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  • Grabowski-GradedClusterAlgebras-1309.6170v3

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

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Graded cluster algebras

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Graded cluster algebras. / Grabowski, Jan.
In: Journal of Algebraic Combinatorics, Vol. 42, No. 4, 12.2015, p. 1111-1134.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Grabowski, J 2015, 'Graded cluster algebras', Journal of Algebraic Combinatorics, vol. 42, no. 4, pp. 1111-1134. https://doi.org/10.1007/s10801-015-0619-9

APA

Grabowski, J. (2015). Graded cluster algebras. Journal of Algebraic Combinatorics, 42(4), 1111-1134. https://doi.org/10.1007/s10801-015-0619-9

Vancouver

Grabowski J. Graded cluster algebras. Journal of Algebraic Combinatorics. 2015 Dec;42(4):1111-1134. Epub 2015 Jul 11. doi: 10.1007/s10801-015-0619-9

Author

Grabowski, Jan. / Graded cluster algebras. In: Journal of Algebraic Combinatorics. 2015 ; Vol. 42, No. 4. pp. 1111-1134.

Bibtex

@article{b39af468185d4953a497179c9006c65a,
title = "Graded cluster algebras",
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.",
keywords = "cluster algebra, grading, cluster category, tropical frieze",
author = "Jan Grabowski",
note = "The final publication is available at Springer via http://dx.doi.org/10.1007/s10801-015-0619-9",
year = "2015",
month = dec,
doi = "10.1007/s10801-015-0619-9",
language = "English",
volume = "42",
pages = "1111--1134",
journal = "Journal of Algebraic Combinatorics",
issn = "0925-9899",
publisher = "Springer Netherlands",
number = "4",

}

RIS

TY - JOUR

T1 - Graded cluster algebras

AU - Grabowski, Jan

N1 - The final publication is available at Springer via http://dx.doi.org/10.1007/s10801-015-0619-9

PY - 2015/12

Y1 - 2015/12

N2 - 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.

AB - 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.

KW - cluster algebra

KW - grading

KW - cluster category

KW - tropical frieze

U2 - 10.1007/s10801-015-0619-9

DO - 10.1007/s10801-015-0619-9

M3 - Journal article

VL - 42

SP - 1111

EP - 1134

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

IS - 4

ER -