As noticed by Brauer, Nesbitt and Nakayama in the late 1930s and the early 1940s, modular representation theory of finite groups seemed to be dependent on the structure of non-semisimple algebras. They were able to prove some main results in the general setting of non-semisimple rings, generalizing E. Noether’s results on semisimple rings, for example giving a bijection between the indecomposable summands of the regular representation of an Artinian ring and the isomorphism classes of simple modules.
To understand the structure of non- semisimple (group) algebras, Brauer, Nesbitt and Nakayama used an idea introduced by Frobenius in a pair of papers from 1903 characterizing when the left and right regular representations of a finite group are equivalent. Frobenius’ classical result amounts to showing that these two representations are equivalent precisely when some special matrices, that Frobenius called parastrophic, intertwine the representations. Brauer, Nesbitt Nakayama called the corresponding group algebra a Frobenius algebra, and were able to show that, for any field k, a finite dimensional k-algebra A is a Frobenius algebra if and only if there is an hyperplane in A with no nonzero right ideals.
Later on, Nakayama proved that these conditions are equivalent to the existence of a nondegenerate bilinear form f in A such that f(ab,c)=f(a,bc). Frobenius algebras include matrix algebras over any field (where the bilinear form is given by the trace of the product of two matrices), semisimple algebras, group algebras of finite groups (even if they are not semisimple, which includes the crucial case for modular representation theory, when the characteristic of the field divides the order of the group).
From these concrete beginnings, a beautiful theory was born: the representation theory of finite dimensional algebras over fields. The book under review has as its main goal to give an introduction to this theory using the representation theory of Frobenius algebras.
The first three chapters give a self-contained and detailed introduction to modern representation theory of finite dimensional algebras over fields, emphasizing the case of Frobenius algebras. Thus, Chapter I start with the definition of algebras, modules and their morphisms. Several examples are fully detailed, from the classical group algebra of a finite group to the important path algebra of a quiver. Next, representations of finite groups are introduced, first in a classical setting and later on as modules over the group algebra. In the next example, following Gabriel, representations of finite quivers are given plenty of space, with many concrete examples explicitly computed. This chapter includes general properties of semisimple modules and algebras, Loewy length, the Jordan-Holder theorem, projective and injective modules, hereditary algebras, Nakayama algebras and the Grothendieck group of an algebra, all illustrated with examples using the path algebra of a quiver.
Chapter II is devoted to the Morita theory of equivalence, A categories, and Morita-Azumaya duality. The exposition is self-contained, ranging from the basic definitions of category theory, adjunctions, natural transformations, tensor products of modules to generators of a category.
Chapter III introduces the Auslander-Reiten theory of finite dimensional modules over a finite dimensional algebra over a field: the Jacobson radical of a category of modules, irreducible morphisms, almost split exact sequences and Auslander-Reiten quivers. The main results are the existence of almost split sequences, Auslander-Reiten formulas, the first Brauer-Thrall conjecture, Nakayama functors, a criterion for an indecomposable finite dimensional algebra to be of finite representation type (Auslander’s theorem) and the Bautista-Smalø theorem (a sufficient condition for the non nullity of a composition of morphisms between indecomposable modules).
The second part of the book, chapters four to six, are more advanced. Chapter IV treats the important case of selfinjective algebras and their module categories; since Frobenius algebras are selfinjective (projective and injective modules coincide), this is a natural generalization. This chapter starts with a reformulation and proof of the classical result of Frobenius that characterizes the finite dimensional algebras over a field for which the left and right regular representations are equivalent, giving the Brauer-Nesbitt-Nakayama criteria for a finite dimensional algebra to be a Frobenius algebra or a symmetric one. Next, we have the classical theorems of Nakayama characterizing selfinjective or symmetric algebras and the Galois bijection between right and left ideals of a finite dimensional selfinjective algebra. It is also shown that there are selfinjective algebras that are not Frobenius. The chapter ends with the important class of Frobenius algebras associated to quivers given by Dynkin and Euclidean graphs and the Riedtmann-Todorov theorem characterizing the quiver of a selfinjective algebra of finite representation type.
The last two chapters are devoted to some special Frobenius algebras. Chapter V treats the case of Hecke algebras associated to a finite group of reflections of a real Euclidean space, proving and using Coxeter classification of these groups and their associated Coxeter graphs. Chapter VI deals with the finite dimensional Hopf algebras, ubiquitous in algebra, topology and combinatorics. The authors show that these are Frobenius algebras and prove some of their important properties.
In addition to classical texts, such as Curtis and Reiner Representation Theory of Finite Groups and Associative Algebras (Wiley, 1962, reprinted by AMS/Chelsea in 2006), there are now several textbooks and monographs on modern representation theory: From introductory textbooks such as Introduction to Representation Theory by P. Etingof et al (AMS, 2011), to advanced treatises as the three volume Elements of Representation Theory, Volume I by I. Assem, D. Simson and A. Skowroński, and Volume 2, Volume 3 by D. Simson and A. Skowroński (Cambridge, 2006 and 2007) or the excellent monographs by some of the leading figures, P. Gabriel and A. Roiter’s Representations of Finite-Dimensional Algebras, (Springer, 1992) and M. Aulander, I. Reiten and S. Smalø’s Representation Theory of Artin Algebras (Cambridge, 1995).
For a graduate student who wants to study representation theory, this is a well-written text, filled with examples, carefully written proofs, and well-chosen sets of exercises at the end of every one of its six chapters. It is a fine introduction to the field, and promises to be the first of two volumes.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.