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 poincarÃ© disk model of the hyperbolic plane. ***lorentz group **spinor group *symplectic group *exceptional groups **g2 **f4 **e6 **e7 **e8 *affine group *euclidean group *poincarÃ© group *heisenberg group ==lie algebras== *commutator *jacobi identity *universal enveloping algebra *campbell-hausdorff formula *casimir invariant *killing form *kacâmoody 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 *maurerâcartan 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) *peterâweyl theorem *borelâweil 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)
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