Lie Theory
| Lie groups |
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Classical groups
General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) |
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Simple Lie groups
List of simple Lie groups Classical: An, Bn, Cn, Dn Exceptional: G2, F4, E6, E7, E8 |
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Other Lie groups
Circle group Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group |
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Lie algebras
Exponential map Adjoint representation of a Lie group Adjoint representation of a Lie algebra Killing form Lie point symmetry |
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Semi-simple Lie groups
Dynkin diagrams Cartan subalgebra Root system Real form Complexification Split Lie algebra Compact Lie algebra |
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Representation theory
Representation of a Lie group Representation of a Lie algebra |
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Lie groups in Physics
Particle physics and representation theory Representation theory of the Lorentz group Representation theory of the Poincaré group Representation theory of the Galilean group |
In the theory of Lie groups the exponential map is a map from the Lie algebra of a Lie group to the group which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary justifications for the study of Lie groups at the level of Lie algebras.
The ordinary exponential function of mathematical analysis is a special case of the exponential map when G is the multiplicative group of non-zero real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects.
Read more about this topic: Exponential Map
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