The Exponential Map

exp: Lie algebra → Lie group — tangent vectors generate group elements

From Algebra to Group

The exponential map is the bridge between a Lie algebra (the tangent space at the identity) and the Lie group itself. Given a tangent vector X, the exponential exp(X) “flows” along the group for unit time, producing a group element. For matrix groups, this is literally the matrix exponential: exp(X) = I + X + X²/2! + X³/3! + …

Near the identity, the exponential map is a diffeomorphism — it gives a perfect local coordinate system. But globally it can fail to be surjective: some group elements may not be reachable by a single exponential. Understanding where and why this fails reveals deep structure in the group.

Exponential Flow on SO(2)

On SO(2), the Lie algebra is just ℝ (angular velocities). The exponential map sends ω to rotation by angle ω. Adjust the angular velocity and watch a glowing particle flow from the tangent line along the circle, wrapping around as the angle increases past 2π.

ω > 0 counterclockwise, ω < 0 clockwise

Key insight: The exponential map for SO(2) is surjective — every rotation is reachable. It wraps the real line (the algebra) around the circle (the group) like thread on a spool.

Exponential Map on SO(3)

In SO(3), the Lie algebra so(3) consists of 3×3 skew-symmetric matrices, parameterized by &Ropf;³. Each algebra element ω generates a one-parameter subgroup: a family of rotations around the axis ω/|ω|. Drag the algebra vector to see the corresponding one-parameter subgroup traced on the SO(3) ball, with afterimage trails on the rotating frame.

Taylor Series: Building exp(X) Term by Term

The matrix exponential exp(X) = I + X + X²/2! + X³/3! + … is an infinite series that always converges. Watch the partial sums converge as terms are added: each order gets a spectral color, and the grid transformation approaches the exact exponential.

The Logarithm and Its Limits

The matrix logarithm is the inverse of the exponential — when it exists. Near the identity, log is well-defined and smooth. But near rotation by π, the logarithm becomes multi-valued: two candidate algebra elements map to the same group element. Click to place a target rotation and see the logarithm computed. The “fold” near π glows as a seam on the ball surface.

Click inside the ball to place a target rotationUnique log

Key Takeaways

  • exp: g → G — the exponential map sends Lie algebra elements (tangent vectors) to Lie group elements
  • Matrix exponential — for matrix groups, exp is the familiar power series I + X + X²/2! + …
  • One-parameter subgroups — exp(tX) traces a curve through the group, the “flow” generated by X
  • Non-surjectivity — the exponential map may not reach every group element; the logarithm can be multi-valued