Quantum cluster algebras of type A and the dual canonical basis

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The article concerns the subalgebra U_v^+(w) of the quantized universal enveloping algebra of the complex Lie algebra sl_{n+1} associated with a particular Weyl group element of length 2n.

• We verify that U_v^+(w) can be endowed with the structure of a quantum cluster algebra of type A_n.
• The quantum cluster algebra is a deformation of the ordinary cluster algebra Geiss-Leclerc-Schroeer attached to w using the representation theory of the preprojective algebra.
• Furthermore, we prove that the quantum cluster variables are, up to a power of v, elements in the dual of Lusztig's canonical basis under Kashiwara's bilinear form.