Lie Group Simulator — From SU(2) and SO(3) to SO(10)

The Name of a Group = A Declaration of “What Stays Unchanged”

Now viewing: the groupUse the sliders on the right and watch, as numbers, the quantities that do not change under the transformation (invariants).

Same Lie Algebra, Different Group — What Does Not Return After One Full Turn

Now viewing: two groups and the one algebra they shareLeft: the direction of the spin (the world seen by SO(3)). Right: the complex phases of the two spinor components (the world seen by SU(2)). Turn θ up to 360°, then compare the two.

Irreducible Representation = “A Set of States That Cannot Be Split Any Further”

Now viewing: the representationThe raising and lowering operators J_± take you up and down the states like the rungs of a ladder. The coefficient is exactly 0 at the top and bottom of the ladder — and that is what fixes the dimension 2j+1.

Because the Rank Is 2, a State Becomes “a Point on a Plane”

Now viewing: the representation (weights)In SU(3) there are two generators that can be diagonalized simultaneously (I₃ and Y). So each state carries two quantum-number labels, and a multiplet becomes a figure in the plane. Click a point to see where the six raising and lowering operators send it.

Root Systems — Only Four Angles Are Possible Between Simple Roots

Now viewing: the Lie algebra (roots)A root = a weight of the adjoint representation = “the direction in which a raising or lowering operator moves a state.” Because the angle between roots is limited to 90°, 120°, 135°, and 150°, all possible simple Lie algebras are completely determined (Cartan’s classification). Click a root to run its reflection (a Weyl transformation).

16 Particles per Generation — From Scattered Pieces to a Single Irreducible Representation

Now viewing: the representation (one generation = one irreducible representation)Switch with the “How to bundle” buttons, and the same 16 particles regroup. In the Standard Model there are 6 unrelated pieces; in SU(5), 3; in SO(10), just one.