The Exponential Map - Key Properties

The exponential map possesses several crucial properties that determine its behavior and effectiveness as a tool for studying Lie groups. Understanding when the exponential map is bijective, surjective, or has other special properties is essential for applications.

Theorem

Local Diffeomorphism Property

The exponential map $\exp: \mathfrak{g} \to G$ is a local diffeomorphism near $0 \in \mathfrak{g}$ . That is, there exists a neighborhood $U$ of $0$ in $\mathfrak{g}$ such that $\exp|_U: U \to \exp(U)$ is a diffeomorphism.

This follows from the inverse function theorem, since $d(\exp)_0 = \text{id}$ is invertible. Consequently, every point in $G$ sufficiently close to the identity can be uniquely written as $\exp(X)$ for some $X$ near $0$ in $\mathfrak{g}$ .

Example

For $SU(2)$ , which is diffeomorphic to the 3-sphere $S^3$ , the exponential map $\exp: \mathfrak{su}(2) \to SU(2)$ is surjective but not injective globally. The Lie algebra $\mathfrak{su}(2) \cong \mathbb{R}^3$ maps onto all of $SU(2)$ , with different elements of $\mathfrak{su}(2)$ potentially mapping to the same group element.

Remark

Global behavior varies:

For $G = \mathbb{R}^n$ (additive group), $\exp$ is the identity map, hence a global diffeomorphism
For compact Lie groups, $\exp$ is always surjective but typically not injective
For $GL_n(\mathbb{R})$ , the exponential map is neither surjective nor injective globally
For nilpotent Lie groups, $\exp$ is a global diffeomorphism

Theorem

Baker-Campbell-Hausdorff Formula

For $X, Y \in \mathfrak{g}$ sufficiently small, there exists $Z \in \mathfrak{g}$ such that: $\exp(X)\exp(Y) = \exp(Z)$ where $Z = X + Y + \frac{1}{2}[X, Y] + \frac{1}{12}[X, [X, Y]] - \frac{1}{12}[Y, [X, Y]] + \cdots$

The series for $Z$ involves only Lie brackets of $X$ and $Y$ .

The BCH formula shows that the group multiplication near the identity is completely determined by the Lie algebra structure. The formula converges when $\|X\|$ and $\|Y\|$ are sufficiently small.

Example

For $\mathfrak{sl}_2$ , if $X, Y$ satisfy $[X, Y] = Y$ and $[[X, Y], Y] = 0$ , then: $\exp(X)\exp(Y) = \exp(X + Y + \frac{1}{2}Y)$ This simplification occurs because higher-order brackets vanish.

Adjoint representation compatibility: The exponential map intertwines the adjoint representations of $G$ and $\mathfrak{g}$ : $\text{Ad}_{\exp(X)} = e^{\text{ad}_X}$ where $\text{Ad}_g(Y) = gYg^{-1}$ for matrix groups, and $\text{ad}_X(Y) = [X, Y]$ .

Remark

For connected Lie groups, the exponential map is surjective if and only if $G$ is compact. For non-compact connected groups (like $SL_n(\mathbb{R})$ with $n \geq 3$ ), there exist elements not in the image of $\exp$ .

The differential of the exponential map at $X \in \mathfrak{g}$ is given by: $d(\exp)_X = (L_{\exp(X)})_* \circ \frac{1 - e^{-\text{ad}_X}}{\text{ad}_X}$ This formula involves the operator $\frac{1 - e^{-\text{ad}_X}}{\text{ad}_X} = \sum_{k=0}^\infty \frac{(-\text{ad}_X)^k}{(k+1)!}$ , which is well-defined as a power series.