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Qsystem Cluster Algebras, Paths and Total Positivity
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2010
"... In the first part of this paper, we provide a concise review of our method of solution of the Ar Qsystems in terms of the partition function of paths on a weighted graph. In the second part, we show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connecti ..."
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In the first part of this paper, we provide a concise review of our method of solution of the Ar Qsystems in terms of the partition function of paths on a weighted graph. In the second part, we show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connection to the theory of planar networks introduced in the context of totally positive matrices by Fomin and Zelevinsky. As an illustration of the further generality of our method, we apply it to give a simple solution for the rank 2 affine cluster algebras studied by Caldero and Zelevinsky.
Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster
 variables, International Mathematics Research Notices 016 (2013) 22 pp., arXiv :1111.3963
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Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials
, 2012
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Matrix formulae and skein relations for cluster algebras from surfaces
 ARXIV:1108.3382. BASES FOR CLUSTER ALGEBRAS FROM SURFACES 53
, 2011
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Canonically positive basis of cluster algebras of type A (1) 2
"... with every choice of coefficients. We find linear basis of such algebras which are called canonically positive: positive linear combinations of the elements of such basis coincide with the cone of positive elements of the cluster algebra. Abstract. We study cluster algebras of type A (1) 2 Contents ..."
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with every choice of coefficients. We find linear basis of such algebras which are called canonically positive: positive linear combinations of the elements of such basis coincide with the cone of positive elements of the cluster algebra. Abstract. We study cluster algebras of type A (1) 2 Contents
Skein algebras and cluster algebras of marked surfaces
"... Abstract. This paper defines several algebras associated to an oriented surface Σ with a finite set of marked points on the boundary. The first is the skein algebra Skq(Σ), which is spanned by links in the surface which are allowed to have endpoints at the marked points, modulo several locally defi ..."
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Abstract. This paper defines several algebras associated to an oriented surface Σ with a finite set of marked points on the boundary. The first is the skein algebra Skq(Σ), which is spanned by links in the surface which are allowed to have endpoints at the marked points, modulo several locally defined relations. The product is given by superposition of links. A basis of this algebra is given, as well as several algebraic results. When Σ is triangulable, the quantum cluster algebra Aq(Σ) and quantum upper cluster algebra Uq(Σ) can be defined. These are algebras coming from the triangulations of Σ and the elementary moves between them.