Positivity and canonical bases in rank 2 cluster algebras of finite and affine types

By Paul Sherman, Andrei Zelevinsky. 2003

162 citations recorded by the source corpus.

The main motivation for the study of cluster algebras initiated in math.RT/0104151, math.RA/0208229 and math.RT/0305434 was to design an algebraic framework for understanding total positivity and canonical bases in semisimple algebraic groups. In this paper, we introduce and explicitly construct the canonical basis for a special family of cluster algebras of rank 2.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d0b7e054913935235a9b4b8e1bca966dd5553ea5

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Citations in the corpus

From triangulated categories to cluster algebras II (2005, methodology): “It was obtained in a direct way in [20].”
On Cluster Algebras Arising from Unpunctured Surfaces (2007, background): “In [ SZ , CZ, Z, MP] cluster expansions for cluster algebras of rank 2 are given, in [Pr, CP, FZ3] the case A is considered and in [M] a cluster expansion for cluster algebras of finite type is given for clusters that lie in a bipartite seed.”
Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials (2011, background): “The existence of atomic basis has been proved only for a few types of cluster algebras ([11],[7], [6],[29]).”
Quiver varieties and cluster algebras (2009, background): “There are many attempts to construct a base for the cluster algebra corresponding to this example in the cluster algebra literature ([56, 9, 19, 18] and [25] in a wider context).”
A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis (2010, methodology): “In this subsection we describe the semicanonical basis of Caldero-Zelevinsky ([6]), the canonical basis of Sherman-Zelevinsky ([36]), and the dual semicanonical basis of Geiß-Leclerc-Schr¨oer ([15]).”
Greedy elements in rank 2 cluster algebras (2012, methodology): “In an unpublished follow-up to [11] they have introduced a special family of greedy elements in (the completion of) an arbitrary rank 2 cluster algebra A, and made several conjectures about them, including the claim that all these elements are indecomposable positive elements, and that they form a Z”
Cluster algebras and derived categories (2012, background): “(16) Notice that in total, we find 6 distinct c-vectors and that these are in natural bijection with the (positive and negative) roots of the root system corresponding to the underlying graph A2 of the quiver Q: We simply map a c-vector with components c1 and c2 to the root c1α1 + c2α2, where α1…”
Tensor diagrams and cluster algebras (2012, background): “Zelevinsky [50], while the square lies in the dual semicanonical basis, see [17,40].”