Lagmental Vicfred

The Exponential Map Sends Lie Algebra Directions to One-Parameter Subgroups by Vicfred

tags: mathematics, Lie groups, hyperbolic geometry, Mobius transformations

Each tangent vector at the identity integrates to a homomorphism from the additive real line. This is a compact note, but the quantifiers and hypotheses stay on the page.

Statement

A Lie group \(G\) is simultaneously a smooth manifold and a group. Hyperbolic metrics on \(\mathbf H\) and \(\mathbf D\) have Lie groups of Möbius isometries.

$$ \exp:\mathfrak g\to G $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ \exp((s+t)X)=\exp(sX)\exp(tX) $$

Worked algebra

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ X=\begin{pmatrix}0&-\theta\\\theta&0\end{pmatrix}\Longrightarrow e^X=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix} $$

Conceptual compression

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad \exp:\mathfrak g\to G,\\[5pt] \mathsf{C}\;&:\quad \exp((s+t)X)=\exp(sX)\exp(tX). \end{aligned} $$

Caveat

Euclidean drawings distort hyperbolic distance and angle behavior unless the chosen model is conformal. Boundary points are not interior points at finite distance.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \exp((s+t)X)=\exp(sX)\exp(tX) \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Thu 08 August 2013. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.