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

A quantum cluster algebra of Kronecker type and the dual canonical basis