4. Graded Frobenius Cluster Categories

  • Jan E. Grabowski Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, United Kingdom
  • Matthew Pressland Institut f\"ur Algebra und Zahlentheorie, Universit\"at Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Keywords: Cluster algebra, Frobenius category, grading, Grothendieck group


Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi--Yau triangulated categories with a choice of cluster-tilting object.  The grading on the category corresponds to a grading on the cluster algebra without coefficients categorified by the cluster category and hence knowledge of one of these structures can help us study the other.

In this work, we extend the above to certain Frobenius categories that categorify cluster algebras with coefficients.  We interpret the grading K-theoretically and prove similar results to the triangulated case, in particular obtaining that degrees are additive on exact sequences.

We show that the categories of Buan, Iyama, Reiten and Scott, some of which were used by Gei\ss, Leclerc and Schr\"oer to categorify cells in partial flag varieties, and those of Jensen, King and Su, categorifying Grassmannians, are examples of graded Frobenius cluster categories.


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