List of Lie groups topics

Short description: Wikipedia list article

This is a list of Lie group topics, by Wikipedia page.

## Contents

* 1 Examples
* 2 Lie algebras
* 3 Foundational results
* 4 Semisimple theory
* 5 Representation theory
* 6 Applications
* 6.1 Physical theories
* 6.2 Geometry
* 6.3 Discrete groups
* 6.4 Algebraic groups
* 7 Special functions
* 7.1 Automorphic forms
* 8 People

## Examples

See Table of Lie groups for a list

* General linear group, special linear group
* SL2(R)
* SL2(C)
* Unitary group, special unitary group
* SU(2)
* SU(3)
* Orthogonal group, special orthogonal group
* Rotation group SO(3)
* SO(8)
* Generalized orthogonal group, generalized special orthogonal group
* The special unitary group SU(1,1) is the unit sphere in the ring of coquaternions. It is the group of hyperbolic motions of the Poincaré disk model of the Hyperbolic plane.
* Lorentz group
* Spinor group
* Symplectic group
* Exceptional groups
* G2
* F4
* E6
* E7
* E8
* Affine group
* Euclidean group
* Poincaré group
* Heisenberg group

## Lie algebras

* Commutator
* Jacobi identity
* Universal enveloping algebra
* Baker-Campbell-Hausdorff formula
* Casimir invariant
* Killing form
* Kac–Moody algebra
* Affine Lie algebra
* Loop algebra
* Graded Lie algebra

## Foundational results

* One-parameter group, One-parameter subgroup
* Matrix exponential
* Infinitesimal transformation
* Lie's third theorem
* Maurer–Cartan form
* Cartan's theorem
* Cartan's criterion
* Local Lie group
* Formal group law
* Hilbert's fifth problem
* Hilbert-Smith conjecture
* Lie group decompositions
* Real form (Lie theory)
* Complex Lie group
* Complexification (Lie group)

## Semisimple theory

* Simple Lie group
* Compact Lie group, Compact real form
* Semisimple Lie algebra
* Root system
* Simply laced group
* ADE classification
* Maximal torus
* Weyl group
* Dynkin diagram
* Weyl character formula

## Representation theory

* Representation of a Lie group
* Representation of a Lie algebra
* Adjoint representation of a Lie group
* Adjoint representation of a Lie algebra
* Unitary representation
* Weight (representation theory)
* Peter–Weyl theorem
* Borel–Weil theorem
* Kirillov character formula
* Representation theory of SU(2)
* Representation theory of SL2(R)

## Applications

### Physical theories

* Pauli matrices
* Gell-Mann matrices
* Poisson bracket
* Noether's theorem
* Wigner's classification
* Gauge theory
* Grand unification theory
* Supergroup
* Lie superalgebra
* Twistor theory
* Anyon
* Witt algebra
* Virasoro algebra

### Geometry

* Erlangen programme
* Homogeneous space
* Principal homogeneous space
* Invariant theory
* Lie derivative
* Darboux derivative
* Lie groupoid
* Lie algebroid

### Discrete groups

* Lattice (group)
* Lattice (discrete subgroup)
* Frieze group
* Wallpaper group
* Space group
* Crystallographic group
* Fuchsian group
* Modular group
* Congruence subgroup
* Kleinian group
* Discrete Heisenberg group
* Clifford–Klein form

### Algebraic groups

* Borel subgroup
* Parabolic subgroup
* Arithmetic group

## Special functions

* Dunkl operator

### Automorphic forms

* Modular form
* Langlands program

## People

* Sophus Lie (1842 - 1899)
* Wilhelm Killing (1847 - 1923)
* Élie Cartan (1869 - 1951)
* Hermann Weyl (1885 - 1955)
* Harish-Chandra (1923 - 1983)
* Lajos Pukánszky (1928 - 1996)
* Bertram Kostant (1928 - 2017)

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