LieGroup - crowlogic/arb4j GitHub Wiki

A Lie group is a mathematical object that combines the properties of a group and a differentiable manifold. It is named after the Norwegian mathematician Sophus Lie, who introduced Lie groups to study the symmetries of differential equations. Lie groups have since become central objects in many areas of mathematics and physics.

Here are the two key concepts that form a Lie group:

1. Group: A group is a set with an operation (usually called multiplication) that satisfies four properties: closure, associativity, existence of an identity element, and existence of an inverse for each element. Groups are used to capture the algebraic structure of symmetries and transformations.

2. Differentiable manifold: A differentiable manifold is a topological space that locally looks like a Euclidean space and has a differentiable structure, meaning that the functions and maps defined on the manifold can be differentiated. Manifolds are used to model a wide range of mathematical objects, such as curves, surfaces, and higher-dimensional spaces.

A Lie group is a group that is also a differentiable manifold, with the group operations (multiplication and inversion) being smooth (differentiable) functions. In other words, a Lie group is a group whose elements can be parameterized by continuous variables, and the group operations can be described using smooth functions.

Lie groups are particularly important in the study of continuous symmetries and transformations, as they naturally capture the geometric and algebraic aspects of these phenomena. Examples of Lie groups include the general linear group $\text{GL}(n)$, the special orthogonal group $\text{SO}(n)$, and the unitary group $\text{U}(n)$. Lie groups play a crucial role in many areas of mathematics, such as differential geometry, algebraic topology, and representation theory, as well as in physics, particularly in the study of symmetries and conservation laws in classical and quantum mechanics.