Triangular bases in quantum cluster algebras and monoidal categorification conjectures

Fan Qin Shanghai Jiao Tong University

Rings and Algebras mathscidoc:1703.31001

to appear in Duke Math Journal, 2017.1
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 parametrized 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-Leclerc and Fomin-Zelevinsky in the corresponding cases: all cluster monomials correspond to simple modules.
cluster algebras, monoidal categorification
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@inproceedings{fan2017triangular,
  title={Triangular bases in quantum cluster algebras and monoidal categorification conjectures},
  author={Fan Qin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170321222507260146721},
  booktitle={to appear in Duke Math Journal},
  year={2017},
}
Fan Qin. Triangular bases in quantum cluster algebras and monoidal categorification conjectures. 2017. In to appear in Duke Math Journal. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170321222507260146721.
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