Rhombus Filtrations and Rauzy Algebras

Details Rhombus Filtrations and Rauzy Algebras Rhombus Filtrations and Rauzy Algebras

2007-10-15

arxiv.org/abs/0710.2781v1

Mathematics

Representation Theory

30 pages, 14 figures

Scientific paper

Peach introduced rhombal algebras associated to quivers given by tilings of the plane by rhombi. We develop general techniques to analyse rhombal algebras, including a filtration by what we call rhombus modules. We introduce a way to relate the infinite-dimensional rhombal algebra corresponding to a complete tiling of the plane to finite-dimensional algebras corresponding to finite portions of the tiling. Throughout, we apply our general techniques to the special case of the Rauzy tiling, which is built in stages reflecting an underlying self-similarity. Exploiting this self-similar structure allows us to uncover interesting features of the associated finite-dimensional algebras, including some of the tree classes in the stable Auslander-Reiten quiver.

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