From the quantum group U_q1/2( sl₂ ) to the Kronecker quantum cluster algebra

Let Aq be the Kronecker quantum cluster algebra with principal coefficients. We construct an explicit algebra homomorphism from the whole quantum group U_q1/2( sl₂ ) to Aq, whose images together with either of the two frozen variables generate Aq. As an application, we obtain algebra homomorphisms from the split quantum groups of affine sl₂ to Aq. Moreover, we explicitly determine the images of the positive and negative real and imaginary root vectors of U_q1/2( sl₂ ) in terms of quantum cluster variables.