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

<br /> f:G \rightarrow G; x \mapsto x^{-1}<br />

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

<br /> \psi \circ f \circ \phi^{-1}<br />

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.