# Bases of the quantum cluster algebra of the Kronecker quiver - Mathematics > Representation Theory

Abstract: We construct bar-invariant $\mathbb{Z}q^{\pm 1-2}-$bases of the quantumcluster algebra of the Kronecker quiver which are quantum analogues of thecanonical basis, semicanonical basis and dual semicanonical basis of thecluster algebra of the Kronecker quiver in the sense of\cite{sherzel},\cite{calzel} and \cite{gls} respectively. As a byproduct, weprove the positivity of the elements in these bases.

Author: Ming Ding, Fan Xu

Source: https://arxiv.org/