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Affine Grassmannian
In mathematics, the affine Grassmannian of an algebraic group ''G'' over a field ''k'' is an ind-scheme—a colimit of finite-dimensional 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 ''L''''G'' through what is known as the geometric Satake correspondence. Definition of Gr via functor of points Let ''k'' be a field, and denote by k\text and \mathrm 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 X:k\text \to \mathrm which takes ''A'' to the set ''X''(''A'') of ''A''-points of ''X''. We then say that this functor is 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 by representable functors. As ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ind-scheme
In algebraic geometry, an ind-scheme is a set-valued functor that can be written (represented) as a direct limit (i.e., inductive limit) of closed embedding of schemes. Examples *\mathbbP^ = \varinjlim \mathbbP^N is an ind-scheme. *Perhaps the most famous example of an ind-scheme is an infinite grassmannian (which is a quotient of the loop group of an algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ... ''G''.) See also * formal scheme References *A. Beilinson, Vladimir Drinfel'd, Quantization of Hitchin’s integrable system and Hecke eigensheaves on Hitchin system, preliminary versio*V.Drinfeld, Infinite-dimensional vector bundles in algebraic geometry, notes of the talk at the `Unity of Mathematics' conferenceExpanded version*http://ncatlab.org/nlab/show/ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flag Variety
In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties. Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space ''V'' over a field F, which is a flag variety for the special linear group over F. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags. In the most general sense, a generalized flag variety is defined to mean a projective ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langlands Dual
In representation theory, a branch of mathematics, the Langlands dual ''L''''G'' of a reductive algebraic group ''G'' (also called the ''L''-group of ''G'') is a group that controls the representation theory of ''G''. If ''G'' is defined over a field ''k'', then ''L''''G'' is an extension of the absolute Galois group of ''k'' by a complex Lie group. There is also a variation called the Weil form of the ''L''-group, where the Galois group is replaced by a Weil group. Here, the letter ''L'' in the name also indicates the connection with the theory of L-functions, particularly the ''automorphic'' L-functions. The Langlands dual was introduced by in a letter to A. Weil. The ''L''-group is used heavily in the Langlands conjectures of Robert Langlands. It is used to make precise statements from ideas that automorphic forms are in a sense functorial in the group ''G'', when ''k'' is a global field. It is not exactly ''G'' with respect to which automorphic forms and representations are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Satake Correspondence
In mathematics, the Satake isomorphism, introduced by , identifies the Hecke algebra of a reductive group over a local field with a ring of invariants of the Weyl group. The geometric Satake equivalence is a geometric version of the Satake isomorphism, proved by . Statement Classical Satake isomorphism. Let G be a semisimple algebraic group, K be a non-Archimedean local field and O be its ring of integers. It's easy to see that Gr = G(K)/G(O) is grassmannian. For simplicity, we can think that K = \Z/p\Z((x)) and O = \Z/p\Z x , p a prime number; in this case, Gr is a infinite dimensional algebraic variety . One denotes the category of all compactly supported spherical functions on G(K) biinvariant under the action of G(O) as \Complex_c (O) \backslash G(K)/G(O), \Complex the field of complex numbers, which is a Hecke algebra and can be also treated as a group scheme over \Complex . Let T(\Complex) be the maximal torus of G(\Complex) , W be the Weyl group of G . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yoneda Lemma
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It allows the embedding of any locally small category into a category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda. Generalities The Yoneda lemma suggests that instead of studying the locally small category \mathcal , one should study the category of all functors of \mathcal into \mathbf (the category of sets with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor Of Points
In algebraic geometry, a functor represented by a scheme ''X'' is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme ''S'' is (up to natural bijections) the set of all morphisms S \to X. The scheme ''X'' is then said to ''represent'' the functor and that ''classify'' geometric objects over ''S'' given by ''F''. The best known example is the Hilbert scheme of a scheme ''X'' (over some fixed base scheme), which, when it exists, represents a functor sending a scheme ''S'' to a flat family of closed subschemes of X \times S. In some applications, it may not be possible to find a scheme that represents a given functor. This led to the notion of a stack, which is not quite a functor but can still be treated as if it were a geometric space. (A Hilbert scheme is a scheme, but not a stack because, very roughly speaking, deformation theory is simpler for closed schemes.) Some moduli problems are solved by giving formal solutions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representable Functor
In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings. From another point of view, representable functors for a category ''C'' are the functors ''given'' with ''C''. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory. Definition Let C be a locally small category and let Set be the category of sets. For each object ''A'' of C let Hom(''A'',–) be the hom functor that maps object ''X'' to the set Hom(''A'',''X''). A functor ''F'' : C → Set is said to be representable if it is naturally isomorphic to Hom(''A'',–) for some object ''A'' of C. A representation of ''F'' is a pair (''A'', Φ) where :Φ : Hom(''A'',&ndash ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtration (mathematics)
In mathematics, a filtration \mathcal is an indexed family (S_i)_ of subobjects of a given algebraic structure S, with the index i running over some totally ordered index set I, subject to the condition that ::if i\leq j in I, then S_i\subseteq S_j. If the index i is the time parameter of some stochastic process, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic structure S_i gaining in complexity with time. Hence, a process that is adapted to a filtration \mathcal is also called non-anticipating, because it cannot "see into the future". Sometimes, as in a filtered algebra, there is instead the requirement that the S_i be subalgebras with respect to some operations (say, vector addition), but not with respect to other operations (say, multiplication) that satisfy only S_i \cdot S_j \subseteq S_, where the index set is the natural numbers; this is by analogy with a graded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Homogeneous Space
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-empty set ''X'' on which ''G'' acts freely and transitively (meaning that, for any ''x'', ''y'' in ''X'', there exists a unique ''g'' in ''G'' such that , where · denotes the (right) action of ''G'' on ''X''). An analogous definition holds in other categories, where, for example, *''G'' is a topological group, ''X'' is a topological space and the action is continuous, *''G'' is a Lie group, ''X'' is a smooth manifold and the action is smooth, *''G'' is an algebraic group, ''X'' is an algebraic variety and the action is regular. Definition If ''G'' is nonabelian then one must distinguish between left and right torsors according to whether the action is on the left or right. In this article, we will use right actions. To state the defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
