Elements of the Representation Theory of Associative Algebras. 3: Representation-Infinite Tilted Algebras
These are the second and third volumes of a long awaited modern treatment of representation theory of finite dimensional algebras, written by some of the leading experts in the area. The first volume dealt with fundamentals of the theory, introducing Auslander--Reiten quivers, tilting theory and classification of representations of finite algebras. The main goal of the second and the third volumes is to study representations of infinite tilted algebras B = End TKQ for a Euclidean diagram Q and an algebraically closed field K and give a complete description of their finite dimensional indecomposable modules, their modules categories mod B and the Auslander-Reiten quivers Γ (mod B).
Volume 2 starts with a chapter on tubes and then develops in detail the structure theory for regular components of concealed algebras of Euclidean type. This is then applied to a complete classification of all indecomposable modules over tame hereditary algebras. In the final chapter, a criterion for infinite representation type is proved and then applied to the Bongartz-Happel-Vossieck classification of all concealed algebras of Euclidean type in terms of quivers and relations.
The first part of Volume 3 culminates in the classification of all tilted algebras of Euclidean type due to Ringel. The next two chapters are dedicated to wild hereditary algebras and to a proof of the Drozd tame-wild dichotomy. In the final chapter, a number of recent results pertaining to the topic are listed without proof; as the authors point out, the extent of the volumes did not allow for presentation of all the contemporary tools (in particular covering techniques and derived categories). Each chapter of both volumes ends with a number of exercises; moreover, there are many examples worked out in detail throughout the text. The volumes are indispensable both for researchers and for graduate students interested in modern representation theory.