STARLIGHT
THEORY

Smooth Manifolds #8

Introduction to Lie Groups

Lie Groups: Where Algebra Meets Geometry

Lie groups represent one of the most beautiful syntheses in mathematics, combining the algebraic structure of groups with the geometric structure of manifolds. They are fundamental in understanding symmetry in mathematics and physics.

What is a Lie Group?

A Lie group is a smooth manifold that is also a group, where the group operations (multiplication and inversion) are smooth maps. This simple definition leads to a rich theory with deep applications.

Key Topics:

  1. Basic definitions and examples
  2. The exponential map
  3. Lie algebras
  4. Matrix Lie groups
  5. Applications to physics and geometry

Understanding Lie groups is essential for:

  • Quantum mechanics
  • Particle physics
  • Robotics
  • Computer vision
  • Differential equations

Join us as we explore these fascinating mathematical objects!