In the theory ofLie groups, theexponential map is a map from theLie algebra${\displaystyle \mathfrak{g}}$ of a Lie group${\displaystyle G}$ 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 ordinaryexponential function of mathematical analysis is a special case of the exponential map when${\displaystyle G}$ is the multiplicative group ofpositive 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${\displaystyle G}$ be aLie group and${\displaystyle \mathfrak{g}}$ be itsLie algebra (thought of as thetangent space to theidentity element of${\displaystyle G}$). Theexponential map is a map

${\displaystyle \exp :\mathfrak{g}\to G}$

which can be defined in several different ways. The typical modern definition is this:

Definition: The exponential of${\displaystyle X\in \mathfrak{g}}$ is given by${\displaystyle \exp (X)=\gamma (1)}$ where

${\displaystyle \gamma :\mathbb{R}\to G}$

is the uniqueone-parameter subgroup of${\displaystyle G}$ whosetangent vector at the identity is equal to${\displaystyle X}$.

It follows easily from thechain rule that${\displaystyle \exp (tX)=\gamma (t)}$. The map${\displaystyle \gamma }$, a group homomorphism from${\displaystyle (\mathbb{R},+)}$ to${\displaystyle G}$, may be constructed as theintegral curve of either the right- or left-invariantvector field associated with${\displaystyle X}$. 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 amatrix Lie group. The exponential map coincides with thematrix exponential and is given by the ordinary series expansion:

${\displaystyle \exp (X)=\sum _{k=0}^{\mathrm{\infty }}\frac{X^k}{k!}=I+X+\frac{1}{2}{X}^2+\frac{1}{6}{X}^3+\cdots }$,

where${\displaystyle I}$ is theidentity matrix. Thus, in the setting of matrix Lie groups, the exponential map is the restriction of the matrix exponential to the Lie algebra${\displaystyle \mathfrak{g}}$ of${\displaystyle G}$.

If${\displaystyle G}$ is compact, it has a Riemannian metric invariant under leftand right translations, then the Lie-theoretic exponential map for${\displaystyle G}$ coincides with theexponential map of this Riemannian metric.

For a general${\displaystyle G}$, 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 willnot in general agree with the exponential map in the Lie group sense. That is to say, if${\displaystyle G}$ 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${\displaystyle G}$^{[citation needed]}.

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

• It is the exponential map of a canonical left-invariantaffine connection onG, such thatparallel transport is given by left translation. That is,${\displaystyle \exp (X)=\gamma (1)}$ where${\displaystyle \gamma }$ is the uniquegeodesic with the initial point at the identity element and the initial velocityX (thought of as a tangent vector).
• It is the exponential map of a canonical right-invariant affine connection onG. 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.
• TheLie group–Lie algebra correspondence also gives the definition: for${\displaystyle X\in \mathfrak{g}}$, the mapping${\displaystyle t\mapsto \exp (tX)}$ is the unique Lie group homomorphism${\displaystyle (\mathbb{R},+)\to G}$ corresponding to the Lie algebra homomorphism${\displaystyle \mathbb{R}\to \mathfrak{g}}$,${\displaystyle t\mapsto tX.}$
• The exponential map is characterized by the differential equation$\frac{d}{dt}\exp (tX)=\exp (tX)\cdot X$ (or, equivalently,$\frac{d}{dt}\exp (tX)=X\cdot \exp (tX)$), where the right side uses the translation mapping${\displaystyle \mathfrak{g}=T_{e}G\to T_{g}G,X\mapsto g\cdot X}$ for${\displaystyle g=\exp (tX)}$. In the one-dimensional case, this is equivalent to${\displaystyle {\exp }^{\prime }(x)=\exp (x)}$.

• Theunit circle centered at 0 in thecomplex plane is a Lie group (called thecircle group) whose tangent space at 1 can be identified with the imaginary line in the complex plane,${\displaystyle \{it:t\in \mathbb{R}\}.}$ The exponential map for this Lie group is given by

${\displaystyle it\mapsto \exp (it)=e^{it}=\cos (t)+i\sin (t),}$

that is, the same formula as the ordinarycomplex exponential.

• More generally, forcomplex torus^[1]: 8${\displaystyle X={\mathbb{C}}^n/\mathrm{\Lambda }}$ for some integrallattice${\displaystyle \mathrm{\Lambda }}$ of rank${\displaystyle n}$ (so isomorphic to${\displaystyle {\mathbb{Z}}^n}$) the torus comes equipped with auniversal covering map

${\displaystyle \pi :{\mathbb{C}}^n\to X}$

from the quotient by the lattice. Since${\displaystyle X}$ is locally isomorphic to${\displaystyle {\mathbb{C}}^n}$ ascomplex manifolds, we can identify it with the tangent space${\displaystyle T_{0}X}$, and the map

${\displaystyle \pi :T_{0}X\to X}$

corresponds to the exponential map for the complex Lie group${\displaystyle X}$.

• In thequaternions${\displaystyle \mathbb{H}}$, the set ofquaternions of unit length form a Lie group (isomorphic to the special unitary groupSU(2)) whose tangent space at 1 can be identified with the space of purely imaginary quaternions,${\displaystyle \{it+ju+kv:t,u,v\in \mathbb{R}\}.}$ The exponential map for this Lie group is given by

${\displaystyle \mathbf{w}:=(it+ju+kv)\mapsto \exp (it+ju+kv)=\cos (|\mathbf{w}|)1+\sin (|\mathbf{w}|)\frac{\mathbf{w}}{|\mathbf{w}|}.}$

This map takes the 2-sphere of radiusR inside the purely imaginaryquaternions to${\displaystyle \{s\in S^3\subset \mathbf{H}:\mathrm{Re}(s)=\cos (R)\}}$, a 2-sphere of radius${\displaystyle \sin (R)}$ (cf.Exponential of a Pauli vector). Compare this to the first example above.

• LetV be a finite dimensional real vector space and view it as a Lie group under the operation of vector addition. Then${\displaystyle \mathrm{Lie}(V)=V}$ via the identification ofV with its tangent space at 0, and the exponential map

${\displaystyle \exp :\mathrm{Lie}(V)=V\to V}$

is the identity map, that is,${\displaystyle \exp (v)=v}$.

• In thesplit-complex number plane${\displaystyle z=x+yȷ,{ȷ}^2=+1,}$ the imaginary line${\displaystyle \{ȷt:t\in \mathbb{R}\}}$ forms the Lie algebra of theunit hyperbola group${\displaystyle \{\cosh t+ȷ\sinh t:t\in \mathbb{R}\}}$ since the exponential map is given by

${\displaystyle ȷt\mapsto \exp (ȷt)=\cosh t+ȷ\sinh t.}$

For all${\displaystyle X\in \mathfrak{g}}$, the map${\displaystyle \gamma (t)=\exp (tX)}$ is the uniqueone-parameter subgroup of${\displaystyle G}$ whosetangent vector at the identity is${\displaystyle X}$. It follows that:

• ${\displaystyle \exp ((t+s)X)=\exp (tX)\exp (sX)}$
• ${\displaystyle \exp (-X)=\exp (X{)}^{-1}.}$

More generally:

• ${\displaystyle \exp (X+Y)=\exp (X)\exp (Y),\text{if }[X,Y]=0}$.^[2]

The preceding identity does not hold in general; the assumption that${\displaystyle X}$ and${\displaystyle Y}$ commute is important.

The image of the exponential map always lies in theidentity component of${\displaystyle G}$.

The exponential map${\displaystyle \exp :\mathfrak{g}\to G}$ is asmooth map. Itsdifferential at zero,${\displaystyle {\exp }_{\ast }:\mathfrak{g}\to \mathfrak{g}}$, is the identity map (with the usual identifications).

It follows from the inverse function theorem that the exponential map, therefore, restricts to adiffeomorphism from some neighborhood of 0 in${\displaystyle \mathfrak{g}}$ to a neighborhood of 1 in${\displaystyle G}$.^[3]

It is then not difficult to show that ifG is connected, every elementg ofG is aproduct of exponentials of elements of${\displaystyle \mathfrak{g}}$:^[4]${\displaystyle g=\exp (X_1)\exp (X_2)\cdots \exp (X_n),X_j\in \mathfrak{g}}$.

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${\displaystyle \mathfrak{s}\mathfrak{o}}$(3) toSO(3) is not a local diffeomorphism; see alsocut locus on this failure. Seederivative of the exponential map for more information.

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

• G is connected and compact,^[5]
• G is connected andnilpotent (for example,G connected and abelian), or
• ${\displaystyle G=G{L}_n(\mathbb{C})}$.^[6]

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 groupSL₂(R) is not the whole group. Its image consists ofC-diagonalizable matrices with eigenvalues either positive or with modulus 1, and of non-diagonalizable matrices with a repeated eigenvalue 1, and the matrix${\displaystyle -I}$. (Thus, the image excludes matrices with real, negative eigenvalues, other than${\displaystyle -I}$.)^[7]

Let${\displaystyle \varphi :G\to H}$ be a Lie group homomorphism and let${\displaystyle {\varphi }_{\ast }}$ be itsderivative at the identity. Then the following diagramcommutes:^[8]

In particular, when applied to theadjoint action of a Lie group${\displaystyle G}$, since${\displaystyle {\mathrm{Ad}}_{\ast }=\mathrm{ad}}$, we have the useful identity:^[9]

${\displaystyle {\mathrm{A}\mathrm{d}}_{\exp X}(Y)=\exp ({\mathrm{a}\mathrm{d}}_X)(Y)=Y+[X,Y]+\frac{1}{2!}[X,[X,Y]]+\frac{1}{3!}[X,[X,[X,Y]]]+\cdots }$.

Given a Lie group${\displaystyle G}$ with Lie algebra${\displaystyle \mathfrak{g}}$, each choice of a basis${\displaystyle X_1,\dots ,X_n}$ of${\displaystyle \mathfrak{g}}$ determines a coordinate system near the identity elemente forG, as follows. By theinverse function theorem, the exponential map${\displaystyle \exp :N\stackrel{\sim }{\to }U}$ is a diffeomorphism from some neighborhood${\displaystyle N\subset \mathfrak{g}\simeq {\mathbb{R}}^n}$ of the origin to a neighborhood${\displaystyle U}$ of${\displaystyle e\in G}$. Its inverse:

${\displaystyle \log :U\stackrel{\sim }{\to }N\subset {\mathbb{R}}^n}$

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

Remark: The open cover${\displaystyle \{Ug|g\in G\}}$ gives a structure of areal-analytic manifold toG such that the group operation${\displaystyle (g,h)\mapsto g{h}^{-1}}$ is real-analytic.^[10]

• List of exponential topics
• Derivative of the exponential map
• Matrix exponential

• Lie algebras
• Lie groups

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