Geodesic
The Multiplication Formulas of Weighted Quantum Cluster Functions
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

By applying the property of Ext-symmetry and the affine space structure of certain fibers, we introduce the notion of weighted quantum cluster functions and prove their multiplication formulas associated to abelian categories with Ext-symmetry and 2-Calabi–Yau triangulated categories with cluster-tilting objects.
Keywords: weighted quantum cluster functions, cluster categories, 2-Calabi–Yau triangulated categories, preprojective algebras. 
@article{SIGMA_2023_19_a96,
     author = {Zhimin Chen and Jie Xiao and Fan Xu},
     title = {The {Multiplication} {Formulas} of {Weighted} {Quantum} {Cluster} {Functions}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2023},
     volume = {19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a96/}
}
TY  - JOUR
AU  - Zhimin Chen
AU  - Jie Xiao
AU  - Fan Xu
TI  - The Multiplication Formulas of Weighted Quantum Cluster Functions
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2023
VL  - 19
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a96/
LA  - en
ID  - SIGMA_2023_19_a96
ER  - 
%0 Journal Article
%A Zhimin Chen
%A Jie Xiao
%A Fan Xu
%T The Multiplication Formulas of Weighted Quantum Cluster Functions
%J Symmetry, integrability and geometry: methods and applications
%D 2023
%V 19
%U http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a96/
%G en
%F SIGMA_2023_19_a96
Zhimin Chen; Jie Xiao; Fan Xu. The Multiplication Formulas of Weighted Quantum Cluster Functions. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a96/

[1] Berenstein A., Zelevinsky A., “Quantum cluster algebras”, Adv. Math., 195 (2005), 405–455, arXiv: math.QA/0404446 | DOI | MR | Zbl

[2] Buan A.B., Iyama O., Reiten I., Scott J., “Cluster structures for 2-Calabi–Yau categories and unipotent groups”, Compos. Math., 145 (2009), 1035–1079, arXiv: math.RT/0701557 | DOI | MR | Zbl

[3] Buan A.B., Marsh R., Reineke M., Reiten I., Todorov G., “Tilting theory and cluster combinatorics”, Adv. Math., 204 (2006), 572–618, arXiv: math.RT/0402054 | DOI | MR | Zbl

[4] Caldero P., Chapoton F., “Cluster algebras as Hall algebras of quiver representations”, Comment. Math. Helv., 81 (2006), 595–616, arXiv: math.RT/0410187 | DOI | MR | Zbl

[5] Caldero P., Keller B., “From triangulated categories to cluster algebras. II”, Ann. Sci. École Norm. Sup. (4), 39 (2006), 983–1009, arXiv: math.RT/0510251 | DOI | MR | Zbl

[6] Caldero P., Keller B., “From triangulated categories to cluster algebras”, Invent. Math., 172 (2008), 169–211, arXiv: math.RT/0506018 | DOI | MR | Zbl

[7] Chen X., Ding M., Zhang H., “The cluster multiplication theorem for acyclic quantum cluster algebras”, Int. Math. Res. Not., 2023 (2023), 20533–20573, arXiv: 2108.03558 | DOI | MR

[8] Ding M., Xu F., “A quantum analogue of generic bases for affine cluster algebras”, Sci. China Math., 55 (2012), 2045–2066, arXiv: 1105.2421 | DOI | MR | Zbl

[9] Fomin S., Zelevinsky A., “Cluster algebras. I Foundations”, J. Amer. Math. Soc., 15 (2002), 497–529, arXiv: math.RT/0104151 | DOI | MR | Zbl

[10] Geiss C., Leclerc B., Schröer J., “Semicanonical bases and preprojective algebras. II A multiplication formula”, Compos. Math., 143 (2007), 1313–1334, arXiv: math.RT/0509483 | DOI | MR | Zbl

[11] Geiss C., Leclerc B., Schröer J., “Kac–Moody groups and cluster algebras”, Adv. Math., 228 (2011), 329–433, arXiv: 1001.3545 | DOI | MR | Zbl

[12] Geiss C., Leclerc B., Schröer J., “Generic bases for cluster algebras and the Chamber ansatz”, J. Amer. Math. Soc., 25 (2012), 21–76, arXiv: 1004.2781 | DOI | MR | Zbl

[13] Hubery A., Acyclic cluster algebras via Ringel–Hall algebras, Preprint

[14] Keller B., “On triangulated orbit categories”, Doc. Math., 10 (2005), 551–581, arXiv: math.RT/0503240 | DOI | MR | Zbl

[15] Keller B., Plamondon P.G., Qin F., A refined multiplication formula for cluster characters

[16] Palu Y., “Cluster characters for 2-Calabi–Yau triangulated categories”, Ann. Inst. Fourier (Grenoble), 58 (2008), 2221–2248, arXiv: math.RT/0703540 | DOI | MR | Zbl

[17] Palu Y., “Cluster characters II: a multiplication formula”, Proc. Lond. Math. Soc. (3), 104 (2012), 57–78, arXiv: 0903.3281 | DOI | MR | Zbl

[18] Qin F., “Quantum cluster variables via Serre polynomials”, J. Reine Angew. Math., 668 (2012), 149–190, arXiv: 1004.4171 | DOI | MR | Zbl

[19] Qin F., “Triangular bases in quantum cluster algebras and monoidal categorification conjectures”, Duke Math. J., 166 (2017), 2337–2442, arXiv: 1501.04085 | DOI | MR | Zbl

[20] Riedtmann C., “Lie algebras generated by indecomposables”, J. Algebra, 170 (1994), 526–546 | DOI | MR | Zbl

[21] Rupel D., “On a quantum analog of the Caldero–Chapoton formula”, Int. Math. Res. Not., 2011 (2011), 3207–3236, arXiv: 1003.2652 | DOI | MR | Zbl

[22] Xiao J., Xu F., “Green's formula with ${\mathbb C}^*$-action and Caldero–Keller's formula for cluster algebras”, Representation Theory of Algebraic Groups and Quantum Groups, Progr. Math., 284, Birkhäuser, New York, 2010, 313–348, arXiv: 0707.1175 | DOI | MR | Zbl

[23] Xiao J., Xu F., Yang F., Motivic cluster multiplication formulas in 2-Calabi–Yau categories

[24] Xu F., “On the cluster multiplication theorem for acyclic cluster algebras”, Trans. Amer. Math. Soc., 362 (2010), 753–776, arXiv: 0711.3255 | DOI | MR | Zbl