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
- SL<sub>2</sub>(R)
- SL<sub>2</sub>(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
- G<sub>2</sub>
- F<sub>4</sub>
- E<sub>6</sub>
- E<sub>7</sub>
- E<sub>8</sub>
- 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 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 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
- 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)