Jacobi–Madden equation: Difference between revisions

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{{for|the variety of all ''k''-dimensional affine subspaces of a finite-dimensional vector space (a smooth finite-dimensional variety over ''k'')|affine Grassmannian (manifold)}}
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In [[mathematics]], the term '''affine Grassmannian''' has two distinct meanings; the concept treated in this article is the '''affine Grassmannian''' of an [[algebraic group]] ''G'' over a field ''k''. It is an [[ind-scheme]] -- a limit of finite-dimensional [[scheme (mathematics)|schemes]] -- which can be thought of as a [[flag variety]] for the loop group ''G(k((t)))'' and which describes the representation theory of the [[Langlands dual]] group <sup>''L''</sup>''G'' through what is known as the [[geometric Satake correspondence]].
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== Definition of Gr via functor of points ==
 
Let ''k'' be a field, and denote by <math>k-\mathrm{Alg}</math> and <math>\mathrm{Set}</math> the category of commutative ''k''-algebras and the category of sets respectively. Through the [[Yoneda lemma]], a scheme ''X'' over a field ''k'' is determined by its [[functor of points]], which is the functor <math>X:k-\mathrm{Alg}\to\mathrm{Set}</math> which takes ''A'' to the set ''X(A)'' of ''A''-points of ''X''. We then say that this functor is [[representable functor|representable]] by the scheme ''X''. The affine Grassmannian is a functor from ''k''-algebras to sets which is not itself representable, but which has a [[filtration (mathematics)|filtration]] by representable functors. As such, although it is not a scheme, it may be thought of as a union of schemes, and this is enough to profitably apply geometric methods to study it.
 
Let ''G'' be an algebraic group over ''k''. The '''affine Grassmannian''' Gr<sub>''G''</sub> is the functor that associates to a ''k''-algebra ''A'' the set of isomorphism classes of pairs (''E'', ''φ''), where ''E'' is a [[principal homogeneous space]] for ''G'' over Spec ''A''[<span/>[''t'']] and ''φ'' is an isomorphism, defined over Spec ''A''((''t'')), of ''E'' with the trivial ''G''-bundle ''G'' × 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 ''X''<sub>''A''</sub> and ''φ'' a trivialization on (''X''&nbsp;−&nbsp;''x'')<sub>''A''</sub>. When ''G'' is a [[reductive group]], Gr<sub>''G''</sub> is in fact ind-projective, i.e., an inductive limit of projective schemes.
 
== Definition as a coset space ==
 
Let us denote by <math>\mathcal K = k((t))</math> the field of [[formal Laurent series]] over ''k'', and by <math>\mathcal O = k[[t]]</math> the ring of formal power series over ''k''. By choosing a trivialization of ''E'' over all of Spec <math>\mathcal O</math>, the set of ''k''-points of Gr<sub>''G''</sub> is identified with the coset space <math>G(\mathcal K)/G(\mathcal O)</math>.
 
==References==
*{{cite book|author=Alexander Schmitt|title=Affine Flag Manifolds and Principal Bundles|url=http://books.google.com/books?id=xrPoBAdiVdQC&pg=PA1|accessdate=1 November 2012|date=11 August 2010|publisher=Springer|isbn=978-3-0346-0287-7|pages=3–6}}
 
{{algebra-stub}}
[[Category:Algebraic geometry]]

Latest revision as of 00:13, 23 December 2014

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