Representation theory of group graded algebras (Q2734760)

The theory of rings graded by groups and semigroups and their representations is a very important theory which has become very useful in many branches of algebra, algebraic geometry and number theory. This book is concentrated on the part of the theory closely connected with some aspects of the finite group representation theory. The book is divided into five chapters as follows: 1.~Modules over graded rings; 2.~Clifford theory; 3.~Clifford extensions; 4.~Auslander-Reiten theory; 5.~Equivalences induced by graded bimodules.NEWLINENEWLINENEWLINEThe first chapter is dedicated to the study of the category of modules graded by a \(G\)-set presenting the basic properties. In Chapter 2, the author studies the endomorphism ring \(E\) of a graded module \(M\) and also the functors \(\Hom_R(M,-)\) and \(M\otimes_E-\). Several applications to Clifford theory are obtained. An important problem in this theory is to study the structure of a graded simple (graded indecomposable) module when it is considered without gradation. The chapter ends with the Clifford theory of bimodules and blocks. The next chapter investigates the problem of extendability for strongly graded rings over a finite group. Chapter 4 contains the extension of the results of the preceding chapters to functor categories. The last chapter contains the characterization of several types of equivalences (Morita equivalences, Rickard's derived equivalences and stable Morita equivalences) between two fully \(G\)-graded \(k\)-algebras projective over \(k\) with \(G\) finite induced by \(G\)-graded bimodules.NEWLINENEWLINENEWLINEThe book is very well written. It is suitable for a graduate course on graded rings and significant and important for researchers in this area.