\(F\)-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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.