Quantum cluster algebras of type A and the dual canonical basis

Abstract

The article concerns the subalgebra Uv+(w) of the quantized universal enveloping algebra of the complex Lie algebra 𝔰𝔩n+1 associated with a particular Weyl group element of length 2n. We verify that Uv+(w) can be endowed with the structure of a quantum cluster algebra of type An. The quantum cluster algebra is a deformation of the ordinary cluster algebra Geiß–Leclerc–Schröer 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.