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Graded cluster algebras

Research output: Contribution to journalJournal article

Forthcoming

<mark>Journal publication date</mark>2015
<mark>Journal</mark>Journal of Algebraic Combinatorics
Number of pages22
<mark>Original language</mark>English

Abstract

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.

Bibliographic note

Date of Acceptance: 30/06/2015