Locally acyclic cluster algebras

By Greg Muller. 2011

102 citations recorded by the source corpus.

Abstract This paper studies cluster algebras locally, by identifying a special class of localizations which are themselves cluster algebras. A ‘locally acyclic cluster algebra’ is a cluster algebra which admits a finite cover (in a geometric sense) by acyclic cluster algebras. Many important results about acyclic cluster algebras extend to locally acyclic cluster algebras (such as being finitely generated, integrally closed, and equaling their upper cluster algebra), as well as a result which is new even for acyclic cluster algebras (regularity over Q when the exchange matrix has full rank). Several techniques are developed for determining whether a cluster algebra is locally acyclic. Cluster algebras of marked surfaces with at least two boundary marked points are shown to be locally acyclic, providing a large class of examples of cluster algebras which are locally acyclic but not acyclic. Some specific examples are worked out in detail.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/48f46a18fd5f6e398e44f5add6783ebea9e0c338

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Cohomology of cluster varieties, I: Locally acyclic case (2016, methodology): “ster localization. The subvariety A(B˜S) ⊂ A(B˜) is called a cluster chart of A. We say that a cluster algebra A(B˜) is locally acyclic if A can be covered by ﬁnitely many acyclic cluster charts; see [Mul13]. We deﬁne an edge e= i→ j of ~Γ to be a separating edge if there is no bi-inﬁnite path throu”
Positive geometries and canonical forms (2017, background): “metric type) and let X = Spec(A) be the corresponding ane cluster variety [15]; thus the ring of regular functions on X is equal to A. We will assume that X is a smooth complex manifold, see e.g. [16,17] for some discussion of this. The generators of Aas a ring are grouped into clusters (x 1;x 2;:::”
Skein algebras and cluster algebras of marked surfaces (2012, methodology): “Theorem 11.2. [Mul13, Theorems 10.6, 10.10] The cluster algebra A 1() is locally acyclic if and only if has at least two marked points in each connected component of . Remark 11.3. Marked surfaces in [Mul13] are allowed to have interior marked points, so the statements there are more general. 11.3. ”
Positroid varieties and cluster algebras (2019, methodology): “prove the rst inclusion A(Q D) C[ v;w] in Section 3, see Corollary 3.7(ii). We show the second inclusion A(Q D) C[ v;w] in Section 4, using the results of Muller{ Speyer [MS17, MS16], of Muller [Mul13], and of Berenstein{Fomin{Zelevinsky [BFZ05]. Throughout the paper, we x a positive integer n, an”