List of Lie groups topics

Examples

• General linear group, special linear group
• *SL₂
• *SL₂
• Unitary group, special unitary group
• *SU
• *SU
• Orthogonal group, special orthogonal group
• *Rotation group SO
• *SO
• *Generalized orthogonal group, generalized special orthogonal group
• **The special unitary group SU 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 algebra]s

• 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
• Complex Lie group
• Complexification

  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
• Peter–Weyl theorem
• Borel–Weil theorem
• Kirillov character formula
• Representation theory of SU
• Representation theory of SL2

  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 group]s

• Lattice
• Lattice
• Frieze group
• Wallpaper group
• Space group
• Crystallographic group
• Fuchsian group
• Modular group
• Congruence subgroup
• Kleinian group
• Discrete Heisenberg group
• Clifford–Klein form

  [Algebraic group]s

• Borel subgroup
• Parabolic subgroup
• Arithmetic group

  [Special functions]

• Dunkl operator

  Automorphic forms

• Modular form
• Langlands program

  People

• Sophus Lie
• Wilhelm Killing
• Élie Cartan
• Hermann Weyl
• Harish-Chandra
• Lajos Pukánszky
• Bertram Kostant