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Rhombus filtrations and Rauzy algebras
Authors:Alex Clark
Institution:a Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK
b Mathematical Institute, 24-29 St Giles', Oxford OX1 3LB, UK
Abstract:Peach introduced rhombal algebras associated to quivers given by tilings of the plane by rhombi. We develop general techniques to analyze 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.
Keywords:primary  16G20  secondary  16G70  52C23
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