Exponential Map - Lie Theory

Lie Theory

Lie groups
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)
Simple Lie groups List of simple Lie groups
Classical: An, Bn, Cn, Dn
Exceptional: G2, F4, E6, E7, E8
Other Lie groups Circle group
Lorentz group
Poincaré group
Conformal group
Diffeomorphism group
Loop group
Lie algebras Exponential map
Adjoint representation of a Lie group
Adjoint representation of a Lie algebra
Killing form
Lie point symmetry
Semi-simple Lie groups Dynkin diagrams
Cartan subalgebra
Root system
Real form
Complexification
Split Lie algebra
Compact Lie algebra
Representation theory Representation of a Lie group
Representation of a Lie algebra
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

Famous quotes containing the words lie and/or theory:

    Ye wise wives, that conne understonde,
    Thus sholde ye speke and bere him wrong on honde—
    For half so boldely can ther no man
    Swere and lie as a woman can.
    Geoffrey Chaucer (1340?–1400)

    Everything to which we concede existence is a posit from the standpoint of a description of the theory-building process, and simultaneously real from the standpoint of the theory that is being built. Nor let us look down on the standpoint of the theory as make-believe; for we can never do better than occupy the standpoint of some theory or other, the best we can muster at the time.
    Willard Van Orman Quine (b. 1908)