List of Lie group topics - Definition and Overview

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

Examples

• General linear group, special linear group
• Unitary group, special unitary group
  • SU(2)
  • SU(3)
• Orthogonal group, special orthogonal group
  • Rotation group = SO(3)
    • Charts on SO(3)
    • SO(8)
    • Triality
  • Generalized orthogonal group, generalized special orthogonal group
    • Lorentz group
    • SU(1,1) is the unit sphere in coquaternions
  • 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

Foundational results

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

Semisimple theory

• Simple Lie group, compact Lie group
• Root system
• Simply laced group
  • ADE classification
• Maximal torus
• Weyl group
• Dynkin diagram
• Coxeter group

Lie group decompositions

Representation theory

• Representations of Lie groups
• Representation of Lie algebras
• Adjoint representation
• Unitary representation
• Weight
• Peter-Weyl theorem

Applications

Physical theories

• Pauli matrices
• Gell-Mann matrices
• Poisson bracket
• Noether's theorem
• Wigner's classification
• Casimir invariant
• Gauge theory
  • SU(3)×SU(2)×U(1)
  • Flipped SU(5)
• Anti de Sitter space
• De Sitter space
• Supergroup
• Lie superalgebra
• Twistor
• Anyon

Geometry

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

Discrete groups

• Lattice (group)
• 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)