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  • Print publication year: 2007
  • Online publication date: January 2010

XVII - Tilted algebras of Euclidean type

Summary

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.