We consider the Yang--Mills phase space on a cylindrical spacetime () and the associated algebra of gauge-invariant functions, the T-variables. We solve the Mandelstam identities both classically and quantum mechanically by considering the T-variables as functions of the eigenvalues of the holonomy and their associated momenta. It is shown that there are two inequivalent representations of the quantum T-algebra. Then we compare this reduced phase-space approach to Dirac quantization and find it gives essentially equivalent results. We proceed to define a loop representation in each of these two cases. One of these loop representations (for N=2) is more or less equivalent to the usual loop representation.