An Elementary Introduction to the Hopf Fibration

TL;DR

This paper provides an accessible introduction to the Hopf fibration, explaining its mathematical properties and physical applications using quaternion algebra, aimed at undergraduate students.

Contribution

It offers a clear, quaternion-based explanation of the Hopf fibration's algebraic and geometric aspects, connecting it to rotations in 3-space for educational purposes.

Findings

Uses quaternion algebra to illustrate Hopf fibration properties

Connects Hopf fibration with physical applications like rotations and quantum information

Provides an accessible introduction suitable for undergraduates

Abstract

The Hopf fibration is an important object in mathematics and physics. A landmark discovery in topology and a fundamental object in the theory of Lie groups, the Hopf fibration has a wide variety of physical applications including magnetic monopoles, rigid body mechanics, and quantum information theory. This expository article presents an introduction to the Hopf fibration that is accessible to undergraduate students. We use the algebra of quaternions to illustrate algebraic and geometric properties of the Hopf fibration and the connection with rotations of 3-space that is the basis for natural applications to physics and engineering.