Graded cluster algebras
WebMay 23, 2014 · 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 … WebThe property of degeneration of modular graded Lie algebras, first investigated by B. Weisfeiler, is analyzed. Transitive irreducible graded Lie algebras over algebraically closed fields of characteristic with classi…
Graded cluster algebras
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WebJul 4, 2024 · Abstract. Let A be a path A∞ -algebra over a positively graded quiver Q: We prove that the derived category of A is triangulated equivalent to the derived category of kQ; which is viewed as a DG algebra with trivial differential. The main technique used in the proof is Koszul duality for DG algebras. Download to read the full article text. WebWe 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.
WebNov 4, 2024 · We also study the connection between cluster algebras with universal coefficients and cluster complexes. We show that for a full rank positively graded cluster algebra $\mathcal{A}$ of geometric and finite cluster type, the cluster algebra $\mathcal{A}^{\mathrm{univ}}$ with universal coefficients may be recovered as the … 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 …
WebSep 24, 2013 · Abstract: 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 … Webgraded cluster algebras are particularly important when studying cluster algebra structures on the (multi-)homogeneous coordinate rings of projective varieties, as one naturally expects. Another important generalisation concerns the quantization of cluster algebras, that is, a noncommutative version of the theory.
WebIt 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
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: fish tank newcastleWebSep 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 … candy cane christmas cast membersWebOct 15, 2024 · In the graded cluster algebra A ( h n, m, Q), the cardinality of the set of cluster variables of degree ( d 1 d 2) is determined by d 1: it is equal to the cardinality of … candy cane christmas candlesWebThis article is organized as follows. §§ 2, 4 are preliminaries for cluster algebras and graded quiver varieties respectively. In § 5 we introduce the category 𝒞 1 subscript 𝒞 1 \mathscr{C}_{1} script_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT following [ 31 ] and study the corresponding graded quiver varieties. fish tank nettingWebFor 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 ... candy cane christmas cookieWeb3 Multi-graded seeds and cluster algebras The natural definition for a multi-graded seed is as follows. Definition 3.1 A multi-graded seed is a triple (x,B,G) such that (a) (x = (X1,...,Xr),B) is a seed of cardinality r and (b) G is an r ×d integer matrix such that BT G = 0. From now on, we use the term “graded” to encompass multi-graded ... candy cane chocolate cherry pepper seedsWeb4 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 ... fish tank news