Quantum Unipotent Subgroup and Dual Canonical Basis

In a series of works [18, 21, 19, 20, 23, 22], Geiß-Leclerc-Schröer defined the cluster algebra structure on the coordinate ring C[N(w)] of the unipotent subgroup, associated with a Weyl group element w. And they proved cluster monomials are contained in Lusztig’s dual semicanonical basis S∗. We give a set up for the quantization of their results and propose a conjecture which relates the quantum cluster algebras in [4] to the dual canonical basis B. In particular, we prove that the quantum analogue Oq[N(w)] of C[N(w)] has the induced basis from B, which contains quantum flag minors and satisfies a factorization property with respect to the ‘q-center’ of Oq[N(w)]. This generalizes Caldero’s results [7, 8, 9] from ADE cases to an arbitary symmetrizable Kac-Moody Lie algebra.