Cluster algebras via cluster categories with infinite-dimensional morphism spaces

By Pierre-Guy Plamondon. Compositio Mathematica, 2010

130 citations recorded by the source corpus.

Abstract We apply our previous work on cluster characters for Hom-infinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky’s Cluster algebras IV [Compositio Math. 143 (2007), 112–164] for skew-symmetric cluster algebras. We also construct an explicit bijection sending certain objects of the cluster category to the decorated representations of Derksen, Weyman and Zelevinsky, and show that it is compatible with mutations in both settings. Using this map, we give a categorical interpretation of the E-invariant and show that an arbitrary decorated representation with vanishing E-invariant is characterized by its g-vector. Finally, we obtain a substitution formula for cluster characters of not necessarily rigid objects.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/14340dfb92f7218f43b1a322b1724a7247e7230c

How researchers see this paper

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In Algebra and Number Theory

1. B. Keller
2. Hugh Thomas
3. Thomas Brustle
4. Dong Ping Yang
5. Dong Yang
6. Jie Zhang
7. David E. Speyer
8. I. Reiten

In Geometry and Topology

1. B. Keller
2. Hugh Thomas
3. Thomas Brustle
4. Dong Ping Yang
5. Dong Yang
6. Jie Zhang
7. David E. Speyer
8. Osamu Iyama

In Statistical and Nonlinear Physics

1. Hugh Thomas
2. Thomas Brustle
3. Dong Yang
4. Dong Ping Yang
5. David E. Speyer
6. B. Keller
7. Jie Zhang
8. Osamu Iyama

Citations in the corpus

• Derived equivalences from mutations of quivers with potential (2009, background): “Let us point out that mutation of quivers with potential goes back to the insight of mathematical physicists, cf. for example Section 5.2 of [14], Section 2.3 of [6], Section 6 of [15] and Section 3 of [35].”
• Universal geometric cluster algebras (2012, unknown): “(See also [16, 18].)”
• Cluster algebras and derived categories (2012, methodology): “The conjecture was recently proved in [19] using the additive categorification of [117] and techniques from [17] [20].”
• On the cluster category of a marked surface without punctures (2010, background): “Viewing the indecomposable objects in C0 as noncontractible curves and noncontractible closed curves in (S,M), or as indecomposable decorated representation of (Q0,W0), then [Plamondon 2010] shows the compatibility between the equivalence μ̄i and the mutation of decorated representations of (Q0,W0).”
• Ordered Exchange Graphs (2013, methodology): “The following remarkable result was conjectured by Fomin and Zelevinky [34, Conjecture 4.14 (2)] (for all cluster algebras) and proved by Cerulli, Keller, Labardini and Plamondon [23, Corollary 3.6] (for all skew-symmetric cluster algebras) by using additive categorification, see also [42, Theorem 3”