In a series of works, Geiss, Leclerc, and Schroer defined the cluster algebra structure on the coordinate ring C[N(ω)] of the unipotent subgroup, associated with a Weyl group element, w. And they proved that cluster monomials are contained in Lusztig's dual semicanonical basis S We give a setup for the quantization of their results and propose a conjecture that relates the quantum cluster algebras in Berenstein and Zeievinsky's woik to the dual canonical basis BuP In particular, we prove that the quan-tum analogue Oq[N(ω)] of [N(ω)] has the induced basis from Bup, which contains quantum flag minors and satisfies a factorization property with respect to the "q-center" of Oq [N(ω)]. This generalizes Caldero's results from finite type to an arbitrary symme-trizable Kac Moody Lie algebra.
展开▼
