Graded cluster algebras
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In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for finite-type cluster algebras without coefficients, giving a full classification. Translating the definition suitably again, we obtain a notion of multi-grading for (generalised) cluster categories. This setting allows us to prove additional properties of graded cluster algebras in a wider range of cases. We also obtain interesting combinatorics—namely tropical frieze patterns—on the Auslander–Reiten quivers of the categories.
KeywordsCluster algebra Graded Cluster category Tropical frieze
Mathematics Subject Classification13F60 (Primary) 18E30 16G70 (Secondary)
The author would particularly like to thank Stéphane Launois for many helpful discussions throughout the collaboration that this work originated from, Robert Marsh for useful references regarding cluster categories and Thomas Booker-Price for permission to reproduce the results of his calculations in Sects. 4.2 and 4.3. We also thank numerous colleagues with whom preliminary versions of these results were discussed following their presentation at various institutions.
- 4.Booker-Price, T.: Personal communication (2014)Google Scholar
- 14.Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster Algebras and Poisson Geometry, Mathematical Surveys and Monographs, vol. 167. American Mathematical Society, Providence, RI (2010)Google Scholar
- 15.Grabowski, J.E., Launois, S.: Graded quantum cluster algebras and an application to quantum Grassmannians. Proc. Lond. Math. Soc. (3) 109(3), 697–732 (2014)Google Scholar
- 16.Guo, L.: On tropical friezes associated with Dynkin diagrams. Int. Math. Res. Not. IMRN, no. 18, 4243–4284 (2013)Google Scholar
- 19.Keller, B.: Cluster Algebras, Quiver Representations and Triangulated Categories, Triangulated Categories, London Mathematical Society Lecture Note Series, vol. 375. Cambridge University Press, Cambridge, pp. 76–160 (2010)Google Scholar
- 22.Propp, J.: The combinatorics of frieze patterns and Markoff numbers, pp. 29–34 (2008). arXiv:math/0511633v4