# List of Lie group topics

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List of Lie group topics

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
*Campbell-Hausdorff formula
*Casimir invariant
*Kac-Moody algebra
*Affine Lie algebra
*Loop algebra

Foundational results

*One-parameter group, One-parameter subgroup
*Matrix exponential
*Infinitesimal transformation
*Lie's third theorem
*Maurer-Cartan form
*Cartan's theorem
*Local Lie group
*Formal group law
*Hilbert's fifth problem
*Hilbert-Smith conjecture
*Lie group decompositions

emisimple theory

*Simple Lie group, compact Lie group
*Root system
*Simply laced group
*Maximal torus
*Weyl group
*Dynkin diagram
*Coxeter group

Representation theory

*Representation of a Lie group
*Representation of a Lie algebra
*Adjoint representation of a Lie algebra
*Unitary representation
*Weight
*Peter-Weyl 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
*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 &ndash; 1899)
*Wilhelm Killing (1847 &ndash; 1923)
*Élie Cartan (1869 &ndash; 1951)
*Hermann Weyl (1885 &ndash; 1955)

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