Noncommutative geometry of angular momentum space U(su(2))

We study the standard angular momentum algebra [xi,xj]=ıλεijkxk as a noncommutative manifold Rλ3. We show that there is a natural 4D differential calculus and obtain its cohomology and Hodge * operator. We solve the spin 0 wave equation and some aspects of the Maxwell or electromagnetic theory including solutions for a uniform electric current density, and we find a natural Dirac operator ∂/. We embed Rλ3 inside a 4D noncommutative space–time which is the limit q→1 of q-Minkowski space and show that Rλ3 has a natural quantum isometry group given by the quantum double C(SU(2))⋊U(su(2)) which is a singular limit of the q-Lorentz group. We view Rλ3 as a collection of all fuzzy spheres taken together. We also analyze the semiclassical limit via minimum uncertainty states |j,θ,φ〉 approximating classical positions in polar coordinates.