Quantum cluster algebras and the dual canonical basis

Abstract

Let Q be either a Dynkin quiver of type A with alternating orientation or the Kronecker quiver. With the indecomposable injective modules over the path algebra of Q and their Auslander-Reiten translates we associate an element w in the Weyl group of corresponding type. The thesis verifies that the subalgebra U_v(w) of the quantized universal enveloping algebra attached to w carries the structure of a quantum cluster algebra in the sense of Berenstein-Zelevinsky. The quantum cluster algebra is a v-deformation of the cluster algebra A(w) Geiß-Leclerc-Schröer attached to w. Furthermore, we show that all quantum cluster variables are elements in the dual of Lusztig's canonical basis (up to a power of v).