Exponential map (Lie theory)

For the exponential map from a subset of the tangent space of a Riemannian manifold to the manifold, see Exponential map (Riemannian geometry).

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 is the multiplicative group of positive 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.


Let be a Lie group and be its Lie algebra (thought of as the tangent space to the identity element of ). The exponential map is a map

which can be defined in several different ways as follows:

is the unique one-parameter subgroup of whose tangent vector at the identity is equal to . It follows easily from the chain rule that . The map may be constructed as the integral curve of either the right- or left-invariant vector field associated with . That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.
(here is the identity matrix).


that is, the same formula as the ordinary complex exponential.
This map takes the 2-sphere of radius R inside the purely imaginary quaternions to , a 2-sphere of radius when . (cf. Exponential of a Pauli vector.) Compare this to the first example above.
is the identity map.


See also


  1. Hall 2015 Corollary 3.44
  2. Hall 2015 Corollary 11.10
  3. Hall 2015 Exercises 2.9 and 2.10
  4. Hall 2015 Exercise 3.22
  5. Hall 2015 Theorem 3.28


This article is issued from Wikipedia - version of the 10/12/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.