The Exponential Map - Core Definitions

The exponential map is the fundamental bridge connecting Lie groups to their Lie algebras. It provides a way to move from the linear structure of the Lie algebra to the nonlinear structure of the Lie group, enabling the study of Lie groups through infinitesimal methods.

Definition

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ . For each $X \in \mathfrak{g}$ , there exists a unique one-parameter subgroup $\gamma_X: \mathbb{R} \to G$ such that $\gamma_X(0) = e$ and $\gamma_X'(0) = X$ . The exponential map is defined by: $\exp: \mathfrak{g} \to G, \quad \exp(X) = \gamma_X(1)$

A one-parameter subgroup is a smooth group homomorphism $\gamma: \mathbb{R} \to G$ . The condition $\gamma'(0) = X$ means that $\gamma$ is the integral curve of the left-invariant vector field corresponding to $X$ , passing through the identity at $t = 0$ .

Example

For matrix Lie groups $G \subseteq GL_n(\mathbb{C})$ , the exponential map coincides with the matrix exponential: $\exp(X) = e^X = \sum_{k=0}^\infty \frac{X^k}{k!} = I + X + \frac{X^2}{2!} + \frac{X^3}{3!} + \cdots$ This series converges absolutely for all matrices $X \in M_n(\mathbb{C})$ .

The matrix exponential satisfies $\frac{d}{dt}e^{tX} = Xe^{tX} = e^{tX}X$ , confirming that $t \mapsto e^{tX}$ is a one-parameter subgroup with tangent vector $X$ at $t = 0$ .

Remark

The exponential map is natural in the sense that for any Lie group homomorphism $\phi: G \to H$ with induced Lie algebra homomorphism $\phi_*: \mathfrak{g} \to \mathfrak{h}$ , we have: $\phi(\exp_G(X)) = \exp_H(\phi_*(X))$ This commutative diagram property makes the exponential map functorial.

Definition

A vector field $X$ on a Lie group $G$ is left-invariant if $(L_g)_* X_h = X_{gh}$ for all $g, h \in G$ . The flow of a left-invariant vector field is given by $\phi_t(g) = g \cdot \exp(tX)$ for $X \in \mathfrak{g}$ .

The exponential map is smooth (in fact, real analytic) and its differential at the origin is the identity map: $d(\exp)_0 = \text{id}: \mathfrak{g} \to T_e G$ This follows from the fact that $\exp(tX)$ has derivative $X$ at $t = 0$ .

Example

For $G = S^1 = \{z \in \mathbb{C} : |z| = 1\}$ , the Lie algebra is $\mathfrak{g} = \mathbb{R}$ (identified with pure imaginary numbers $i\mathbb{R}$ ), and: $\exp: \mathbb{R} \to S^1, \quad \exp(\theta) = e^{i\theta}$ This is a smooth surjective homomorphism with kernel $2\pi\mathbb{Z}$ .

The exponential map encodes how "infinitesimal transformations" (elements of $\mathfrak{g}$ ) generate "finite transformations" (elements of $G$ ). This perspective is fundamental in physics, where generators of symmetries (Lie algebra elements) produce symmetry transformations (Lie group elements) via exponentiation.