Left Translation Identifies Every Tangent Space of a Lie Group

A tangent vector at the identity extends uniquely to a left-invariant vector field. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

The data

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.

$$ L_g:G\to G,\qquad L_g(h)=gh $$

I keep the defining relation $\mathsf D$ above the derived relation $\mathsf C$. This exposes whether cancellation used $x\ne0$ and whether the conclusion is canonical.

$$ X_g=(dL_g)_eX_e $$

Derivation

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ [X,Y]_e=\left.\frac{d}{dt}\frac{d}{ds}\right|_{0}\exp(tX)\exp(sY)\exp(-tX)\exp(-sY) $$

Invariant content

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad L_g:G\to G,\qquad L_g(h)=gh,\\[5pt] \mathsf{C}\;&:\quad X_g=(dL_g)_eX_e. \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] X_g=(dL_g)_eX_e \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Tue 02 October 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.

Left Translation Identifies Every Tangent Space of a Lie Group by Vicfred

The data

Derivation

Invariant content

Scope