High Quality Content by WIKIPEDIA articles! In mathematics, the term affine Grassmannian has two distinct meanings. In one meaning the affine Grassmannian (manifold) is the manifold of all k-dimensional affine subspaces of a finite dimensional vector space, while (described here) the affine Grassmannian of an algebraic group G over a field k is defined in one of two ways: As the coset space G(K)/G(O), where K = k((t)) is the field of formal Laurent series over k and O = k[[t]] is the ring of formal power series; As the ind-scheme GrG which is described as a functor by the following data: to every k-algebra A, GrG(A) is the set of isomorphism classes of pairs (E, ?), where E is a principal homogeneous space for G over Spec A[[t]] and ? is an isomorphism, defined over Spec A((t)), of E with the trivial G-bundle G x Spec A((t)). By the Beauville–Laszlo theorem, it is also possible to specify this data by fixing an algebraic curve X over k, a k-point x on X, and taking E to be a G bundle on XA and ? a trivialization on (X ? x)A.

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