Representations of the $SU(N)$ $T$-algebra and the loop representation in $1+1$-dimensions

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.