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Graded cluster algebras

Research output: Contribution to journalJournal article

E-pub ahead of print

<mark>Journal publication date</mark>12/2015
<mark>Journal</mark>Journal of Algebraic Combinatorics
Issue number4
Number of pages24
Pages (from-to)1111-1134
<mark>State</mark>E-pub ahead of print
Early online date11/07/15
<mark>Original language</mark>English


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

The final publication is available at Springer via http://dx.doi.org/10.1007/s10801-015-0619-9