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Laurent phenomenon and simple modules of quiver Hecke algebras

Part of: Arithmetic rings and other special rings Groups and algebras in quantum theory Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  04 October 2019

Masaki Kashiwara  and
Myungho Kim
Masaki Kashiwara
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan email masaki@kurims.kyoto-u.ac.jp Korea Institute for Advanced Study, Seoul 02455, Korea email masaki@kurims.kyoto-u.ac.jp
Myungho Kim
Affiliation:
Department of Mathematics, Kyung Hee University, Seoul 02447, Korea email mkim@khu.ac.kr
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Abstract

In this paper we study consequences of the results of Kang et al. [Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), 349–426] on a monoidal categorification of the unipotent quantum coordinate ring $A_{q}(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category ${\mathcal{C}}_{w}$ strongly commutes with all the cluster variables in a cluster $[\mathscr{C}]$, then $[S]$ is a cluster monomial in $[\mathscr{C}]$. If $S$ strongly commutes with cluster variables except for exactly one cluster variable $[M_{k}]$, then $[S]$ is either a cluster monomial in $[\mathscr{C}]$ or a cluster monomial in $\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. 166 (2017), 2337–2442]) of the localization $\widetilde{A}_{q}(\mathfrak{n}(w))$ of $A_{q}(\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[\mathscr{C}]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[\mathscr{C}]$.

Keywords

cluster algebraLaurent phenomenonmonoidal categorificationquiver Hecke algebraunipotent quantum coordinate ring

MSC classification

Primary: 13F60: Cluster algebras 81R50: Quantum groups and related algebraic methods 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 12 , December 2019 , pp. 2263 - 2295
Copyright
© The Authors 2019 

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Footnotes

This work was supported by Grant-in-Aid for Scientific Research (B) 22340005, Japan Society for the Promotion of Science. This work was supported by the National Research Foundation of Korea (NF) grant funded by the Korea government (MSIP) (No. NRF-2017R1C1B2007824).

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