Exponential map: Encyclopedia II - Exponential map - Lie theory

Exponential map - Lie theory

The exponential map is a fundamental construction in the theory of Lie groups. It is a map from the Lie algebra of a Lie group to the group which allows one to completely 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 may be viewed as a special case of the exponential map when G is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The Lie-theoretic exponential map satisfies many properties analogous to those the ordinary exponential function, however, it also differs in many important respects.

Exponential map - Definition

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

given by exp(X) = γ(1) where

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

If G is a matrix Lie group, then the exponential map coincides with the matrix exponential and is given by the ordinary series expansion:

Exponential map - Properties

• For all , the map γ(t) = exp(tX) is the unique one-parameter subgroup of G whose tangent vector at the identity is X. It follows that:
• The exponential map is a smooth map. Its derivative at the identity, , is the identity map (with the usual identifications). The exponential map, therefore, restricts to a diffeomorphism from some neighborhood of 0 in to a neighborhood of 1 in G.
• The image of the exponential map always lies in the identity component of G. When G is compact, the exponential map is surjective onto the identity component.
• The map γ(t) = exp(tX) is the integral curve through the identity of both the right- and left-invariant vector fields associated to X.
• The integral curve through of the left-invariant vector field X^L associated to X is given by gexp(tX). Likewise, the integral curve through g of the right-invariant vector field X^R is given by exp(tX)g. It follows that the flows ξ^L,R generated by the vector fields X^L,R are given by:
  Since these flows are globally defined, every left- and right-invariant vector field on G is complete.
• Let be a Lie group homomorphism and let φ _* be its derivative at the identity. Then the following diagram commutes:

• In particular, when applied to the adjoint action of a group G we have

Adapted from the Wikipedia article "Lie theory", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki