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Representation of a Lie group information


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.

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Representation of a Lie group

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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...

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Adjoint representation

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representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra...

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Lie algebra representation

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field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices...

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Compact group

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Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups...

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Group representation

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mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to...

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Representation theory of the Lorentz group

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The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations...

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Representation theory

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a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory...

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Lie algebra

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between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the representation theory of Lie groups. For...

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Antifundamental representation

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differential geometry, an antifundamental representation of a Lie group is the complex conjugate of the fundamental representation, although the distinction between...

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Lie group

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In mathematics, a Lie group (pronounced /liː/ LEE) is a group that is also a differentiable manifold, such that group multiplication and taking inverses...

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Fundamental representation

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In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple...

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Representation

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entity Representation of a Lie group, a linear action of a Lie group on a vector space Representation (chemistry), graphic representation of the molecular...

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Simple Lie group

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a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can...

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List of Lie groups topics

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This is a list of Lie group topics, by Wikipedia page. See Table of Lie groups for a list General linear group, special linear group SL2(R) SL2(C) Unitary...

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Representation theory of the Galilean group

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representations of the nontrivial central extension of the universal covering group of the Galilean group by the one-dimensional Lie group R, cf. the article...

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Particle physics and representation theory

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particles to the structure of Lie groups and Lie algebras. According to this connection, the different quantum states of an elementary particle give...

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Representation theory of semisimple Lie algebras

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mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory...

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Dual representation

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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...

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Table of Lie groups

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article gives a table of some common Lie groups and their associated Lie algebras. The following are noted: the topological properties of the group (dimension;...

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Lie group decomposition

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tools in the representation theory of Lie groups and Lie algebras; they can also be used to study the algebraic topology of such groups and associated...

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Trivial representation

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identity mapping of V. A trivial representation of an associative or Lie algebra is a (Lie) algebra representation for which all elements of the algebra act...

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Selberg trace formula

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character of the unitary representation of a Lie group G on the space L2(Γ\G) of square-integrable functions, where Γ is a cofinite discrete group. The character...

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Semisimple Lie algebra

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In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any...

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Spin representation

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linear group GL(S) such that the element −1 is not in the kernel of ρ. If S is such a representation, then according to the relation between Lie groups and...

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Affine representation

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affine representation of a topological Lie group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A, the...

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