Representation theory
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Lie group
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Representation theory of groups
Representation of a Lie group
In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a bas...
Representation of a Lie group - Wikipedia
Lie algebra representation
Lie algebra representation
In the mathematical 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 (or endomorphisms of a vector...
Table of Clebsch–Gordan coefficients
This is a table of Clebsch–Gordan coefficients used for adding angular momentum values in quantum mechanics. The overall sign of the coefficients for each set of constant , , is arbitrary to some deg...
Langlands dual
In representation theory, a branch of mathematics, the Langlands dual 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...
Weight (representation theory)
In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is...
Kazhdan–Lusztig polynomial
Kazhdan–Lusztig polynomial - Wikipedia
Wigner 3-j symbols
In quantum mechanics, the Wigner 3-j symbols, also called 3j or 3-jm symbols,are related to Clebsch–Gordan coefficientsthrough
The inverse relation can be found by noting that j1 − j2 − m3 is an i...
Representation theory of the Poincaré group
In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in t...
Representation theory of the Poincaré group - Wikipedia
Orbit method
In mathematics, the orbit method (also known as the Kirillov theory, the method of coadjoint orbits and by a few similar names) establishes a correspondence between irreducible unitary representations...
Theta correspondence
In mathematics, the theta correspondence or Howe correspondence is a correspondence between automorphic forms associated to the two groups of a dual reductive pair, introduced by Howe (1979).
Holomorphic discrete series representation
In mathematics, a holomorphic discrete series representation is a discrete series representation of a semisimple Lie group that can be represented in a natural way as a Hilbert space of holomorphic fu...
Fundamental lemma (Langlands program)
In the mathematical theory of automorphic forms, the fundamental lemma relates orbital integrals on a reductive group over a local field to stable orbital integrals on its endoscopic groups. It was c...
Borel–Weil theorem
In mathematics, in the field of representation theory, the Borel–Weil theorem, named after Armand Borel and André Weil, provides a concrete model for irreducible representations of compact Lie groups ...
Verma module
Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.Verma modules can be used to prove that an irreducible highest weight modu...
Base change lifting
In mathematics, base change lifting is a method of constructing new automorphic forms from old ones, that corresponds in Langlands philosophy to the operation of restricting a representation of a Galo...
Plancherel theorem for spherical functions
In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural ge...
Gelfand pair
In mathematics, the expression Gelfand pair is a pair (G,K) consisting of a group G and a subgroup K that satisfies a certain property on restricted representations. The theory of Gelfand pairs is clo...
Langlands program
In mathematics, the Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algeb...
Affine representation
An 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 affine group Aff(A). Similarly, an a...
Borel–Weil–Bott theorem
In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain c...
Borel–Weil–Bott theorem - Wikipedia
Category O
Category O (or category ) is a mathematical object in representation theory of semisimple Lie algebras. It is a category whose objects arecertain representations of a semisimple Lie algebra and morphi...
Infinitesimal character
In mathematics, the infinitesimal character of an irreducible representation ρ of a semisimple Lie group G on a vector space V is, roughly speaking, a mapping to scalars that encodes the process of fi...
Representation up to homotopy
A Representation up to homotopy is a concept in differential geometry that generalizes the notion of representation of a Lie algebra to Lie algebroids and nontrivial vector bundles. It was introduced...
Principal series representation
In mathematics, the principal series representations of certain kinds of topological group G occur in the case where G is not a compact group. There, by analogy with spectral theory, one expects that ...
Algebraic character
Algebraic character is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite-dimensional representation and is analogo...
Zonal spherical function
In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the mat...
Wigner's classification
In mathematics and theoretical physics, Wigner's classificationis a classification of the nonnegative (E ≥ 0) energy irreducible unitary representations of the Poincaré group, which have sharp mass ei...
(g,K)-module
In mathematics, more specifically in the representation theory of reductive Lie groups, a -module is an algebraic object, first introduced by Harish-Chandra, used to deal with continuous infinite-dime...
Automorphic L-function
In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic form π of a reductive group G over a global field and a finite-dimensional comple...
Antifundamental representation
In mathematics, an antifundamental representation of a lie group is the complex conjugate of the fundamental representation, although the distinction between the fundamental and the antifundamental re...
Anyon
In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than fermions and bosons; the operation of exchanging two identical pa...
Anyon - Wikipedia