# Factoriality and class groups of cluster algebras

@article{Elsener2017FactorialityAC,
title={Factoriality and class groups of cluster algebras},
author={Ana Garcia Elsener and Philipp Lampe and Daniel Smertnig},
journal={arXiv: Commutative Algebra},
year={2017}
}
• Published 18 December 2017
• Mathematics
• arXiv: Commutative Algebra
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Let $\mathcal L^*$ be a family of finite subsets of $\mathbb N_0$ having the following properties. (a). $\{0\}, \{1\} \in \mathcal L^*$ and all other sets of $\mathcal L^*$ lie in $\mathbb N_{\ge ## References SHOWING 1-10 OF 48 REFERENCES Acyclic cluster algebras from a ring theoretic point of view The article gives a ring theoretic perspective on cluster algebras. Gei{\ss}-Leclerc-Schr\"oer prove that all cluster variables in a cluster algebra are irreducible elements. Furthermore, they Cluster algebras III: Upper bounds and double Bruhat cells • Mathematics • 2003 We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of Cluster algebras II: Finite type classification • Mathematics • 2002 This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many Factorial cluster algebras • Mathematics • 2011 We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an Cluster algebra structures on Poisson nilpotent algebras • Mathematics • 2018 Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent Cluster algebras IV: Coefficients • Mathematics Compositio Mathematica • 2007 We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these From triangulated categories to cluster algebras • Mathematics • 2005 The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra$\mathcal{A}\$ of finite type can be realized as a
Quantum Cluster Algebra Structures on Quantum Nilpotent Algebras
• Mathematics
• 2017
All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that