Topological groups, Lie groups Lie groups: basic notions Concept
Lie group
Summary
A Lie group is a group that is also a smooth manifold, in such a way that multiplication and inversion are smooth maps. It is the natural setting for continuous symmetry: rotations, translations, matrix groups and the symmetry groups of physics are all Lie groups. Its structure near the identity is captured by a finite-dimensional Lie algebra, which turns many questions about the group into linear algebra.
Where this leads
63 pages build on this one.
- Semisimple Lie group
- Rotation group SO(3)
- Left-invariant vector field
- Adjoint representation of a Lie group
- List of simple Lie groups
- Representation of a Lie group
- Oscillator representation
- Ratner's theorems
- Closed subgroup theorem
- Exponential map of a Lie group
- Cartan–Iwasawa–Malcev theorem
- Compact symplectic group
- Lie supergroup
- Loop group
- Cohomology of Lie groups
- Poincaré group
- Representation theory of SU(2)
- Real form of a Lie group
- Special unitary group SU(n)
- Solvmanifold
- Cartan's theorem on continuous homomorphisms
- Hilbert–Smith conjecture
- Universal covering group
- Levi–Malcev decomposition
- Complexification of a Lie group
- Wigner's classification
- SU(2)
- Compact Lie group
- Lie group action
- One-parameter subgroup
- SL(2,C)
- Nilmanifold
- Flag manifold
- Spin group
- Lie subgroup
- Lie groupoid
- Lie group–Lie algebra correspondence
- Kleinian group
- Raghunathan's conjecture
- Exceptional Lie group E8
- Simple Lie group
- Hilbert's fifth problem
- Lie's theorems on transformation groups
- Zassenhaus neighbourhood
- Lie's third theorem
- Reductive Lie group
- Lorentz group
- Representation theory of the Lorentz group
- Nilpotent Lie group
- Haar measure on a Lie group
- Clifford–Klein form
- Pin group
- Solvable Lie group
- Diffeomorphism group
- Maurer–Cartan form
- Baker–Campbell–Hausdorff formula
- Matrix Lie group
- Gleason–Yamabe theorem
- Malcev completion
- Exceptional Lie group G2
- Lattice in a Lie group
- Stone–von Neumann theorem
- Heisenberg group
Further reading
About this page
| Tag | 7JBRGCVJRB |
|---|---|
| URI | m2://7JBRGCVJRB |
| Kind | concept |
| Writing | W0 stub |
| MSC | 22E15, 22E10 |
| Specializes | Topological group |
| Same as | |
| Changed | 2 November 2026 |
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What links here
Used by
- Semisimple Lie group
- Rotation group SO(3)
- Left-invariant vector field
- Adjoint representation of a Lie group
- Representation of a Lie group
- Ratner's theorems
- Closed subgroup theorem
- Exponential map of a Lie group
- Cartan–Iwasawa–Malcev theorem
- Lie supergroup
- Loop group
- Cohomology of Lie groups
- Real form of a Lie group
- Special unitary group SU(n)
- Solvmanifold
- Cartan's theorem on continuous homomorphisms
- Hilbert–Smith conjecture
- Universal covering group
- Levi–Malcev decomposition
- Complexification of a Lie group
- Compact Lie group
- Lie group action
- One-parameter subgroup
- Flag manifold
- Lie subgroup
- Lie groupoid
- Lie group–Lie algebra correspondence
- Raghunathan's conjecture
- Simple Lie group
- Hilbert's fifth problem
- Lie's theorems on transformation groups
- Zassenhaus neighbourhood
- Lie's third theorem
- Reductive Lie group
- Nilpotent Lie group
- Haar measure on a Lie group
- Clifford–Klein form
- Solvable Lie group
- Diffeomorphism group
- Maurer–Cartan form
- Baker–Campbell–Hausdorff formula
- Gleason–Yamabe theorem
- Lattice in a Lie group
- Heisenberg group
Special cases
Instances
Results about this
- Cartan's theorem on continuous homomorphisms
- Levi–Malcev decomposition
- Lie group–Lie algebra correspondence
- Hilbert's fifth problem
- Lie's third theorem
- Gleason–Yamabe theorem