interested in the field of finite dimensional algebras.

Chapter I contains the basic concepts and elementary properties of modules including the Jordan-HĂ¶lder Theorem.

In Chapter II the theory of semisimple algebras is presented, in particular the Wedderburn-Artin Theorem is proved and the characterization of modules over semisimple algebras is given.

Chapter III deals with the radical of modules and algebras. Primary algebras are characterized and basic algebras are studied. Moreover, the concept of the diagram (quiver) of an algebra, which is widely used in representation theory of algebras, is explained. The chapter ends with the classification of hereditary algebras (over an algebraically closed field).

Chapter IV is devoted to central simple algebras. Here the techniques of tensor products and bimodules and fundamental facts of the theory of division algebras can be found, the concept of the Brauer group is discussed and the Frobenius Theorem on finite dimensional algebras over \(\mathbb{R}\) is stated.

Chapter V begins with the study of field theory with emphasis on finite fields, including the Wedderburn Theorem on finite skew-fields. Then, using the machinery of bimodules and tensor products, the foundations of Galois theory are developed. As an application of Galois theory to central simple algebras the construction of crossed products is given.

In Chapter VI some characterizations of separable algebras and the Wedderburn-Malcev Theorem can be found.

Based on the theory of semisimple algebras, in Chapter VII the basic results of classical group theory are developed up to the integral theorems and the Burnside Theorem on the solvability of a group of order \(p^ a q^ b\).

Chapter VIII deals with the Morita Theorem on equivalences of module categories. The necessary background from category theory and the techniques of tensor products and exact sequences are presented. Furthermore in this chapter the concept of the tensor algebra of a bimodule as a generalization of the path algebra is explained.

In Chapters IX and X the classes of quasi-Frobenius algebras, uniserial algebras and generalized uniserial (serial) algebras are studied. These algebras are the best understood non-semisimple ones, however some of the results presented here until now have been available only in journal articles.

Chapter XI gives a short introduction to the methods of homological algebra. Apart from the fundamental concepts the topics of homological dimension, almost split sequences and Auslander algebras are briefly examined.

In the Appendix V. Dlab reviews the basic results on quasi-hereditary algebras. These algebras play an important role in the concept of Kazhdan-Lusztig theory as developed by Cline, Parshall and Scott and in the study of the Bernstein-Gelfand-Gelfand category \(\mathcal O\).

The book is very well written. Each chapter contains many interesting exercises covering a wide spectrum of difficulty. The prerequisites are only linear algebra and, in places, a little group theory. In particular no knowledge of any preliminary information about the theory of rings and modules is assumed. Thus the textbook is accessible to a wide audience starting from students of the second and third year. Moreover, it provides a good approach to modern representation theory for the reader. On the other hand the book serves also as an excellent reference for algebraists and scientists interested in the field of finite dimensional algebras.

Reviewer: H.Meltzer (Chemnitz)

### MSC:

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

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

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

16K20 | Finite-dimensional division rings |