For the sake of simplicity one may assume that a cluster algebra of rank n is a subring of a given field of rational functions in n variables generated by all cluster variables.
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This very definition emphasizes the geometric and combinatorial nature of a cluster algebra, which is not surprising if one remembers that cluster algebras were defined in the context of Lie theory. Thus, examples of cluster algebras include coordinate rings of Grassmannians, Lagrangian-Grassmannians, and several other flag varieties important in the representation theory of algebraic groups. There are several connections of cluster algebras to other subjects in mathematics. Reineke and A. Williams Bull. By a natural approach, going from categories of modules over hereditary algebras, cluster categories were introduced to categorify the essential components in the definition of a cluster algebra in some important cases, basically replacing some components of the definition by a corresponding categorical concept which captures the essence of the component.
- Topics in cluster algebras.
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Gekhtman and M. Shapiro AMS, This roll-call is only intended to show the variety of diverse subjects where cluster algebras play a fundamental role. Thus, the task of presenting an introduction to this relatively new field in mathematics involves making several choices. The basic notions and properties of cluster algebras are the content of the first three chapters. Here the emphasis is on motivation, and many explicit examples are discussed at length.
Proofs are given of many of the statements, but the reader is assumed to be willing to fill in some details or go the detailed references to find a proof of a resulted being quoted. The next three chapters treat the finite type classification theory in terms of the ubiquitous Dynkin diagrams and a simplicial complex associated to a cluster algebra of finite type. The last three chapters are devoted to illustrating how cluster algebras arise in some other areas of mathematics. From the Zamolodchikov periodicity conjecture in mathematical physics to quivers associated to marked Riemann surfaces, and to the cluster algebra structure of the coordinate ring of the Grassmannian variety.
As may be gathered from the above remarks, this is a beautiful subject, simple from its definition, and yet so basic that its interactions with many diverse fields of mathematics seem very natural. Marsh, A category of wide subcategories. Preprint , February Updated May Accepted to appear in International Mathematics Research Notices. Marsh and K. The B-model connection and mirror symmetry for Grassmannians. Preprint , July Revised December Marsh and P. Pascal arrays: counting Catalan sets. Preprint , December List of publications ArXiv preprints Google Scholar page.
Marsh and J. Braid groups and quiver mutation. Pacific Journal of Mathematics , Preprint version, March Baur, A. King, and R. Dimer models and cluster categories of Grassmannians. London Math. Preprint version, June Marsh and Y. Nearly Morita equivalences and rigid objects. Nagoya Mathematical Journal, , , Marsh and I.
- Total positivity and cluster algebras.
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Rigid and Schurian modules over cluster-tilted algebras of tame type. Mathematische Zeitschrift, 3 , , Click here for Preprint version. Preprint version M.
Barot and R. Reflection group presentations arising from cluster algebras. Preprint version. Marsh and S. The geometry of Brauer graph algebras and cluster mutations. Algebra , , A circular order on edge-coloured trees and RNA m-diagrams. Adv Appl. Math 54 , Coloured quivers for rigid objects and partial triangulations: The unpunctured case.
Proceedings of the London Mathematical Society 2 , Free-access link.
Baur , A. Torsion pairs and rigid objects in tubes. Algebras and Representation Theory 17 2 : The final publication is available at link.
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From triangulated categories to module categories via localization II: Calculus of fractions. From triangulated categories to module categories via localisation. Baur and R. A geometric model of tube categories. Algebra , Categorification of a frieze pattern determinant. Journal of Combinatorial Theory, Series A , Fordy and R. Cluster mutation-periodic quivers and associated Laurent sequences. Journal of Algebraic Combinatorics, 34 , no.
Positivity for quantum cluster algebras | Annals of Mathematics
Tiling bijections between paths and Brauer diagrams. Journal of Algebraic Combinatorics, 33 , no. Brown , P. Guil Asensio and R. Smith's 65th Birthday.
Denominators in cluster algebras of affine type. Journal of Algebra , no. Buan , R. Marsh and D. Cluster structures from 2-Calabi-Yau categories with loops. Mathematische Zeitschrift no. Denominators of cluster variables.
Journal of the London Mathematical Society 79, no. Frieze patterns for punctured discs. Journal of Algebraic Combinatorics 30 , Issue 3 , Pages A geometric description of m-cluster categories. Cluster mutation via quiver representations. Commentarii Mathematici Helvetici 83 Issue 1 ,