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 reasons that Lie algebras are a useful tool for studying Lie groups.

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. The typical modern definition is this:

Definition: The exponential of is given by where
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.

We have a more concrete definition in the case of a matrix Lie group. The exponential map coincides with the matrix exponential and is given by the ordinary series expansion:


where is the identity matrix. Thus, in the setting of matrix Lie groups, the exponential map is the restriction of the matrix exponential to the Lie algebra of .

Comparison with Riemannian exponential map

If G is compact, it has a Riemannian metric invariant under left and right translations, then the Lie-theoretic exponential map for G coincides with the exponential map of this Riemannian metric.

For a general G, there will not exist a Riemannian metric invariant under both left and right translations. Although there is always a Riemannian metric invariant under, say, left translations, the exponential map in the sense of Riemannian geometry for a left-invariant metric will not in general agree with the exponential map in the Lie group sense. That is to say, if G is a Lie group equipped with a left- but not right-invariant metric, the geodesics through the identity will not be one-parameter subgroups of G[citation needed].

Other definitions

Other equivalent definitions of the Lie-group exponential are as follows:


that is, the same formula as the ordinary complex exponential.

from the quotient by the lattice. Since is locally isomorphic to as complex manifolds, we can identify it with the tangent space , and the map

corresponds to the exponential map for the complex Lie group .

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


Elementary properties of the exponential

For all , the map is the unique one-parameter subgroup of whose tangent vector at the identity is . It follows that:

More generally:

It is important to emphasize that the preceding identity does not hold in general; the assumption that and commute is important.

The image of the exponential map always lies in the identity component of .

The exponential near the identity

The exponential map is a smooth map. Its differential at zero, , is the identity map (with the usual identifications).

It follows from the inverse function theorem that the exponential map, therefore, restricts to a diffeomorphism from some neighborhood of 0 in to a neighborhood of 1 in .[2]

It is then not difficult to show that if G is connected, every element g of G is a product of exponentials of elements of :[3].

Globally, the exponential map is not necessarily surjective. Furthermore, the exponential map may not be a local diffeomorphism at all points. For example, the exponential map from (3) to SO(3) is not a local diffeomorphism; see also cut locus on this failure. See derivative of the exponential map for more information.

Surjectivity of the exponential

In these important special cases, the exponential map is known to always be surjective:

For groups not satisfying any of the above conditions, the exponential map may or may not be surjective.

The image of the exponential map of the connected but non-compact group SL2(R) is not the whole group. Its image consists of C-diagonalizable matrices with eigenvalues either positive or with modulus 1, and of non-diagonalizable matrices with a repeated eigenvalue 1, and the matrix . (Thus, the image excludes matrices with real, negative eigenvalues, other than .)[6]

Exponential map and homomorphisms

Let be a Lie group homomorphism and let be its derivative at the identity. Then the following diagram commutes:[7]

In particular, when applied to the adjoint action of a Lie group , since , we have the useful identity:[8]


Logarithmic coordinates

Given a Lie group with Lie algebra , each choice of a basis of determines a coordinate system near the identity element e for G, as follows. By the inverse function theorem, the exponential map is a diffeomorphism from some neighborhood of the origin to a neighborhood of . Its inverse:

is then a coordinate system on U. It is called by various names such as logarithmic coordinates, exponential coordinates or normal coordinates. See the closed-subgroup theorem for an example of how they are used in applications.

Remark: The open cover gives a structure of a real-analytic manifold to G such that the group operation is real-analytic.[9]

See also


  1. ^ Birkenhake, Christina (2004). Complex Abelian Varieties. Herbert Lange (Second, augmented ed.). Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-662-06307-1. OCLC 851380558.
  2. ^ Hall 2015 Corollary 3.44
  3. ^ Hall 2015 Corollary 3.47
  4. ^ Hall 2015 Corollary 11.10
  5. ^ Hall 2015 Exercises 2.9 and 2.10
  6. ^ Hall 2015 Exercise 3.22
  7. ^ Hall 2015 Theorem 3.28
  8. ^ Hall 2015 Proposition 3.35
  9. ^ Kobayashi & Nomizu 1996, p. 43.

Works cited