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Research Article | Open Access

A generalized quantum cluster algebra of Kronecker type

Liqian Bai1, Xueqing Chen2, Ming Ding3( ), Fan Xu4
Department of Applied Mathematics, Northwestern Polytechnical University, Shaanxi 710072, China
Department of Mathematics, University of Wisconsin-Whitewater, Whitewater WI.53190, USA
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
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Abstract

The notion of generalized quantum cluster algebras was introduced as a natural generalization of Berenstein and Zelevinsky's quantum cluster algebras as well as Chekhov and Shapiro's generalized cluster algebras. In this paper, we focus on a generalized quantum cluster algebra of Kronecker type which possesses infinitely many cluster variables. We obtain the cluster multiplication formulas for this algebra. As an application of these formulas, a positive bar-invariant basis is explicitly constructed. Both results generalize those known for the Kronecker cluster algebra and quantum cluster algebra.

Keywords

generalized quantum cluster algebracluster variablecluster multiplication formulapositive basisKronecker quiver

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Electronic Research Archive
Volume 32 Issue 1,
January 2024
Pages 670-685
DOI: 10.3934/era.2024032

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Cite this article:
Bai L, Chen X, Ding M, et al. A generalized quantum cluster algebra of Kronecker type. Electronic Research Archive, 2024, 32(1): 670-685. https://doi.org/10.3934/era.2024032

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Received: 05 April 2023
Revised: 21 July 2023
Accepted: 02 August 2023
Published: 09 January 2024
©2024 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)