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Graded quantum cluster algebras and an application to quantum Grassmannians

Research output: Contribution to journalJournal article

Published

Journal publication date2014
JournalProceedings of the London Mathematical Society
Journal number3
Volume109
Number of pages36
Pages697-732
Early online date23/05/14
Original languageEnglish

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 Geiß–Leclerc–Schröer and completes earlier work of the authors on the finite-type cases.

Bibliographic note

© 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.