A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis

The article concerns the dual of Lusztig’s canonical basis of a subalgebra of the positive part $U_q(\mathfrak n)$ of the quantized 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 a Weyl group element w of length 4. Geiß–Leclerc–Schröer attached to M a category $\mathcal C_M$ of nilpotent modules over the preprojective algebra of the Kronecker quiver together with an acyclic cluster algebra $\mathcal A(\mathcal C_M)$. The dual semicanonical basis contains all cluster monomials. By construction, the cluster algebra $\mathcal A(\mathcal C_M)$ is a subalgebra of the graded dual of the (nonquantized) universal enveloping algebra $U(\mathfrak n)$. We transfer to the quantized setup. Following Lusztig, we attach to w a subalgebra $U^+_q(w)$ of $U_q(\mathfrak n)$. 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 Poincaré–Birkhoff–Witt basis, a canonical basis, and their duals. We prove recursions for dual canonical basis elements. The recursions imply that every cluster variable in $\mathcal A(\mathcal C_M)$ 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.