  We give several explicit examples of quantum cluster algebra structures, as
introduced by Berenstein and Zelevinsky, on quantized coordinate rings of
partial flag varieties and their associated unipotent radicals. These
structures are shown to be quantizations of the cluster algebra structures
found on the corresponding classical objects by Geiss, Leclerc and Schroer,
whose work generalizes that of several other authors. We also exhibit quantum
cluster algebra structures on the quantized enveloping algebras of the Lie
algebras of the unipotent radicals.
