Czechoslovak Mathematical Journal, Vol. 61, No. 4, pp. 1077-1090, 2011

# Bar-invariant bases of the quantum cluster algebra of type $A^{(2)}_2$

## Xueqing Chen, Ming Ding, Jie Sheng

X. Chen, Department of Mathematical and Computer Sciences, University of Wisconsin-Whitewater, 800 W. Main Street, Whitewater, WI 53190, U.S.A., e-mail: chenx@uww.edu; M. Ding, Center for Advanced Study, Tsinghua University, Beijing 10084, P. R. China, e-mail: m-ding04@mails.tsinghua.edu.cn; J. Sheng, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China, e-mail: shengjie@amss.ac.cn

Abstract: We construct bar-invariant $\mathbb{Z}[q^{\pm1/2}]$-bases of the quantum cluster algebra of the valued quiver $A^{(2)}_2$, one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947-974.

Keywords: quantum cluster algebra, $\mathbb{Z}[q^{\pm1/2}]$-basis, valued quiver

Classification (MSC 2010): 16G20, 20G42, 14M17

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