A deformation of the big cell inside the, grassmannian manifold, G(r,n)

In this paper we construct a quantum analogue of the big cell inside the gr assmannian manifold. Our deformation comes in tandem with a coaction of the upper parabolic subgroup in SLn(k), giving to the big cell the structure o f quantum homogeneous space. At the end we give the De Rham complex of the quantum big cell and we define a ring of differential operators acting on t he quantum big cell.