List of Lie groups topics explained

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 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

See also: List of representation theory topics.

• 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
• 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)