# Proving differentiability of function on a Lie group.

• Daverz
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  to keep inline Latex from appearing above the line.
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?

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

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.

Last edited:
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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