In the theory of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s, the exponential map is a map from the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
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
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
of mathematical analysis is a special case of the exponential map when
is the multiplicative group of
positive real numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used f ...
(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.
Definitions
Let
be a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
and
be its
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
(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
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
:\varphi : \mathbb \rightarrow G
from the real line \mathbb (as an additive group) to some other topological group G.
If \varphi i ...
of
whose
tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
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
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
. 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, and 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'' .
Other definitions
Other equivalent definitions of the Lie-group exponential are as follows:
* It is the exponential map of a canonical left-invariant
affine connection on ''G'', such that
parallel transport is given by left translation. That is,
where
is the unique
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...
with the initial point at the identity element and the initial velocity ''X'' (thought of as a tangent vector).
* It is the exponential map of a canonical right-invariant affine connection on ''G''. This is usually different from the canonical left-invariant connection, but both connections have the same geodesics (orbits of 1-parameter subgroups acting by left or right multiplication) so give the same exponential map.
* The
Lie group–Lie algebra correspondence In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are ...
also gives the definition: for ''X'' in
,
is the unique Lie group homomorphism corresponding to the Lie algebra homomorphism
(note:
.)
Examples
* The
unit circle centered at 0 in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
is a Lie group (called the
circle group) whose tangent space at 1 can be identified with the imaginary line in the complex plane,
The exponential map for this Lie group is given by
::
:that is, the same formula as the ordinary
complex exponential.
* More generally, for
complex toruspg 8 for some integral
lattice of rank
(so isomorphic to
) the torus comes equipped with a
universal covering map
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
.
* In the
quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
, the set of
quaternions of unit length form a Lie group (isomorphic to the special unitary group ) whose tangent space at 1 can be identified with the space of purely imaginary quaternions,
The exponential map for this Lie group is given by
::
: This map takes the 2-sphere of radius inside the purely imaginary
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s to
, a 2-sphere of radius
(cf.
Exponential of a Pauli vector). Compare this to the first example above.
* Let ''V'' be a finite dimensional real vector space and view it as a Lie group under the operation of vector addition. Then
via the identification of ''V'' with its tangent space at 0, and the exponential map
::
:is the identity map, that is,
.
* In the
split-complex number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
plane
the imaginary line
forms the Lie algebra of the
unit hyperbola
In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative ra ...
group
since the exponential map is given by
::
Properties
Elementary properties of the exponential
For all
, the map
is the unique
one-parameter subgroup
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
:\varphi : \mathbb \rightarrow G
from the real line \mathbb (as an additive group) to some other topological group G.
If \varphi i ...
of
whose
tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
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
.
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
:
.
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:
* ''G'' is connected and compact,
* ''G'' is connected and nilpotent (for example, ''G'' connected and abelian), and
*
.
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
''SL''2(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
.)
Exponential map and homomorphisms
Let
be a Lie group homomorphism and let
be its
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
at the identity. Then the following diagram
commutes:
In particular, when applied to the
adjoint action of a Lie group
, since
, we have the useful identity:
[ Proposition 3.35]
:
.
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
In mathematics, an analytic manifold, also known as a C^\omega manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic g ...
to ''G'' such that the group operation
is real-analytic.
See also
*
List of exponential topics
*
Derivative of the exponential map
*
Matrix exponential
Citations
Works cited
*.
*.
*.
*
{{DEFAULTSORT:Exponential Map
Lie algebras
Lie groups