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
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)