Graded quantum cluster algebras and an application to quantum Grassmannians

Grabowski, Jan and Launois, Stéphane (2014) Graded quantum cluster algebras and an application to quantum Grassmannians. Proceedings of the London Mathematical Society, 109 (3). pp. 697-732. ISSN 0024-6115

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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.

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Journal Article
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Proceedings of the London Mathematical Society
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© 2014 London Mathematical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
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23 Apr 2013 11:46
Last Modified:
25 Nov 2023 00:14