Representation of Gaussian Isotropic Spin Random Fields

We develop a technique for the construction of random fields on algebraic structures. We deal with two general situations: random fields on homogeneous spaces of a compact group and in the spin-line bundles of the 2-sphere. In particular, every spin Gaussian isotropic field can be obtained with this construction. Our construction extends P. L\'evy's original idea for the spherical Brownian Motion.