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) WebMar 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 …

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WebTheorem 1. Let Rbe an artinian standard graded algebra and M a nitely gen-erated graded R-module, generated in a single degree. If Exti R (M;M) = 0 = Exti R (M;R) for all i>0, then Mis free. Our second result concerns Gorenstein rings. The hypothesis of the following theorem is known to be satis ed by generic standard graded Gorenstein algebras 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 ... how many ova are present at birth

[0905.0002] Quiver varieties and 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 ... 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 . http://arxiv-export3.library.cornell.edu/pdf/1510.04109v1 how many ovaries do we have