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.
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Abstract: 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 clusters. This classification turns out to be identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems, which is intriguing since in most cases, the symmetry exhibited by the Cartan-Killing type of a cluster algebra is not at all apparent from its geometric origin.
The combinatorial structure behind a cluster algebra of finite type is captured by its cluster complex. We identify this complex as the normal fan of a generalized associahedron introduced and studied in hep-th/0111053 and math.CO/0202004. Another essential combinatorial ingredient of our arguments is a new characterization of the Dynkin diagrams.
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Citations
A non-simply-laced version for cluster structures on 2-Calabi–Yau categories
TL;DR: In this paper, a non-simply-laced version of cluster structures for 2-Calabi-Yau or stably 2-calabi-yau categories over arbitrary fields was investigated.
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Cluster -varieties for dual Poisson–Lie groups. I
TL;DR: In this article, a family of cluster X-varieties is associated to the dual Poisson-Lie group G* of a complex semi-simple Lie group G of adjoint type given with the standard Poisson structure.
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Theta functions and quiver Grassmannians.
TL;DR: In this paper, the authors use the relationship between cluster scattering diagrams and stability scattering diagrams to relate quiver representations with these diagrams, and give the Hall algebra theta functions which recover the cluster character formula by the Euler characteristic map.
6
•Posted Content
Triangulations and soliton graphs for totally positive Grassmannian
Rachel Karpman,Yuji Kodama +1 more
TL;DR: In this paper, the authors considered soliton graphs for the KP hierarchy, a family of commuting flows which are compatible with the KP equation, and showed that soliton graph is in bijection with Postnikov's plabic graphs.
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Cluster Algebras from Surfaces
Ralf Schiffler
- 01 Jan 2018
TL;DR: Cluster algebras were introduced by Fomin and Zelevinsky [17] in 2002 as discussed by the authors, and their original motivation was coming from canonical bases in Lie Theory.
6
References
•Book
Discriminants, Resultants, and Multidimensional Determinants
Izrailʹ Moiseevich Gelʹfand,Mikhail Kapranov,Andrei Zelevinsky +2 more
- 10 May 2013
TL;DR: The Cayley method of studying discriminants was used by Cayley as discussed by the authors to study the Cayley Method of Discriminants and Resultants for Polynomials in One Variable and for forms in Several Variables.
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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.
•Book
A Course in Convexity
Alexander Barvinok
- 01 Jan 2002
TL;DR: Convex sets at large Faces and extreme points ConveX sets in topological vector spaces Polarity, duality and linear programming Convex bodies and ellipsoids Faces of polytopes Lattices and convex bodies Lattice points and polyhedra.
Y-systems and generalized associahedra
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this paper, a simplicial complex A(b) is constructed for an arbitrary finite root system D, which is the face complex of the ordinary associahedron, whereas in type B it produces the Bott-Taubes polytope or cyclohedron.
•Book
Cluster Algebras and Poisson Geometry
Michael Gekhtman,Michael Shapiro,Alek Vainshtein +2 more
- 12 Nov 2010
TL;DR: Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in overlapping subsets of the same cardinality (the rank of the cluster algebra) connected by exchange relations as discussed by the authors.
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