📂Abstract Algebra

Lie Groups

Definition¹

A group $G$ is called a Lie group if it satisfies the following conditions:

1. It has a differentiable structure.

2. The binary operation $\cdot : G \times G \to G$ defined in $G$ is differentiable.

3. The inverse ${}^{-1} : G \to G$ is differentiable.

Explanation

Simply put, a Lie group is a differentiable group.

Examples

$(\mathbb{R}, +)$

1. Euclidean space has a differentiable structure.

2. $f : (x,y) \mapsto x+y \in C^{\infty}$

3. $g : x \mapsto -x \in C^{\infty}$