Exponential map (Lie theory)

For the exponential map from a subset of the tangent space of a Riemannian manifold to the manifold, see Exponential map (Riemannian geometry).

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In the theory of Lie groups, the exponential map is a map from the Lie algebra ${\mathfrak {g}}$ of a Lie group $G$ 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 justifications for the study of Lie groups at the level of Lie algebras.

The ordinary exponential function of mathematical analysis is a special case of the exponential map when $G$ 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.

Definitions

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

\exp\colon \mathfrak g \to G

which can be defined in several different ways as follows:

It is given by $\exp(X) = \gamma(1)$ where

\gamma\colon \mathbb R \to G

is the unique one-parameter subgroup of

G

whose tangent vector at the identity is equal to

X

. It follows easily from the chain rule that

\exp(tX) = \gamma(t)

. The map

\gamma

may be constructed as the integral curve of either the right- or left-invariant vector field associated with

X

. That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.

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, $\exp(X) = \gamma(1)$ where $\gamma$ is the unique geodesic 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.
If $G$ is a matrix Lie group, then the exponential map coincides with the matrix exponential and is given by the ordinary series expansion:

\exp (X) = \sum_{k=0}^\infty\frac{X^k}{k!} = I + X + \frac{1}{2}X^2 + \frac{1}{6}X^3 + \cdots

(here

I

is the identity matrix).

If G is compact, it has a Riemannian metric invariant under left and right translations, and the exponential map is the exponential map of this Riemannian metric.
The Lie group–Lie algebra correspondence also gives the definition: for X in ${\mathfrak {g}}$ , $t\mapsto \exp(tX)$ is the unique Lie group homomorphism correspondening to the Lie algebra homomorphism $t\mapsto tX.$ (note: $\operatorname {Lie}({\mathbb {R}})={\mathbb {R}}$ .)

Examples

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, $\{it:t\in\mathbb R\}.$ The exponential map for this Lie group is given by
$it \mapsto \exp(it) = e^{it} = \cos(t) + i\sin(t),\,$

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

In the split-complex number plane $z = x + y \jmath , \quad \jmath^2 = +1,$ the imaginary line $\lbrace \jmath t : t \in \mathbb R \rbrace$ forms the Lie algebra of the unit hyperbola group $\lbrace \cosh t + \jmath \ \sinh t : t \in \mathbb R \rbrace$ since the exponential map is given by
$\jmath t \mapsto \exp(\jmath t) = \cosh t + \jmath \ \sinh t.$
The unit 3-sphere $S$ ³ centered at 0 in the quaternions H is 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, $\{it+ju + kv :t, u, v\in\mathbb R\}.$ The exponential map for this Lie group in this fundamental representation is given by
$\bold{w} = (it+ju+kv) \mapsto \exp(it+ju+kv) = \cos(|\bold{w}|) + \sin(|\bold{w}|)\frac{\bold{w}}{|\bold{w}|}.\,$

This map takes the 2-sphere of radius

R

inside the purely imaginary quaternions to

\{s\in S^{3}\subset {\mathbf {H}}:\operatorname {Re}(s)=\cos(R)\}

, a 2-sphere of radius

\sin(R)

when

R\not\equiv 0\pmod{2\pi}

. (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 an additive Lie group. Then $\operatorname {Lie}(V)=V$ via the identification of V with its tangent spaces at 0, and the exponential map

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

is the identity map.

Properties

For all $X\in\mathfrak g$ , the map $\gamma(t) = \exp(tX)$ is the unique one-parameter subgroup of $G$ whose tangent vector at the identity is $X$ . It follows that:
- $\exp(t+s)X = (\exp tX)(\exp sX)\,$
- $\exp(-X) = (\exp X)^{-1}.\,$
The exponential map $\exp\colon \mathfrak g \to G$ is a smooth map. Its derivative at the identity, $\exp_{*}\colon \mathfrak g \to \mathfrak g$ , is the identity map (with the usual identifications). The exponential map, therefore, restricts to a diffeomorphism from some neighborhood of 0 in ${\mathfrak {g}}$ to a neighborhood of 1 in $G$ .^[1]
The exponential map is not, however, a covering map in general – it is not a local diffeomorphism at all points. For example, so(3) to SO(3) is not a covering map; see also cut locus on this failure.
The image of the exponential map always lies in the identity component of $G$ . When $G$ is compact, the exponential map is surjective onto the identity component.^[2]
In general, the exponential map is surjective in the following cases: G is connected and compact, G is connected and nilpotent and $G=GL_{n}({\mathbb {C}})$ .^[3]
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 $-I$ . (Thus, the image excludes matrices with real, negative eigenvalues, other than $-I$ .) ^[4]
The map $\gamma(t) = \exp(tX)$ is the integral curve through the identity of both the right- and left-invariant vector fields associated to $X$ .
The integral curve through $g\in G$ of the left-invariant vector field $X^L$ associated to $X$ is given by $g \exp(t X)$ . Likewise, the integral curve through $g$ of the right-invariant vector field $X^R$ is given by $\exp(t X) g$ . It follows that the flows $\xi^{L,R}$ generated by the vector fields $X^{L,R}$ are given by:
- $\xi^L_t = R_{\exp tX}$
- $\xi^R_t = L_{\exp tX}.$
Since these flows are globally defined, every left- and right-invariant vector field on $G$ is complete.
Let $\phi \colon G\to H$ be a Lie group homomorphism and let $\phi _{*}$ be its derivative at the identity. Then the following diagram commutes:^[5]

In particular, when applied to the adjoint action of a group $G$ we have
- $g(\exp X)g^{-1} = \exp(\mathrm{Ad}_gX)\,$
- $\mathrm{Ad}_{\exp X} = \exp(\mathrm{ad}_X).\,$

Notes

↑ Hall 2015 Corollary 3.44
↑ Hall 2015 Corollary 11.10
↑ Hall 2015 Exercises 2.9 and 2.10
↑ Hall 2015 Exercise 3.22
↑ Hall 2015 Theorem 3.28

References

Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 0-387-40122-9 .
Hazewinkel, Michiel, ed. (2001), "Exponential mapping", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2848-9, MR 1834454 .
Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Vol. 1 (New ed.), Wiley-Interscience, ISBN 0-471-15733-3 .

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