scholarly journals On quantum cluster algebras of finite type

2011 ◽  
Vol 6 (2) ◽  
pp. 231-240 ◽  
Author(s):  
Ming Ding
Keyword(s):  
Finite Type ◽  
Cluster Algebras ◽  
Quantum Cluster

scholarly journals Laurent phenomenon and simple modules of quiver Hecke algebras

2019 ◽  
Vol 155 (12) ◽  
pp. 2263-2295 ◽  
Author(s):  
Masaki Kashiwara ◽  
Myungho Kim
Keyword(s):  
Simple Module ◽  
Hecke Algebras ◽  
Duke Math ◽  
Cluster Algebras ◽  
Coordinate Ring ◽  
Simple Modules ◽  
Laurent Phenomenon ◽  
Cluster Variable ◽  
Quantum Cluster

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 $[\mathscr{C}]$.

scholarly journals F-Polynomials in Quantum Cluster Algebras

2010 ◽  
Vol 14 (6) ◽  
pp. 1025-1061 ◽  
Author(s):  
Thao Tran
Keyword(s):  
Cluster Algebras ◽  
Quantum Cluster

Atomic bases of quantum cluster algebras of type A˜2n−1,1

2021 ◽  
Author(s):  
Ming Ding ◽  
Fan Xu ◽  
Xueqing Chen
Keyword(s):  
Type A ◽  
Cluster Algebras ◽  
Quantum Cluster

A 2-Calabi–Yau realization of finite-type cluster algebras with universal coefficients

2019 ◽  
Vol 291 (3-4) ◽  
pp. 1495-1523
Author(s):  
Alfredo Nájera Chávez
Keyword(s):  
Finite Type ◽  
Cluster Algebras

Combinatorics of X-variables in finite type cluster algebras

2019 ◽  
Vol 165 ◽  
pp. 273-298 ◽  
Author(s):  
Melissa Sherman-Bennett
Keyword(s):  
Finite Type ◽  
Cluster Algebras

scholarly journals Polynomial recognition of cluster algebras of finite type

2016 ◽  
Vol 457 ◽  
pp. 457-468
Author(s):  
Elisângela Silva Dias ◽  
Diane Castonguay
Keyword(s):  
Finite Type ◽  
Cluster Algebras

scholarly journals Cluster automorphism groups of cluster algebras of finite type

2016 ◽  
Vol 447 ◽  
pp. 490-515 ◽  
Author(s):  
Wen Chang ◽  
Bin Zhu
Keyword(s):  
Finite Type ◽  
Automorphism Groups ◽  
Cluster Algebras
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