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Harish-Chandra Module
In mathematics, specifically in the representation theory of Lie groups, a Harish-Chandra module, named after the Indian mathematician and physicist Harish-Chandra, is a representation of a real Lie group, associated to a general representation, with regularity and finiteness conditions. When the associated representation is a (\mathfrak,K)-module, then its Harish-Chandra module is a representation with desirable factorization properties. Definition Let ''G'' be a Lie group and ''K'' a compact subgroup of ''G''. If (\pi,V) is a representation of ''G'', then the ''Harish-Chandra module'' of \pi is the subspace ''X'' of ''V'' consisting of the K-finite smooth vectors in ''V''. This means that ''X'' includes exactly those vectors ''v'' such that the map \varphi_v : G \longrightarrow V via :\varphi_v(g) = \pi(g)v is smooth, and the subspace :\text\ is finite-dimensional. Notes In 1973, Lepowsky showed that any irreducible (\mathfrak,K)-module ''X'' is isomorphic to the Harish-Cha ... [...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 mathematical analysis, 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 mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory Of Lie Groups
In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras. Finite-dimensional representations Representations A complex representation of a group is an action by a group on a finite-dimensional vector space over the field \mathbb C. A representation of the Lie group ''G'', acting on an ''n''-dimensional vector space ''V'' over \mathbb C is then a smooth group homomorphism :\Pi:G\rightarrow\operatorname(V), where \operatorname(V) is the general linear group of all invertible linear transformations of V under their composition. Since all ''n''-dimens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra
Harish-Chandra FRS (11 October 1923 – 16 October 1983) was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups. Early life Harish-Chandra was born in Kanpur. He was educated at B.N.S.D. College, Kanpur and at the University of Allahabad. After receiving his master's degree in Physics in 1943, he moved to the Indian Institute of Science, Bangalore for further studies under Homi J. Bhabha. In 1945, he moved to University of Cambridge, and worked as a research student under Paul Dirac. While at Cambridge, he attended lectures by Wolfgang Pauli, and during one of them pointed out a mistake in Pauli's work. The two were to become lifelong friends. During this time he became increasingly interested in mathematics. At Cambridge he obtained his PhD in 1947. Honors and awards He was a member of the National Academy of Sciences and a Fellow of the Royal Society. He was the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup of ''G'' if the restriction of ∗ to is a group operation on ''H''. This is often denoted , read as "''H'' is a subgroup of ''G''". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group ''G'' is a subgroup ''H'' which is a proper subset of ''G'' (that is, ). This is often represented notationally by , read as "''H'' is a proper subgroup of ''G''". Some authors also exclude the trivial group from being proper (that is, ). If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an overgroup of ''H''. The same definitions apply more generally when ''G'' is an arbitrary semigroup, but this article will only deal with subgroups of groups. Subgroup tests Suppose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-finite
In mathematics, a K-finite function is a type of generalized trigonometric polynomial. Here ''K'' is some compact group, and the generalization is from the circle group ''T''. From an abstract point of view, the characterization of trigonometric polynomials amongst other functions ''F'', in the harmonic analysis of the circle, is that for functions ''F'' in any of the typical function spaces, ''F'' is a trigonometric polynomial if and only if its Fourier coefficients :''a''''n'' vanish for , ''n'', large enough, and that this in turn is equivalent to the statement that all the translates :''F''(''t'' + θ) by a fixed angle θ lie in a finite-dimensional subspace. One implication here is trivial, and the other, starting from a finite-dimensional invariant subspace, follows from complete reducibility In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
(g,K)-module
In mathematics, more specifically in the representation theory of reductive Lie groups, a (\mathfrak,K)-module is an algebraic object, first introduced by Harish-Chandra, used to deal with continuous infinite-dimensional representations using algebraic techniques. Harish-Chandra showed that the study of irreducible unitary representations of a real reductive Lie group, ''G'', could be reduced to the study of irreducible (\mathfrak,K)-modules, where \mathfrak is the Lie algebra of ''G'' and ''K'' is a maximal compact subgroup of ''G''. Definition Let ''G'' be a real Lie group. Let \mathfrak be its Lie algebra, and ''K'' a maximal compact subgroup with Lie algebra \mathfrak. A (\mathfrak,K)-module is defined as follows:This is James Lepowsky's more general definition, as given in section 3.3.1 of it is a vector space ''V'' that is both a Lie algebra representation of \mathfrak and a group representation of ''K'' (without regard to the topology of ''K'') satisfying the following thr ... [...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] |
Unitary Representation
In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G'' is a locally compact ( Hausdorff) topological group and the representations are strongly continuous. The theory has been widely applied in quantum mechanics since the 1920s, particularly influenced by Hermann Weyl's 1928 book ''Gruppentheorie und Quantenmechanik''. One of the pioneers in constructing a general theory of unitary representations, for any group ''G'' rather than just for particular groups useful in applications, was George Mackey. Context in harmonic analysis The theory of unitary representations of topological groups is closely connected with harmonic analysis. In the case of an abelian group ''G'', a fairly complete picture of the representation theory of ''G'' is given by Pontryagin duality. In general, the unitary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
