  The article concerns the dual of Lusztig's canonical basis of a subalgebra of
the positive part U_q(n) of the universal enveloping algebra of a Kac-Moody Lie
algebra of type A_1^{(1)}. The examined subalgebra is associated with a
terminal module M over the path algebra of the Kronecker quiver via an Weyl
group element w of length four.
  Geiss-Leclerc-Schroeer attached to M a category C_M of nilpotent modules over
the preprojective algebra of the Kronecker quiver together with an acyclic
cluster algebra A(C_M). The dual semicanonical basis contains all cluster
monomials. By construction, the cluster algebra A(C_M) is a subalgebra of the
graded dual of the (non-quantized) universal enveloping algebra U(n).
  We transfer to the quantized setup. Following Lusztig we attach to w a
subalgebra U_q^+(w) of U_q(n). The subalgebra is generated by four elements
that satisfy straightening relations; it degenerates to a commutative algebra
in the classical limit q=1. The algebra U_q^+(w) possesses four bases, a PBW
basis, a canonical basis, and their duals. We prove recursions for dual
canonical basis elements. The recursions imply that every cluster variable in
A(C_M) is the specialization of the dual of an appropriate canonical basis
element. Therefore, U_q^+(w) is a quantum cluster algebra in the sense of
Berenstein-Zelevinsky. Furthermore, we give explicit formulae for the quantized
cluster variables and for expansions of products of dual canonical basis
elements.
