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 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 A duality map for quantum cluster varieties from surfaces

2017 ◽  
Vol 306 ◽  
pp. 1164-1208 ◽  
Author(s):  
Dylan G.L. Allegretti ◽  
Hyun Kyu Kim
Keyword(s):  
Quantum Cluster ◽  
Duality Map ◽  
Cluster Varieties

scholarly journals Quantum periods and TBA-like equations for a class of Calabi-Yau geometries

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bao-ning Du ◽  
Min-xin Huang
Keyword(s):  
Large Class ◽  
Bethe Ansatz ◽  
Difference Equations ◽  
The Other ◽  
Elementary Functions ◽  
Quantum Dilogarithm ◽  
Thermodynamic Bethe Ansatz ◽  
Dilogarithm Function ◽  
Quantum Operators

Abstract We continue the study of a novel relation between quantum periods and TBA(Thermodynamic Bethe Ansatz)-like difference equations, generalize previous works to a large class of Calabi-Yau geometries described by three-term quantum operators. We give two methods to derive the TBA-like equations. One method uses only elementary functions while the other method uses Faddeev’s quantum dilogarithm function. The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.

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 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
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