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We provide a complete classification of all algebras of generalized dihedral type, which are natural generalizations of algebras which occurred in the study of blocks of group algebras with dihedral defect groups. This gives a description by quivers and relations coming from surface triangulations.
We complete the derived equivalence classification of all symmetric algebras of polynomial growth, by solving the subtle problem of distinguishing the standard and nonstandard nondomestic symmetric algebras of polynomial growth up to derived equivalence.
We have already described in Chapter XII the structure and the combinatorial invariants of the module categories mod A of concealed algebras A of Euclidean type, while in Chapter XIV we give a complete classification of these algebras A by means of quivers and relations.
The main objective of this chapter is to describe the structure and the combinatorial invariants of the module category mod B of an arbitrary representation-infinite tilted algebra B of Euclidean type. Moreover, we show that these algebras B are domestic branch extensions or domestic branch coextensions of concealed algebras A of Euclidean type. In Section 1, we study the distribution of indecomposable direct summands of a splitting tilting module among the hereditary standard stable tubes of the Auslander–Reiten quiver Γ(mod B) of an arbitrary algebra B, while in Section 2 we show how the structure of the hereditary standard stable tubes in Γ(mod B) is changed under the related tilting process of B.
The main result of Section 3 asserts that every representation-infinite tilted algebra B of Euclidean type is a domestic tubular (branch) extension or a domestic tubular (branch) coextension of a concealed algebra A of Euclidean type. The inverse implication is proved in Section 4 by showing that every domestic tubular (branch) extension and every domestic tubular (branch) coextension of a concealed algebra of Euclidean type is a representation-infinite tilted algebra of Euclidean type.
In Section 5, we present a characterisation of representation-infinite tilted algebras B of Euclidean type, and we exhibit their module categories mod B.
The final part of a three-volume set providing a modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field. The subject is presented from the perspective of linear representations of quivers and homological algebra. This volume provides an introduction to the representation theory of representation-infinite tilted algebras from the point of view of the time-wild dichotomy. Also included is a collection of selected results relating to the material discussed in all three volumes. The book is primarily addressed to a graduate student starting research in the representation theory of algebras, but will also be of interest to mathematicians in other fields. Proofs are presented in complete detail, and the text includes many illustrative examples and a large number of exercises at the end of each chapter, making the book suitable for courses, seminars, and self-study.