Quantum Deformation of the Grassmannian Manifold

In this paper we work out a deformation ofG(r, n), the grassmannian ofr-subspaces in a vector space of dimensionnover a fieldkof characteristic 0.G(r, n) is deformed as an homogeneous space forSLn(k), the special linear group ofkn; this means thatk[G(r, n)], the coordinate ring ofG(r, n), gets deformed together with with the coaction ofk[SLn], the coordinate ring ofSLn(k), on it. Our deformation comes together with a coaction of the corresponding deformation ofSLn(k). At the end we give an explicit presentation of the deformed grassmannian, in terms of generators and relations.