Cluster algebras and quantum affine algebras
David Hernandez,Bernard Leclerc +1 more
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In this article, the Grothendieck rings of a quantum affine algebra U q ( g ∧ ) of simply laced type were studied using cluster algebras.Abstract:
Let C be the category of finite-dimensional representations of a quantum affine algebra U q ( g ∧ ) of simply laced type. We introduce certain monoidal subcategories C l ( l ∈ N ) of C , and we study their Grothendieck rings using cluster algebras.read more
Citations
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Cluster algebras
Bernard Leclerc,Lauren Williams +1 more
TL;DR: Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings.
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Positivity for cluster algebras from surfaces
TL;DR: In this article, the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed, is studied.
Journal ArticleDOI
Positivity for cluster algebras
Kyungyong Lee,Ralf Schiffler +1 more
TL;DR: In this paper, the positivity conjecture for all skew-symmetric cluster algebras was shown to hold for all possible skewymmetric clique types, and the conjecture was proved for all clique classes.
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Quiver varieties and cluster algebras
TL;DR: In this article, the authors embed a Fomin-Zelevinsky cluster algebra into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties.
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Quiver varieties and cluster algebras
TL;DR: In this article, the authors embed a Fomin-Zelevinsky cluster algebra into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg) of a symmetric Kac-Moody Lie algebra g, studied earlier by the author via perverse sheaves on graded quiver varieties.
References
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Book
A guide to quantum groups
Vyjayanthi Chari,Andrew Pressley +1 more
TL;DR: In this paper, the Kac-Moody algebras and quasitriangular Hopf algesas were used to represent the universal R-matrix and the root of unity case.
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Cluster algebras I: Foundations
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this article, a new class of commutative algebras was proposed for dual canonical bases and total positivity in semisimple groups. But the study of the algebraic framework is not yet complete.
Journal ArticleDOI
Tilting theory and cluster combinatorics
TL;DR: In this article, a new category C, called the cluster category, is introduced, which is obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field.
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Cluster algebras IV: Coefficients
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this paper, the dependence of a cluster algebra on the choice of coefficients was studied, and it was shown that for cluster algebras with principal coefficients, the exchange graph of the cluster algebra with the same exchange matrix covers the exchange matrix of any cluster algebra of the same type.
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Cluster algebras II: Finite type classification
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this paper, a complete classification of cluster algebras of finite type is presented, i.e., those with finitely many clusters, which is identical to the Cartan-Killing classification of semisimple Lie algebases and finite root systems.