Lie groups

Lie groups are mathematical structures that combine the properties of continuous shapes (like circles or spheres) with algebraic operations (like multiplication or addition). They describe smooth, symmetrical ways objects can be transformed without tearing or gluing. For example, rotations of a circle form a Lie group because you can smoothly rotate the circle by any angle, and these rotations can be combined or reversed. Lie groups are fundamental in understanding symmetry in physics, geometry, and many areas of science, providing a framework for analyzing continuous transformations that preserve structure.