Proving differentiability of function on a Lie group.

In summary, on page 116 of Choquet-Bruhat, Analysis, Manifolds, and Physics, Lie groups are defined and the first exercise asks to prove that for a Lie group G, the function f: G -> G; x -> x^-1 is differentiable. The approach is to use the definition of a Lie group, which states that the map from G x G -> G; (x, y) -> xy^-1 is differentiable, and let x = e, the identity element. Some users suggest using [itex][/itex] to keep inline Latex from appearing above the line.
  • #1
Daverz
1,003
78
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 [itex]G[/itex]

[itex]
f:G \rightarrow G; x \mapsto x^{-1}
[/itex]

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

[itex]
\psi \circ f \circ \phi^{-1}
[/itex]

is differentiable, where [itex](U, \phi)[/itex] and [itex](W, \psi)[/itex] are charts for neighborhoods of [itex] x [/itex] and [itex]y=f(x)=x^{-1}[/itex]. 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?
 
Last edited:
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  • #2
In the books which I have used your exercise is the definition of the Lie group :smile: ?.
 
  • #3
Ah, OK, I see it now, you just prove it from the definition of a Lie group, which is that the map
from [itex]G \times G \rightarrow G; (x, y) \mapsto x y^{-1}[/itex] is differentiable. Just let [itex]x = e[/itex], 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.
 
Last edited:
  • #4
Just use [ itex ] [ /itex ].

Daniel.
 

What is a Lie group?

A Lie group is a mathematical concept that combines the structure of a group (a set with a binary operation) with that of a smooth manifold (a space that locally resembles Euclidean space). It is named after the Norwegian mathematician Sophus Lie, who first studied these types of groups in the 19th century.

Why is proving differentiability important for functions on a Lie group?

Proving differentiability is important because it allows us to study the behavior of functions on a Lie group in a smooth and consistent manner. This is crucial in many applications, such as physics and engineering, where Lie groups are often used to describe symmetries and transformations.

What is the definition of differentiability for a function on a Lie group?

A function on a Lie group is differentiable if it is continuous and has a well-defined tangent space at every point on the group. In other words, it must be smooth and have a well-defined derivative at every point.

How do you prove differentiability for a function on a Lie group?

To prove differentiability, we typically use the tools of differential geometry. This involves showing that the function is smooth and has a well-defined derivative at every point on the group. In some cases, we may also need to use specific techniques for Lie groups, such as the theory of Lie algebras.

What are some applications of proving differentiability for functions on a Lie group?

Functions on a Lie group are used in a wide range of applications, such as in physics to describe symmetries and in computer science for efficient algorithms. Proving differentiability allows us to study these functions in a rigorous and consistent manner, which is important for developing accurate and reliable models and solutions.

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