On Infinite Dimensional Grassmannians and Their Quantum Deformations

An algebraic approach is developed to define and study infinite dimensional grassmannians. Using this approach a quantum deformation is obtained for both the ind-variety union of all finite dimensional grassmannians G∞, and the Sato grassmannian ̃ UGM introduced by Sato in [Sa1], [Sa2]. They are both quantized as homogeneous spaces, that is together with a coaction of a quantum infinite dimensional group. At the end, an infinite dimensional version of the first theorem of invariant theory is discussed for both the infinite dimensional special linear group and its quantization.