03/12/2025
In mathematics, ubiquitous objects called groups display nearly magical powers. Though they’re defined by just a few rules, groups help illuminate an astonishing range of mysteries. They can tell you which polynomial equations are solvable, for instance, or how atoms are arranged in a crystal.
And yet, among all the different kinds of groups, one type stands out. Identified in the early 1870s, Lie groups are crucial to some of the most fundamental theories in physics, and they’ve made lasting contributions to number theory and chemistry. The key to their success is the way they blend group theory, geometry and linear algebra.
What makes a Lie group special is that it can be visualized as a smooth, continuous shape called a manifold. Lie groups might look like a circle, or the surface of a doughnut, or a high-dimensional sphere. Some look even stranger: The group of all rotations of a ball in space, known to mathematicians as SO(3), is a six-dimensional tangle of spheres and circles.
The manifold nature of Lie groups has been an enormous boon to mathematicians and physicists.
🌐 Keep reading: https://www.quantamagazine.org/what-are-lie-groups-20251203/
