Exponential map (Lie theory)

In the theory of Lie groups, the exponential map is a map from the Lie 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 ordinary exponential function of mathematical analysis is a special case of the exponential map when ${\displaystyle G}$ 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 ${\displaystyle G}$ 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 ${\displaystyle \gamma }$, a group homomorphism from ${\displaystyle (\mathbb {R} ,+)}$ to ${\displaystyle G}$, 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]: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 ${\displaystyle e}$ for ${\displaystyle G}$, as follows. By the inverse function theorem, the exponential map ${\displaystyle \exp :N\xrightarrow {\sim } U}$ 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: $${\displaystyle \log :U\xrightarrow {\sim } N\subset \mathbb {R} ^{n}}$$ is then a coordinate system on ${\displaystyle U}$. 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 ${\displaystyle \{Ug\mid g\in G\}}$ gives a structure of a real-analytic manifold to ${\displaystyle G}$ such that the group operation ${\displaystyle (g,h)\mapsto gh^{-1}}$ is real-analytic.^[10]

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