ActaMathematicaSinica,.PDF

Acta Mathematica Sinica, English Series Jun., 2012, Vol. 28, No. 6, pp. 1169–1178 Acta Mathematica Sinica, Published online: December 7, 2011 English Series DOI: 10.1007/s10114-011-0344-9 Springer-Verlag Berlin Heidelberg Http://www.ActaM The Editorial Office of AMS 2012 Bases of the Quantum Cluster Algebra of the Kronecker Quiver Ming DING School of Mathematical Sciences, Nankai University, Tianjin 300071, P. R. China E-mail : m-ding 04 @ Fan XU Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China E-mail : fanxu@ ± 1 Abstract We construct bar-invariant Z [q 2 ]-bases of the quantum cluster algebra of Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the corresponding cluster algebra. As a byproduct, we prove positivity of the elements in these bases. ± 1 Keywords Quantum cluster algebra, Z [q 2 ]-basis, positivity MR(2000) Subject Classification 16G20, 20G42 1 Introduction Cluster algebras were introduced by Fomin and Zelevinsky [1, 2] in order to study total posi- tivity in algebraic groups and the specialization of canonical bases of quantum groups at q = 1. The study of Z-bases of cluster algebras is important. There are many results involving the con- struction of Z-bases of cluster algebras (for example, see [3, 4] for cluster algebras of rank 2, [5]