Lie Group

Introduction

A Lie group is a mathematical structure that combines the properties of a differentiable manifold and a group in a compatible way. Named after the Norwegian mathematician Sophus Lie, Lie groups play a fundamental role in modern geometry, particularly in the study of continuous transformation groups.

Definition

Formally, a Lie group is a group that is also a finite-dimensional real or complex differentiable manifold, in which the group operations of multiplication and inversion are smooth maps. More specifically, if G is a group and G is also a differentiable manifold such that the group operations

m : G × G → G (multiplication)

i : G → G (inversion)

are smooth, then G is a Lie group.

A smooth, continuous surface representing a Lie group.

Structure of Lie Groups

Lie groups can be thought of as smooth manifolds that have a group structure. This means that they have a topological structure and a smooth structure that is compatible with the group operations. The smooth structure allows us to talk about smooth maps and the tangent space at a point, which are essential in the study of Lie groups.

Topological Structure

The topological structure of a Lie group is given by the topology induced by the manifold structure. This means that a set is open in the group if and only if it is open in the manifold. The group operations of multiplication and inversion are continuous with respect to this topology.

Smooth Structure

The smooth structure of a Lie group is given by the differentiable structure of the manifold. This allows us to talk about smooth maps between Lie groups, and to define the tangent space at a point. The group operations of multiplication and inversion are smooth with respect to this structure.

Lie Algebra

To each Lie group, we can associate a Lie algebra that encapsulates the local structure of the group. The Lie algebra of a Lie group G is the tangent space at the identity element of G, equipped with a binary operation called the Lie bracket. This Lie bracket is defined in terms of the group operation and the differential of the group operation.

The study of Lie algebras is a central part of the study of Lie groups, as they allow us to understand the local behavior of the group around the identity element. In many cases, the global structure of the group can be understood from its Lie algebra.

Examples of Lie Groups

There are many examples of Lie groups that arise in various areas of mathematics and physics. Some of the most important examples include:

- The general linear group GL(n, R) of invertible n × n matrices with real entries. - The special linear group SL(n, R) of n × n matrices with real entries and determinant 1. - The orthogonal group O(n) of n × n orthogonal matrices. - The special orthogonal group SO(n) of n × n orthogonal matrices with determinant 1. - The unitary group U(n) of n × n unitary matrices. - The special unitary group SU(n) of n × n unitary matrices with determinant 1.

Applications of Lie Groups

Lie groups have wide applications in many areas of mathematics and physics. They are used to study symmetry in physics, differential equations, and geometry. In physics, Lie groups are used to describe the symmetry of physical systems, such as the rotational symmetry of a sphere or the gauge symmetry of a quantum field theory. In mathematics, they are used to study differential equations and geometric structures.