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On infinite-dimensional Grassmannians and their quantum deformations. (English) Zbl 1167.14325

Summary: An algebraic approach is developed to define and study infinite-dimensional Grassmannians. Using this approach, a quantum deformation (i.e. a deformation of the coordinate ring) is obtained for both the ind-variety union of all finite-dimensional Grassmannians \(G_{\infty}\), and the Sato Grassmannian \(\widetilde{UGM}\) introduced by Sato. 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.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
58B25 Group structures and generalizations on infinite-dimensional manifolds
58B32 Geometry of quantum groups
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