2024 Graded cluster algebras

Graded cluster algebras

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WebSep 15, 2024 · Graded cluster algebras. Let A be a cluster algebra A (B ˜) or a quantum cluster algebra A (Λ, B ˜). Then A is a Z-graded cluster algebra or Z-graded quantum cluster algebra, respectively, if the following hold: (i) There is a direct sum decomposition A = ⨁ g ∈ Z A g such that A g ⋅ A h ⊆ A g + h for all g, h ∈ Z; (ii) WebOct 15, 2024 · Graded cluster algebras arising from marked surfaces Preliminaries. We refer the reader to [6] for the general background on cluster algebras in the context we …

MR2948875 (Review) 13F60 (81R50) (F-PARIS7-IMJ)

WebIn the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra … WebWe begin by creating a simple cluster algebra and printing its initial exchange matrix: sage: A = ClusterAlgebra( ['A', 2]); A A Cluster Algebra with cluster variables x0, x1 and no coefficients over Integer Ring sage: A.b_matrix() [ 0 1] [-1 0] A is of finite type so we can explore all its exchange graph: edny shields

Graded quantum cluster algebras of infinite rank as colimits

WebIn this section, we consider two constructions that produce graded quantum cluster algebra structures on skew-Laurent extensions of a given graded quantum cluster algebra. The … WebThe most influential and powerful invariant is the Chekanov-Eliashberg differential graded algebra, which set apart the first non-classical Legendrian pair and stimulated many subsequent developments. ... microlocal sheaf theory and cluster algebras to find the first examples of Legendrian links with infinitely many Lagrangian fillings. Watch ... WebOct 14, 2015 · Similarly, those interested in graded classical cluster algebras may recover that case by con- sidering the multiplicatively skew-symmetric matrix with r ( M ) k j = 1 for all k ≤ j . edny sdny local rules

Graded cluster algebras arising from marked surfaces

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Web4 N. Hanihara Theorem 1.2 (Theorem 4.6(4)). Let R be a graded bimodule (𝑑+1)-CY algebra of a-invariant a satisfying (R0) and (R1). Then the subcategory add{𝑅(−𝑖)[𝑖] 𝑖∈Z}⊂qperZ𝑅is a (𝑑+𝑎)-cluster tilting subcategory. For example, by setting R to be the polynomial ring with standard positive grading, we deduce that the derived category of coherent sheaves over the ... WebWe consider gradings on cluster algebras generated by 3 × 3 skew-symmetric matrices. We show that, except in one particular case, mutation-cyclic matrices give rise to gradings in which all occurring degrees are positive and have … edny tiscioneWebMay 23, 2014 · This completes our general theory of graded quantum cluster algebras. Now we turn to our application, the existence of a quantum cluster algebra structure on … constantine the 11

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Graded cluster algebras

WebIn the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a ... WebGraded seeds and graded quantum cluster algebras. Berenstein and Zelevinsky [3, Definition 6.5] have given a definition of graded quantum seeds, which give rise to module gradings but not algebra gradings. In what follows, we will have need of algebra gradings on quantum cluster algebras and so we now give a different definition of a graded ...

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WebSep 24, 2013 · In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of … WebFor finite type, it can be regarded as an un-graded analogue of Jacobian algebra of certain quiver with potential appeared in the theory of (monoidal) categorification of cluster algebras. In this talk, we will present an interpretation of graded structures of the preprojective algebra of general type, in terms of a multi-parameter deformation ...

WebMar 22, 2024 · graded cluster algebras arising from marked surf aces with punctures 3 and constructs an initial ideal triangulation with the required properties, which in- 1 dicates that any ideal triangulation... WebIn the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster …

WebJan 10, 2013 · In the quantum setting, we use this grading framework to give a construction that behaves somewhat like twisting, in that it produces a new quantum cluster algebra … Webgraded (dg) categories. It is motivated by the study of Calabi-Yau categoriﬁcations of cluster algebras. We refer the reader to [B. Keller, in Triangulated categories, 76–160, London Math. Soc. ... Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier 59 (2009), no. 6, 2525–2590 ...

WebJul 11, 2015 · Graded cluster algebras Abstract. In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since …

http://arxiv-export3.library.cornell.edu/pdf/1510.04109v1 edny summonsWebMar 1, 2014 · In this work, our main aim is to bring these three parts of cluster algebra theory together, to study graded quantum cluster algebras of infinite rank. More specifically, Assem, Dupont and Schiffler ([1]) have introduced a category Clus of rooted cluster algebras. The objects of Clus are pairs consisting of a cluster algebra and a … edny recordsWebIt is shown that many results on cluster algebras (the Laurent phenomenon, the classiﬁcation of ﬁnite-type cluster algebras) extend to the quantum case. The important open problem is to show that quantum cluster monomials have non-negative coefﬁcients. The paper under review deals with skew-symmetric acyclic quantum cluster algebras. The constantine the first popeWebIn this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a de nition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for nite type cluster algebras without coe cients, giving a full classi ... edny related caseWebMay 9, 2012 · cluster algebras, graded quiver varieties, deformed Grothendieck rings, and t-analogues of q -characters. More details can b e found in [ BZ05 ] [ Nak01 ] [ Nak04 ] [ Nak11 ], or in [ Qin10 ] edny officeWebWe then study gradings for finite type cluster algebras without coefficients, giving a full classification. Translating the definition suitably again, we obtain a notion of multi-grading for (generalised) cluster categories. This setting allows us to prove additional properties of graded cluster algebras in a wider range of cases. constantine the grWebSep 27, 2013 · A categorification of Grassmannian cluster algebras. We describe a ring whose category of Cohen–Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k ‐planes in n ‐space. More precisely, there is a cluster character defined on the category … constantine the first christian emperor