Cluster characters for 2-Calabi–Yau triangulated categories

Yann Palu[1]

  • [1] Université Paris 7 - Denis Diderot UMR 7586 du CNRS, case 7012 2 place Jussieu 75251 Paris Cedex 05 (France)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 6, page 2221-2248
  • ISSN: 0373-0956

Abstract

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Starting from an arbitrary cluster-tilting object T in a 2-Calabi–Yau triangulated category over an algebraically closed field, as in the setting of Keller and Reiten, we define, for each object L , a fraction X ( T , L ) using a formula proposed by Caldero and Keller. We show that the map taking L to X ( T , L ) is a cluster character, i.e. that it satisfies a certain multiplication formula. We deduce that it induces a bijection, in the finite and the acyclic case, between the indecomposable rigid objects of the cluster category and the cluster variables, which confirms a conjecture of Caldero and Keller.

How to cite

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Palu, Yann. "Cluster characters for 2-Calabi–Yau triangulated categories." Annales de l’institut Fourier 58.6 (2008): 2221-2248. <http://eudml.org/doc/10376>.

@article{Palu2008,
abstract = {Starting from an arbitrary cluster-tilting object $T$ in a 2-Calabi–Yau triangulated category over an algebraically closed field, as in the setting of Keller and Reiten, we define, for each object $L$, a fraction $X(T,L)$ using a formula proposed by Caldero and Keller. We show that the map taking $L$ to $X(T,L)$ is a cluster character, i.e. that it satisfies a certain multiplication formula. We deduce that it induces a bijection, in the finite and the acyclic case, between the indecomposable rigid objects of the cluster category and the cluster variables, which confirms a conjecture of Caldero and Keller.},
affiliation = {Université Paris 7 - Denis Diderot UMR 7586 du CNRS, case 7012 2 place Jussieu 75251 Paris Cedex 05 (France)},
author = {Palu, Yann},
journal = {Annales de l’institut Fourier},
keywords = {Calabi–Yau triangulated category; cluster algebra; cluster category; cluster-tilting object; Calabi-Yau triangulated categories; cluster algebras; cluster-tilting objects},
language = {eng},
number = {6},
pages = {2221-2248},
publisher = {Association des Annales de l’institut Fourier},
title = {Cluster characters for 2-Calabi–Yau triangulated categories},
url = {http://eudml.org/doc/10376},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Palu, Yann
TI - Cluster characters for 2-Calabi–Yau triangulated categories
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 6
SP - 2221
EP - 2248
AB - Starting from an arbitrary cluster-tilting object $T$ in a 2-Calabi–Yau triangulated category over an algebraically closed field, as in the setting of Keller and Reiten, we define, for each object $L$, a fraction $X(T,L)$ using a formula proposed by Caldero and Keller. We show that the map taking $L$ to $X(T,L)$ is a cluster character, i.e. that it satisfies a certain multiplication formula. We deduce that it induces a bijection, in the finite and the acyclic case, between the indecomposable rigid objects of the cluster category and the cluster variables, which confirms a conjecture of Caldero and Keller.
LA - eng
KW - Calabi–Yau triangulated category; cluster algebra; cluster category; cluster-tilting object; Calabi-Yau triangulated categories; cluster algebras; cluster-tilting objects
UR - http://eudml.org/doc/10376
ER -

References

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