Lie Group Flow on a Torus
Visualizing Lie algebra generator-induced flow around the torus:
group elements wind around & spiral in dual directions.
Lie Groups – The Flow of Continuous Symmetries
What Is a Lie Group?
A Lie Group is a structure where group operations and smooth geometry coexist. Every element flows continuously from the identity — like rotating a shape by ever finer angles.
Lie Algebra & Generators
At the infinitesimal level, the Lie Algebra provides directions of flow. These are the building blocks for transformations across the group via exponentiation.
Why It Matters
From quantum spin to robotic arms, Lie Groups structure all continuous symmetries. The Standard Model is built on SU(3) × SU(2) × U(1) — all Lie Groups at heart.
Live Simulation
Lie Algebra – Bracket, Flow, and Transformation
What Is a Lie Algebra?
A Lie Algebra is the tangent space at the identity of a Lie group. It encodes how infinitesimal elements interact through the Lie Bracket, a bilinear operation capturing directional curl and deformation.
The Lie Bracket [X, Y]
Think of it as the failure of vector fields to commute. The Lie bracket shows:
[X, Y] = XY – YX
It measures curvature of flow — the residual twist after going around a small square.
Jacobi Identity
The Lie bracket obeys:
[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0
This ensures consistent algebraic structure — echoing conservation in physics and structure in geometry.
