Generic bases of skew-symmetrisable affine type cluster algebras

Lang Mou; Xiuping Su

doi:10.1007/s00029-026-01159-5

Generic bases of skew-symmetrisable affine type cluster algebras

Lang Mou
, Xiuping Su

Department of Mathematical Sciences

University of Cologne

Research output: Contribution to journal › Article › peer-review

Abstract

Geiss, Leclerc and Schroeer introduced a class of 1-Iwanaga--Gorenstein algebras $H$ associated to symmetrizable Cartan matrices with acyclic orientations, generalizing the path algebras of acyclic quivers. Indecomposable rigid $H$-modules of finite projective dimension are known in bijection with non-initial cluster variables of the corresponding Fomin--Zelevinsky cluster algebra. We prove in all affine types that a Caldero--Chapoton type function on these modules proposed by Geiss--Leclerc--Schr\"oer coincide with cluster variables. By taking generic Caldero--Chapoton functions on varieties of modules of finite projective dimension, we obtain bases for affine type cluster algebras with full-rank coefficients containing all cluster monomials.

Original language	English
Number of pages	35
Journal	Selecta Mathematica
Volume	32
Early online date	12 Jun 2026
DOIs	https://doi.org/10.1007/s00029-026-01159-5
Publication status	Published - 12 Jun 2026

Acknowledgements

LM thanks Jan Schröer for valuable conversations. LM was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 281071066 – TRR 191. LM and XS received support from the Royal Society by the Newton International Fellowships Alumni AL\231002.

Keywords

Affine cluster algebra
GLS algebra
Caldero–Chapoton function
Generic basis

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10.1007/s00029-026-01159-5

https://arxiv.org/abs/2409.03954

Cite this

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title = "Generic bases of skew-symmetrisable affine type cluster algebras",

abstract = "Geiss, Leclerc and Schroeer introduced a class of 1-Iwanaga--Gorenstein algebras \$H\$ associated to symmetrizable Cartan matrices with acyclic orientations, generalizing the path algebras of acyclic quivers. Indecomposable rigid \$H\$-modules of finite projective dimension are known in bijection with non-initial cluster variables of the corresponding Fomin--Zelevinsky cluster algebra. We prove in all affine types that a Caldero--Chapoton type function on these modules proposed by Geiss--Leclerc--Schr\textbackslash{}{"}oer coincide with cluster variables. By taking generic Caldero--Chapoton functions on varieties of modules of finite projective dimension, we obtain bases for affine type cluster algebras with full-rank coefficients containing all cluster monomials.",

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AU - Su, Xiuping

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N2 - Geiss, Leclerc and Schroeer introduced a class of 1-Iwanaga--Gorenstein algebras $H$ associated to symmetrizable Cartan matrices with acyclic orientations, generalizing the path algebras of acyclic quivers. Indecomposable rigid $H$-modules of finite projective dimension are known in bijection with non-initial cluster variables of the corresponding Fomin--Zelevinsky cluster algebra. We prove in all affine types that a Caldero--Chapoton type function on these modules proposed by Geiss--Leclerc--Schr\"oer coincide with cluster variables. By taking generic Caldero--Chapoton functions on varieties of modules of finite projective dimension, we obtain bases for affine type cluster algebras with full-rank coefficients containing all cluster monomials.

AB - Geiss, Leclerc and Schroeer introduced a class of 1-Iwanaga--Gorenstein algebras $H$ associated to symmetrizable Cartan matrices with acyclic orientations, generalizing the path algebras of acyclic quivers. Indecomposable rigid $H$-modules of finite projective dimension are known in bijection with non-initial cluster variables of the corresponding Fomin--Zelevinsky cluster algebra. We prove in all affine types that a Caldero--Chapoton type function on these modules proposed by Geiss--Leclerc--Schr\"oer coincide with cluster variables. By taking generic Caldero--Chapoton functions on varieties of modules of finite projective dimension, we obtain bases for affine type cluster algebras with full-rank coefficients containing all cluster monomials.

KW - Affine cluster algebra

KW - GLS algebra

KW - Caldero–Chapoton function

KW - Generic basis

U2 - 10.1007/s00029-026-01159-5

DO - 10.1007/s00029-026-01159-5

M3 - Article

SN - 1022-1824

VL - 32

JO - Selecta Mathematica

JF - Selecta Mathematica

ER -

Generic bases of skew-symmetrisable affine type cluster algebras

Abstract

Acknowledgements

Keywords

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