  We consider the phase-space of Yang-Mills on a cylindrical space-time ($S^1
\times {\bf R}$) 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 to
give 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.
