Translator Disclaimer
1 September 2017 Triangular bases in quantum cluster algebras and monoidal categorification conjectures
Fan Qin
Duke Math. J. 166(12): 2337-2442 (1 September 2017). DOI: 10.1215/00127094-2017-0006

Abstract

We consider the quantum cluster algebras which are injective-reachable and introduce a triangular basis in every seed. We prove that, under some initial conditions, there exists a unique common triangular basis with respect to all seeds. This basis is parameterized by tropical points as expected in the Fock–Goncharov conjecture.

As an application, we prove the existence of the common triangular bases for the quantum cluster algebras arising from representations of quantum affine algebras and partially for those arising from quantum unipotent subgroups. This result implies monoidal categorification conjectures of Hernandez and Leclerc and Fomin and Zelevinsky in the corresponding cases: all cluster monomials correspond to simple modules.

Citation

Download Citation

Fan Qin. "Triangular bases in quantum cluster algebras and monoidal categorification conjectures." Duke Math. J. 166 (12) 2337 - 2442, 1 September 2017. https://doi.org/10.1215/00127094-2017-0006

Information

Received: 14 March 2015; Revised: 14 September 2016; Published: 1 September 2017
First available in Project Euclid: 26 May 2017

zbMATH: 06783125
MathSciNet: MR3694569
Digital Object Identifier: 10.1215/00127094-2017-0006

Subjects:
Primary: 13F60
Secondary: 17B37

Keywords: categorification , dual canonical basis , positivity , quantum cluster algebra , quiver variety

Rights: Copyright © 2017 Duke University Press

JOURNAL ARTICLE
106 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.166 • No. 12 • 1 September 2017
Back to Top