Exponential map (Lie theory)

Let ${\displaystyle G}$ be a Lie group and ${\displaystyle {\mathfrak {g}}}$ be its Lie algebra (thought of as the tangent space to the identity element of ${\displaystyle G}$). The exponential map is a map

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

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

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

is the unique one-parameter subgroup of ${\displaystyle G}$ whose tangent vector at the identity is equal to ${\displaystyle X}$.

It follows easily from the chain rule that ${\displaystyle \exp(tX)=\gamma (t)}$. The map ${\displaystyle \gamma }$ may be constructed as the integral curve of either the right- or left-invariant vector 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 a matrix Lie group. The exponential map coincides with the matrix exponential and is given by the ordinary series expansion:

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

where ${\displaystyle I}$ 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 ${\displaystyle {\mathfrak {g}}}$ of ${\displaystyle G}$.

• The unit circle centered at 0 in the complex plane is a Lie group (called the circle 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 ordinary complex exponential.

• More generally, for complex torus[1]^{pg 8} ${\displaystyle X=\mathbb {C} ^{n}/\Lambda }$ for some integral lattice ${\displaystyle \Lambda }$ of rank ${\displaystyle n}$ (so isomorphic to ${\displaystyle \mathbb {Z} ^{n}}$) the torus comes equipped with a universal covering map

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

from the quotient by the lattice. Since ${\displaystyle X}$ is locally isomorphic to ${\displaystyle \mathbb {C} ^{n}}$ as complex 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 the quaternions ${\displaystyle \mathbb {H} }$, the set of quaternions of unit length form a Lie group (isomorphic to the special unitary group SU(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 radius R inside the purely imaginary quaternions to ${\displaystyle \{s\in S^{3}\subset \mathbf {H}$ :\operatorname {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.

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

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

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

• In the split-complex number plane ${\displaystyle z=x+y\jmath ,\quad \jmath ^{2}=+1,}$ the imaginary line ${\displaystyle \lbrace \jmath t:t\in \mathbb {R} \rbrace }$ forms the Lie algebra of the unit hyperbola group ${\displaystyle \lbrace \cosh t+\jmath \ \sinh t:t\in \mathbb {R} \rbrace }$ since the exponential map is given by

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

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 element e for G, as follows. By the inverse function theorem, the exponential map ${\displaystyle \operatorname {exp} :N{\overset {\sim }{\to }}U}$ 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:

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

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 ${\displaystyle \{Ug|g\in G\}}$ gives a structure of a real-analytic manifold to G such that the group operation ${\displaystyle (g,h)\mapsto gh^{-1}}$ is real-analytic.[9]

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