A generalized quantum cluster algebra of Kronecker type

Liqian Bai; Xueqing Chen; Ming Ding; Fan Xu; Liqian Bai; Xueqing Chen; Ming Ding; Fan Xu

doi:10.3934/era.2024032

Electronic Research Archive

2024, Volume 32, Issue 1: 670-685. doi: 10.3934/era.2024032

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Research article

A generalized quantum cluster algebra of Kronecker type

1.
Department of Applied Mathematics, Northwestern Polytechnical University, Shaanxi 710072, China
2.
Department of Mathematics, University of Wisconsin-Whitewater, Whitewater WI.53190, USA
3.
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
4.
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received: 05 April 2023 Revised: 21 July 2023 Accepted: 02 August 2023 Published: 09 January 2024

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.
- generalized quantum cluster algebra,
- cluster variable,
- cluster multiplication formula,
- positive basis,
- Kronecker quiver
Citation: Liqian Bai, Xueqing Chen, Ming Ding, Fan Xu. A generalized quantum cluster algebra of Kronecker type[J]. Electronic Research Archive, 2024, 32(1): 670-685. doi: 10.3934/era.2024032

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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.

References

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