Lie group

Definition

In mathematics, a Lie group is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.

Related concepts

3D rotation group Abelian Lie algebra Abelian Lie group Abelian group Abelian variety Absolute value Academic Press Additive group Adele ring Adjoint bundle Adjoint representation Adjoint representation of a Lie group Ado's theorem Affine Lie algebra Affine bundle Affine connection Affine group Affine transformation Albert Nijenhuis Algebraic equation Algebraic geometry Algebraic group Algebraic structure Almost-contact manifold Almost complex manifold Almost flat manifold Almost symplectic manifold Alternating group American Mathematical Society Analytic map Andrew Gleason Anthony W. Knapp Archiv for Mathematik og Naturvidenskab Arithmetic group Armand Borel Associated bundle Atiyah–Singer index theorem Atlas (topology)August Ferdinand Möbius Automorphic form Baker–Campbell–Hausdorff formula Banach manifold Banach space Bernhard Riemann Bijective Bilinear operator Binary operation Borel–Weil–Bott theorem Bulletin of the American Mathematical Society Calabi–Yau manifold Cambridge University Press Carl Gustav Jacobi Cartan connection Cartan subalgebra Category (mathematics)Category theory Cauchy's theorem (group theory)Circle Circle group Classical group Classical mechanics Classification of finite simple groups Classification of manifolds Claude Chevalley Clebsch–Gordan coefficients for SU(3)Closed and exact differential forms Closed manifold Closed set Closed subgroup theorem Closure (topology)Cofibration Collapsing manifold Commutative diagram Commutator Compact Lie algebra Compact Lie group Compact group Compact space Complete manifold Complete metric space Complex Lie group Complex differential form Complex manifold Complex number Complex plane Complexification (Lie group)Conformal geometry Conformal group Conjugate transpose Connected space Connectedness Connection (fibred manifold)Connection (mathematics)Connection (principal bundle)Connection (vector bundle)Connection form Contact manifold Continuous group Continuous symmetry Convenient vector space Cotangent bundle Cotangent space Covariant derivative Cyclic group Darboux's theorem David Hilbert De Rham cohomology Deane Montgomery Dense set Derivation (abstract algebra)Derivative of the exponential map Determinant Diffeology Diffeomorphic Diffeomorphism Differentiable curve Differentiable manifold Differential Galois theory Differential calculus Differential calculus over commutative algebras Differential equation Differential equations Differential form Differential structure Diffiety Dihedral group Direct product of groups Direct sum of groups Direct sum of modules Discrete group Discrete topology Distribution (differential geometry)Donaldson theory Dover Publications Dual bundle Dynkin diagram E6 (mathematics)E7 (mathematics)E8 (mathematics)Ehresmann connection Einstein manifold Elementary abelian group Elliptic curve Ellis Kolchin Embedding Erlangen program Euclidean geometry Euclidean group Euclidean space Exceptional Lie group Exponential function Exponential map (Lie theory)Exponential map (Riemannian geometry)Exterior derivative F4 (mathematics)Felix Klein Fiber bundle Fibered manifold Fibration Fibre bundle Field (mathematics)Finite group Finite groups Finite simple group Finsler manifold Flat manifold Foliation Formal Lie group Fraktur (typeface sub-classification)Free group Free product Friedrich Engel (mathematician)Frobenius group Frobenius theorem (differential topology)Fréchet manifold Fréchet space Functor Fundamental group G-structure G-structure on a manifold G2 (mathematics)Galois group Galois theory Gauge group Gauge theory (mathematics)General linear group Generalized Stokes theorem Geodesic Geometry Global analysis Glossary of Lie groups and Lie algebras Glossary of differential geometry and topology Glossary of group theory Graduate Texts in Mathematics Grassmann Group (mathematics)Group action Group action (mathematics)Group extension Group homomorphism Group object Group of Lie type Group representation Group theory Groupoid object Haar measure Hadamard manifold Half-integer spin Hall subgroup Harish-Chandra Hausdorff space Heisenberg group Henri Poincaré Hermann Weyl Hermitian manifold Highest weight Hilbert's fifth problem Hilbert manifold Hilbert–Smith conjecture History of manifolds and varieties Holomorphic map Homogeneous space Homomorphism Hopf–Rinow theorem Hydrogen-like atom Hydrogen atom Hyperbolic group Hyperbolic manifold Hypercomplex manifold Hyperkähler manifold Identity component Image (mathematics)Immersion (mathematics)Inclusion map Indefinite integral Index of a Lie algebra Infinite dimensional Lie group Infinite group Infinitesimal Injective Integer Integer spin Integral curve Interior product International Congress of Mathematicians Invariant (mathematics)Invertible matrix Irrational number Isometry J.-P. Serre Jet bundle Joe Harris (mathematician)John Frank Adams John Stillwell Julian Coolidge Julius Plücker K-theory Kac–Moody algebra Kenmotsu manifold Kernel (algebra)Killing form Koszul connection Kuiper's theorem Kähler manifold Lagrange's theorem (group theory)Lattice (discrete subgroup)Lattice (group)Leo Zippin Levi-Civita connection Levi decomposition Lie's third theorem Lie algebra Lie algebra bundle Lie algebra extension Lie algebra homomorphism Lie algebra representation Lie bracket Lie bracket of vector fields Lie derivative Lie group action Lie groupoid Lie group–Lie algebra correspondence Lie point symmetry Linear algebraic group Linear flow on the torus Linear map List of Lie group topics List of group theory topics List of manifolds List of simple Lie groups Local diffeomorphism Local property Locally convex space Loop algebra Loop group Lorentz group M-theory Manifold Manifold with boundary Mathematics Matrix (math)Matrix (mathematics)Matrix exponential Matrix group Minkowski space Modular form Modular group Morse theory Moving frame Multiplication Multiplicative group Natural transformation Neighborhood (mathematics)Neighborhood (topology)Nicolas Bourbaki Nilmanifold Nilpotent group Noether's theorem Non-Hausdorff manifold Normal bundle Normal coordinates Normal subgroup Number theory One-form One-parameter subgroup Orbifold Ordinary differential equation Orientability Orientable manifold Orthogonal group Orthogonal polynomials P-adic number P-group Parallel transport Parallelizable Parallelizable manifold Parametric equations Partial differential equation Particle physics and representation theory Paul Cohn Physics Poincaré group Pointwise multiplication Poisson manifold Polynomial equation Positive real numbers Power series Prime manifold Principal bundle Projective geometry Projective group Pseudo-Riemannian manifold Pullback (differential geometry)Pushforward (differential)Quadrature (mathematics)Quantization (physics)Quantum field theory Quantum mechanics Quaternion group Quaternionic manifold Quotient group Real-analytic manifold Real form (Lie theory)Real number Reductive group Ree group Representation of a Lie group Representation theory Representation theory of SL2(R)Representation theory of semisimple Lie algebras Representation theory of the Galilean group Representation theory of the Lorentz group Representation theory of the Poincaré group Representations of Lie groups Representations of classical Lie groups Ricci-flat manifold Ricci curvature Ricci flow Riemann curvature tensor Riemannian manifold Riemannian symmetric space Rizza manifold Root system Rotation (mathematics)Rotation group SO(3)SL2(R)SU(n)Sard's theorem Sasakian manifold Schrödinger equation Schur multiplier Secondary calculus and cohomological physics Section (fiber bundle)Semidirect product Semisimple Lie algebra Semisimple group Sheaf (mathematics)Simple Lie algebra Simple Lie group Simple group Simply connected Simply connected space Siméon Denis Poisson Singularity theory Smooth function Smooth map Smooth structure Smoothness Solenoid (mathematics)Solvable group Sophus Lie Special function Special linear group Special orthogonal group Special unitary group Spin(n)Spinor bundle Split Lie algebra Sporadic group Springer-Verlag Stable normal bundle Stein manifold Stratified space Stratifold Sub-Riemannian manifold Subbundle Subgroup Submanifold Submersion (mathematics)Subset Subspace topology Supergroup (physics)Supermanifold Surjective function Sylow theorems Symmetric group Symmetry Symmetry in quantum mechanics Symplectic group Symplectic manifold Symplectic vector space Table of Lie groups Tame manifold Tangent bundle Tangent space Tensor Tensor bundle Tensor density Tensor field Theorem of the highest weight Topological group Topological manifold Torsion tensor Torus Toshiyuki Kobayashi Totally disconnected group Transformation group Transpose Two-dimensional conformal field theory Uncountable set Unit quaternion Unitary group Universal cover Vector-valued differential form Vector bundle Vector field Vector flow Virasoro algebra Volume form Wedge product Weyl group Whitney embedding theorem Wigner's classification Wilhelm Killing William Fulton (mathematician)Witt algebra Wreath product Élie Cartan Évariste Galois

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