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Graded quantum cluster algebras and an application to quantum Grassmannians

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Graded quantum cluster algebras and an application to quantum Grassmannians. / Grabowski, Jan; Launois, Stéphane.

In: Proceedings of the London Mathematical Society, Vol. 109, No. 3, 2014, p. 697-732.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

Harvard

Grabowski, J & Launois, S 2014, 'Graded quantum cluster algebras and an application to quantum Grassmannians', Proceedings of the London Mathematical Society, vol. 109, no. 3, pp. 697-732. https://doi.org/10.1112/plms/pdu018

APA

Grabowski, J., & Launois, S. (2014). Graded quantum cluster algebras and an application to quantum Grassmannians. Proceedings of the London Mathematical Society, 109(3), 697-732. https://doi.org/10.1112/plms/pdu018

Vancouver

Grabowski J, Launois S. Graded quantum cluster algebras and an application to quantum Grassmannians. Proceedings of the London Mathematical Society. 2014;109(3):697-732. https://doi.org/10.1112/plms/pdu018

Author

Grabowski, Jan ; Launois, Stéphane. / Graded quantum cluster algebras and an application to quantum Grassmannians. In: Proceedings of the London Mathematical Society. 2014 ; Vol. 109, No. 3. pp. 697-732.

Bibtex

@article{e81b22fdf16d40be83c5ff70e3d7d109,
title = "Graded quantum cluster algebras and an application to quantum Grassmannians",
abstract = "We introduce a framework for Z-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then one extends this to all cluster variables by mutation. The resulting grading has the property that every (quantum) cluster variable is homogeneous.In the quantum setting, we use this grading framework to give a construction that behaves somewhat like twisting, in that it produces a new quantum cluster algebra with the same cluster combinatorics but with different quasi-commutation relations between the cluster variables.We apply these results to show that the quantum Grassmannians $K_q[Gr(k, n)]$ admit quantum cluster algebra structures, as quantizations of the cluster algebra structures on the classical Grassmannian coordinate ring found by Scott. This is done by lifting the quantum cluster algebra structure on quantum matrices due to Gei{\ss}–Leclerc–Schr{\"o}er and completes earlier work of the authors on the finite-type cases.",
author = "Jan Grabowski and St{\'e}phane Launois",
note = "{\textcopyright} 2014 London Mathematical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.",
year = "2014",
doi = "10.1112/plms/pdu018",
language = "English",
volume = "109",
pages = "697--732",
journal = "Proceedings of the London Mathematical Society",
issn = "0024-6115",
publisher = "Oxford University Press",
number = "3",

}

RIS

TY - JOUR

T1 - Graded quantum cluster algebras and an application to quantum Grassmannians

AU - Grabowski, Jan

AU - Launois, Stéphane

N1 - © 2014 London Mathematical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

PY - 2014

Y1 - 2014

N2 - We introduce a framework for Z-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then one extends this to all cluster variables by mutation. The resulting grading has the property that every (quantum) cluster variable is homogeneous.In the quantum setting, we use this grading framework to give a construction that behaves somewhat like twisting, in that it produces a new quantum cluster algebra with the same cluster combinatorics but with different quasi-commutation relations between the cluster variables.We apply these results to show that the quantum Grassmannians $K_q[Gr(k, n)]$ admit quantum cluster algebra structures, as quantizations of the cluster algebra structures on the classical Grassmannian coordinate ring found by Scott. This is done by lifting the quantum cluster algebra structure on quantum matrices due to Geiß–Leclerc–Schröer and completes earlier work of the authors on the finite-type cases.

AB - We introduce a framework for Z-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then one extends this to all cluster variables by mutation. The resulting grading has the property that every (quantum) cluster variable is homogeneous.In the quantum setting, we use this grading framework to give a construction that behaves somewhat like twisting, in that it produces a new quantum cluster algebra with the same cluster combinatorics but with different quasi-commutation relations between the cluster variables.We apply these results to show that the quantum Grassmannians $K_q[Gr(k, n)]$ admit quantum cluster algebra structures, as quantizations of the cluster algebra structures on the classical Grassmannian coordinate ring found by Scott. This is done by lifting the quantum cluster algebra structure on quantum matrices due to Geiß–Leclerc–Schröer and completes earlier work of the authors on the finite-type cases.

U2 - 10.1112/plms/pdu018

DO - 10.1112/plms/pdu018

M3 - Journal article

VL - 109

SP - 697

EP - 732

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 3

ER -