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

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 algebraLie 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 Unified 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 groupDiscrete 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 groupAlgebraic groups

• Borel subgroup

• Arithmetic group

Special functionsSpecial 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)