On quantum cluster algebras of finite type

Ming Ding

doi:10.1007/s11464-011-0104-2

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Front. Math. China ›› 2011, Vol. 6 ›› Issue (2) : 231-240. DOI: 10.1007/s11464-011-0104-2

Research Article

RESEARCH ARTICLE

On quantum cluster algebras of finite type

Ming Ding¹^,^a

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Abstract

We extend the definition of a quantum analogue of the Caldero-Chapoton map defined by D. Rupel. When Q is a quiver of finite type, we prove that the algebra [inline-graphic not available: see fulltext](Q) generated by all cluster characters is exactly the quantum cluster algebra [inline-graphic not available: see fulltext](Q).

Keywords

Cluster variable / quantum cluster algebra

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Ming Ding. On quantum cluster algebras of finite type. Front. Math. China, 2011, 6(2): 231‒240 https://doi.org/10.1007/s11464-011-0104-2

References

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[1.]	Berenstein A., Fomin S., Zelevinsky A. Cluster algebras III: Upper bounds and double Bruhat cells. Duke Math J, 2005, 126, 1-52 CrossRef Google scholar

[2.]	Buan A., Marsh R., Reineke M., Reiten I., Todorov G. Tilting theory and cluster combinatorics. Adv Math, 2006, 204, 572-618 CrossRef Google scholar

[3.]	Berenstein A., Zelevinsky A. Quantum cluster algebras. Adv Math, 2005, 195, 405-455 CrossRef Google scholar

[4.]	Caldero P., Chapoton F. Cluster algebras as Hall algebras of quiver representations. Comm Math Helv, 2006, 81, 595-616 CrossRef Google scholar

[5.]	Caldero P., Keller B. From triangulated categories to cluster algebras. Invent Math, 2008, 172 1 169-211 CrossRef Google scholar

[6.]	Ding M, Xiao J, Xu F. Integral bases of cluster algebras and representations of tame quivers. arXiv:0901.1937 [math.RT]

[7.]	Ding M, Xu F. Bases of the quantum cluster algebra of the Kronecker quiver. arXiv: 1004.2349v4 [math.RT]

[8.]	Ding M, Xu F. The multiplication theorem and bases in finite and affine quantum cluster algebras. arXiv:1006.3928v3 [math.RT]

[9.]	Fomin S., Zelevinsky A. Cluster algebras. I. Foundations. J Amer Math Soc, 2002, 15 2 497-529 CrossRef Google scholar

[10.]

Fomin

, Zelevinsky

Cluster algebras. II. Finite type classification. Invent Math, 2003, 154 1 63-121

CrossRef Google scholar

[11.]

Geiss C, Leclerc B, Schröer J. Kac-Moody groups and cluster algebras. arXiv: 1001.3545v2 [math.RT]

[12.]

Geiss C, Leclerc B, Schröer J. Generic bases for cluster algebras and the Chamber Ansatz. arXiv:1004.2781v2 [math.RT]

[13.]

Grabowski J, Launois S. Quantum cluster algebra structures on quantum Grassmannians and their quantum Schubert cells: the finite-type cases. Int Math Res Notices, 2010, doi: 10.1093/imrn/rnq153

[14.]

Hubery A. Acyclic cluster algebras via Ringel-Hall algebras. Preprint, 2005

[15.]

Lampe P. A quantum cluster algebra of Kronecker type and the dual canonical basis. Int Math Res Notices, 2010, doi: 10.1093/imrn/rnq162

[16.]

Qin F. Quantum cluster variables via Serre polynomials. arXiv:1004.4171v2 [math.QA]

[17.]

Rupel D. On a quantum analogue of the Caldero-Chapoton Formula. Int Math Res Notices, 2010, doi:10.1093/imrn/rnq192