Table of contents

1 Examples 2 Lie algebras 3 Semisimple theory 4 Lie group decompositions 5 Representation theory 6 Applications 7 History

Examples

• General linear group, special linear group
• Unitary group, special unitary group
• Orthogonal group, special orthogonal group
  • Generalized orthogonal group, generalized special orthogonal group
  • Spinor group
  • Charts on SO(3)
• Symplectic group
• Exceptional groups
  • G₂
  • F₄
  • E₆
  • E₇
  • E₈
• Affine group
• Lorentz group
• Poincaré group

• Commutator
• Jacobi identity
• Adjoint representation
• Universal enveloping algebra
• Campbell-Hausdorff formula

• Simple Lie group
• Root system
• Maximal torus
• Weyl group
• Dynkin diagram
• Coxeter group

Representation theory

• Representation of Lie algebras
• Representations of Lie groups and algebras
• Weight (representation theory)
• Unitary representation
• Peter-Weyl theorem

• Physical theories
  • Poisson bracket
  • Noether's theorem
  • Wigner's classification
  • 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
• Automorphic forms
  • Modular form
  • Langlands program