Proving differentiability of function on a Lie group.

[Daverz]

On page 116 of Choquet-Bruhat, Analysis, Manifolds, and Physics, Lie groups are defined, and the first exercise after that asks you to prove that
for a Lie group $G$

$f:G \rightarrow G; x \mapsto x^{-1}$

is differentiable. I know from the previous definitions that a function $f$ on a manifold is differentiable at $x$ if

$\psi \circ f \circ \phi^{-1}$

is differentiable, where $(U, \phi)$ and $(W, \psi)$ are charts for neighborhoods of $x$ and $y=f(x)=x^{-1}$. It's probably an indication of how weak my analysis is, but I don't see how to proceed from there. Can anyone point me in the right direction?

 

[Los Bobos]

In the books which I have used your exercise is the definition of the Lie group ?.

 

[Daverz]

Ah, OK, I see it now, you just prove it from the definition of a Lie group, which is that the map
from $G \times G \rightarrow G; (x, y) \mapsto x y^{-1}$ is differentiable. Just let $x = e$, the identity element. I was making it way too difficult. BTW, does anyone know how to keep inline Latex from showing up above the line like that.

 

[dextercioby]

Just use [ itex ] [ /itex ].

Daniel.