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Locally Profinite Group
In mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup. Equivalently, a locally profinite group is a topological group that is Hausdorff, locally compact, and totally disconnected. Moreover, a locally profinite group is compact if and only if it is profinite; this explains the terminology. Basic examples of locally profinite groups are discrete groups and the ''p''-adic Lie groups. Non-examples are real Lie groups, which have the no small subgroup property. In a locally profinite group, a closed subgroup is locally profinite, and every compact subgroup is contained in an open compact subgroup. Examples Important examples of locally profinite groups come from algebraic number theory. Let ''F'' be a non-archimedean local field. Then both ''F'' and F^\times are locally profinite. More generally, the matrix ring \operatorname_n(F) and the general linear group \operatorname_n( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-Archimedean Ordered Field
In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Examples are the Levi-Civita field, the hyperreal numbers, the surreal numbers, the Dehn field, and the field of rational functions with real coefficients with a suitable order. Definition The Archimedean property is a property of certain ordered fields such as the rational numbers or the real numbers, stating that every two elements are within an integer multiple of each other. If a field contains two positive elements for which this is not true, then must be an infinitesimal, greater than zero but smaller than any integer unit fraction. Therefore, the negation of the Archimedean property is equivalent to the existence of infinitesimals. Applications Hyperreal fields, non-Archimedean ordered fields containing the real numbers as a subfield, may be used to provide a mathematical foundation for nonstandard analysis. Max Dehn used the Dehn field, an example of a non ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contragredient Representation
In mathematics, if is a group and is a linear representation of it on the vector space , then the dual representation is defined over the dual vector space as follows: : is the transpose of , that is, = for all . The dual representation is also known as the contragredient representation. If is a Lie algebra and is a representation of it on the vector space , then the dual representation is defined over the dual vector space as follows: : = for all . The motivation for this definition is that Lie algebra representation associated to the dual of a Lie group representation is computed by the above formula. But the definition of the dual of a Lie algebra representation makes sense even if it does not come from a Lie group representation. In both cases, the dual representation is a representation in the usual sense. Properties Irreducibility and second dual If a (finite-dimensional) representation is irreducible, then the dual representation is also irreducible—bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Admissible Representation
In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra. Real or complex reductive Lie groups Let ''G'' be a connected reductive (real or complex) Lie group. Let ''K'' be a maximal compact subgroup. A continuous representation (π, ''V'') of ''G'' on a complex Hilbert space ''V''I.e. a homomorphism (where GL(''V'') is the group of bounded linear operators on ''V'' whose inverse is also bounded and linear) such that the associated map is continuous. is called admissible if π restricted to ''K'' is unitary and each irreducible unitary representation of ''K'' occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation of ''G''. An admissible representation π induces a (\mathfrak,K)-module which is easier to deal with as it is an algebraic object. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Representation
In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra. Real or complex reductive Lie groups Let ''G'' be a connected reductive (real or complex) Lie group. Let ''K'' be a maximal compact subgroup. A continuous representation (π, ''V'') of ''G'' on a complex Hilbert space ''V''I.e. a homomorphism (where GL(''V'') is the group of bounded linear operators on ''V'' whose inverse is also bounded and linear) such that the associated map is continuous. is called admissible if π restricted to ''K'' is unitary and each irreducible unitary representation of ''K'' occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation of ''G''. An admissible representation π induces a (\mathfrak,K)-module which is easier to deal with as it is an algebraic obje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Galois Group
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' that fix ''K''. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group. (When ''K'' is a perfect field, ''K''sep is the same as an algebraic closure ''K''alg of ''K''. This holds e.g. for ''K'' of characteristic zero, or ''K'' a finite field.) Examples * The absolute Galois group of an algebraically closed field is trivial. * The absolute Galois group of the real numbers is a cyclic group of two elements (complex conjugation and the identity map), since C is the separable closure of R and ''C:Rnbsp;= 2. * The absolute Galois group of a finite field ''K'' is isomorphic to the group :: \hat = \varprojlim \mathbf/n\mathbf. (For the notation, see Inverse limit.) :The Frobenius automorphism Fr is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Group
In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field ''F'', its Weil group is generally denoted ''WF''. There also exists "finite level" modifications of the Galois groups: if ''E''/''F'' is a finite extension, then the relative Weil group of ''E''/''F'' is ''W''''E''/''F'' = ''WF''/ (where the superscript ''c'' denotes the commutator subgroup). For more details about Weil groups see or or . Weil group of a class formation The Weil group of a class formation with fundamental classes ''u''''E''/''F'' ∈ ''H''2(''E''/''F'', ''A''''F'') is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the Langlands program. If ''E''/''F'' is a normal layer, then the (relative) Weil group ''W''''E''/''F'' of ''E''/''F'' is the extension :1 → ''A''''F'' → ''W''''E''/''F'' → Gal(''E''/''F'') → 1 co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Linear Group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns (and also the rows) of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers) is the group of invertible matrices of real numbers, and is denoted by GL''n''(R) or . More generally, the general linear group of degree ''n'' over any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology. Sometimes, real numbers R, and the complex numbers C (with their standard topologies) are also defined to be local fields; this is the convention we will adopt below. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields. While Archimedean local fields have been quite well known in mathematics for at lea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and Algebraic function field, function fields. These properties, such as whether a ring (mathematics), ring admits unique factorization, the behavior of ideal (ring theory), ideals, and the Galois groups of field (mathematics), fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History of algebraic number theory Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis. Formal definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
No Small Subgroup Property
In mathematics, especially in topology, a topological group G is said to have no small subgroup if there exists a neighborhood U of the identity that contains no nontrivial subgroup of G. An abbreviation '"NSS"' is sometimes used. A basic example of a topological group with no small subgroup is the general linear group over the complex numbers. A locally compact, separable metric, locally connected group with no small subgroup is a Lie group. (cf. Hilbert's fifth problem Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathemat ....) See also * References * M. Goto, H., YamabeOn some properties of locally compact groups with no small group Group theory #05 Lie groups Topological groups {{math-hist-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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