a Lie algebra
/ LEE /
Continuous symmetries -- rotating an object by any angle, translating it by any distance -- form smooth families called Lie groups. But a whole group of rotations is unwieldy. The trick is to zoom in on the tiny motions near 'do nothing': the infinitesimal generators. These generators, with a rule for combining them, form the Lie algebra -- a small, linear object that captures the entire local structure of the symmetry.
A Lie algebra is a vector space equipped with a bracket [A, B] (an antisymmetric product satisfying the Jacobi identity) that measures the failure of two infinitesimal transformations to commute. For a matrix group the bracket is the commutator [A, B] = AB - BA. The algebra is fixed by its structure constants f^c_(ab) in [X_a, X_b] = i f^c_(ab) X_c (physics convention, with an i so the generators are Hermitian). Exponentiating a generator recovers a group element: a finite rotation is exp(-i theta J) built from the angular-momentum generator J. So the algebra is the group linearized at the identity.
This is the language of symmetry in physics. The angular-momentum commutators [J_x, J_y] = i hbar J_z are exactly the Lie algebra su(2); its representation theory dictates the allowed spin and orbital quantum numbers and the whole ladder-operator structure of angular momentum. The same machinery classifies the gauge symmetries of the Standard Model -- SU(3) x SU(2) x U(1) -- and the Lorentz and Poincare symmetries of spacetime; the particles themselves are labelled by representations of these algebras.
Rotations in 3D have three generators J_x, J_y, J_z obeying [J_i, J_j] = i hbar epsilon_(ijk) J_k. Those three brackets ARE the Lie algebra of angular momentum, and from them alone the entire spectrum of allowed spins (0, 1/2, 1, 3/2, ...) is derived.
The angular-momentum commutators are a Lie algebra, and they fix the spins.
The group and its algebra are not the same: distinct groups can share one algebra. SU(2) and SO(3) have the identical Lie algebra yet differ globally, and that global difference is exactly why a spin-1/2 state picks up a minus sign under a 360-degree rotation.