From rotation matrices to the classification of simple Lie algebras through 42 interactive demonstrations. Fly through SO(3), twist quaternion belts, build Dynkin diagrams, and see how continuous symmetry underpins all of modern physics.
See also: Group Theory for the abstract group axioms that Lie groups satisfy, Differential Geometry for the manifold structure that makes them smooth, and Representation Theory for the linear representations central to physics applications.
Groups that are also smooth manifolds — where algebra meets geometry
GL(n), SL(n), O(n), SO(n), U(n), SU(n) as geometric objects
exp: Lie algebra → Lie group — tangent vectors generate group elements
Axis-angle, Euler angles, gimbal lock, and SLERP interpolation
Double cover of SO(3), the belt trick, and quaternion rotations
Tangent space at identity, the Lie bracket [X,Y], and structure constants
Ad and ad — the group acts on its own algebra
Classification of simple Lie algebras through crystallographic patterns
Gauge symmetries, Noether's theorem, and the Standard Model
Orbits, stabilizers, and homogeneous spaces