scholarly journals Two Quantum Cluster Approximations

2003 ◽  
pp. 106-120
Author(s):  
T. A. Maier ◽  
O. Gonzalez ◽  
M. Jarrell ◽  
T. Schulthess
Keyword(s):  
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 Laurent Positivity of Quantized Canonical Bases for Quantum Cluster Varieties from Surfaces

2019 ◽  
Vol 373 (2) ◽  
pp. 655-705 ◽  
Author(s):  
So Young Cho ◽  
Hyuna Kim ◽  
Hyun Kyu Kim ◽  
Doeun Oh
Keyword(s):  
Canonical Bases ◽  
Quantum Cluster ◽  
Cluster Varieties

scholarly journals The quantum dilogarithm and representations of quantum cluster varieties

2008 ◽  
Vol 175 (2) ◽  
pp. 223-286 ◽  
Author(s):  
V.V. Fock ◽  
A.B. Goncharov
Keyword(s):  
Quantum Dilogarithm ◽  
Quantum Cluster ◽  
Cluster Varieties

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

Ag7Au6: A 13-Atom Alloy Quantum Cluster

2012 ◽  
Vol 51 (9) ◽  
pp. 2155-2159 ◽  
Author(s):  
Thumu Udayabhaskararao ◽  
Yan Sun ◽  
Nirmal Goswami ◽  
Samir K. Pal ◽  
K. Balasubramanian ◽  
...  
Keyword(s):  
Quantum Cluster

scholarly journals Comparison of two-quantum-cluster approximations

2002 ◽  
Vol 65 (4) ◽  
Author(s):  
Th. A. Maier ◽  
M. Jarrell
Keyword(s):  
Quantum Cluster

scholarly journals $(q,t)$-Characters of Kirillov-Reshetikhin Modules of Type $A_r$ as Quantum Cluster Variables

10.37236/7188 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Bolor Turmunkh
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Cluster Algebra ◽  
Quiver Varieties ◽  
Quantum Cluster ◽  
T System

Nakajima (2003) introduced a $t$-deformation of $q$-characters, $(q,t)$-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima $(q,t)$-characters of Kirillov-Reshetikhin modules satisfy a $t$-deformed $T$-system. The $T$-system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this paper, we show that the Nakajima $t$-deformed $T$-system of type $A_r$ forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the $T$-system.

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