\(F\)-polynomials in quantum cluster algebras. (English) Zbl 1253.16016
A quantum cluster algebra is a noncommutative deformation of a cluster algebra. In the present paper the author shows existence of quantum \(F\)-polynomials for quantum cluster algebras. These polynomials have the property that any cluster variable in a quantum cluster algebra may be computed (up to a power of the deformation parameter \(q\)) from the initial extended cluster using quantum \(F\)-polynomials. Specialization \(q=1\) recovers the classical case. The paper also provides a recursive formula for computation of quantum \(F\)-polynomials. For type \(A\) quantum cluster algebras the paper gives an explicit formula for all corresponding quantum \(F\)-polynomials.
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 16G20 | Representations of quivers and partially ordered sets |
| 13F60 | Cluster algebras |
| 16S80 | Deformations of associative rings |
| 16T20 | Ring-theoretic aspects of quantum groups |
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
| 20G42 | Quantum groups (quantized function algebras) and their representations |
Keywords:
quantum cluster algebras; quantum polynomials; deformations; principal coefficients; \(\mathbf g\)-vectorsReferences:
| [1] | Berenstein, A., Zelevinsky, A.: Quantum cluster algebras. Adv. Math. 195, 405–455 (2005) · Zbl 1124.20028 · doi:10.1016/j.aim.2004.08.003 |
| [2] | Buan, A., Vatne, D.: Derived equivalence classification for cluster-tilted algebras of type $\(\backslash\)mbox{A}_{n}$ . J. Algebra 319, 2723–2738 (2008) · Zbl 1155.16010 · doi:10.1016/j.jalgebra.2008.01.007 |
| [3] | Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations II: applications to cluster algebras. J. Am. Math. Soc. 23, 749–790 (2010) · Zbl 1208.16017 · doi:10.1090/S0894-0347-10-00662-4 |
| [4] | Fock, V.V., Goncharov, A.B.: Cluster ensembles, quantization and the dilogarithm. Annales scientifiques de l’ENS 42 6, 865–930 (2009) · Zbl 1180.53081 |
| [5] | Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces. Part I: cluster complexes. Acta Math. 201, 83–146 (2008) · Zbl 1263.13023 · doi:10.1007/s11511-008-0030-7 |
| [6] | Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Am. Math. Soc. 15, 497–529 (2002) · Zbl 1021.16017 · doi:10.1090/S0894-0347-01-00385-X |
| [7] | Fomin, S., Zelevinsky, A.: Cluster algebras II: finite type classification. Invent. Math. 154, 63–121 (2003) · Zbl 1054.17024 · doi:10.1007/s00222-003-0302-y |
| [8] | Fomin, S., Zelevinsky, A.: Cluster algebras IV: coefficients. Compos. Math. 143, 112–164 (2007) · Zbl 1127.16023 · doi:10.1112/S0010437X06002521 |
| [9] | Fu, C., Keller, B.: On cluster algebras with coefficients and 2-Calabi-Yau categories. Trans. Am. Math. Soc. 362(2), 859–895 (2010) · Zbl 1201.18007 · doi:10.1090/S0002-9947-09-04979-4 |
| [10] | Musiker, G., Williams, L.: Combinatorial formulas for F-polynomials and g-vectors for cluster algebras of classical type (unpublished) |
| [11] | Yang, S., Zelevinsky, A.: Cluster algebras of finite type via Coxeter elements and principal minors. Transform. Groups 13, 855–895 (2008) · Zbl 1177.16010 · doi:10.1007/s00031-008-9025-x |
| [12] | Zelevinsky, A.: Quantum Cluster Algebras: Oberwolfach Talk. arXiv:math/0502260 (2005) · Zbl 1124.20028 |
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