A generalized quantum cluster algebra of Kronecker type-Reference-Cited by-同舟云学术

A generalized quantum cluster algebra of Kronecker type

Published:2024 Issue:1 Volume:32 Page:670-685
ISSN:2688-1594
Container-title:Electronic Research Archive
language:
Short-container-title:era

Author:

Bai Liqian¹,Chen Xueqing²,Ding Ming³,Xu Fan⁴

Affiliation:

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

Abstract

<abstract><p>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.</p></abstract>

Publisher

American Institute of Mathematical Sciences (AIMS)

Reference24 articles.

1. S. Fomin, A. Zelevinsky, Cluster algebras Ⅰ. foundations, J. Amer. Math. Soc., 15 (2002), 497–529. https://doi.org/10.1090/S0894-0347-01-00385-X

2. S. Fomin, A. Zelevinsky, Cluster algebras Ⅱ. finite type classification, Invent. Math., 154 (2003), 63–121. https://doi.org/10.1007/s00222-003-0302-y

3. A. Berenstein, A. Zelevinsky, Quantum cluster algebras, Adv. Math., 195 (2005), 405–455. https://doi.org/10.1016/j.aim.2004.08.003

4. L. Chekhov, M. Shapiro, Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables, Int. Math. Res. Notices, 2014 (2014), 2746–2772. https://doi.org/10.1093/imrn/rnt016

5. T. Nakanishi, Structure of seeds in generalized cluster algebras, Pacific J. Math., 277 (2015), 201–218. https://doi.org/10.2140/pjm.2015.277.201