these we obtain from $(\ref{genCCR})$ the commutation formulae in [@KR; @KR1] for the colour electrical field. Of particular importance for the dynamics, is the operator $E_{ij}(\ell) E_{ji}(\ell)$ (summation convention). We have $$E_{ij}(\ell) E_{ji}(\ell)=\sum_{r,s}(Y_r)_{ij}P\s{Y_r}.(Y_s)_{ji}P\s{Y_s}.
=\sum_{r,s}(Y_rY_s)_{ii}P\s{Y_r}.P\s{Y_s}.=n\sum_rP\s{Y_r}.^2$$ i.e. it is the Laplacian for the left regular representation $U:G\to\cl U.(L^2(G))$ which therefore commutes with all $U_g$. Below in Equation (\[PTfell\]) we will see that a gauge transform just transforms the Laplacian to one w.r.t. a transformed basis of $\mathfrak{g}$, which leaves the Laplacian invariant.

The full collection of operators which comprises the set of dynamical variables of the model is as follows. The representation Hilbert space is $$\cl H.=\cl H._F\otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}L^2(G)\qquad\hbox{ where}\qquad
{\pi_F:{\mathfrak F}_{\Lambda}}\to\cl B.(\cl H._F)$$ is any irreducible representation of ${\mathfrak F}_{\Lambda}$. As $\Lambda^1$ is finite, $\cl H.$ is well–defined. Then $\pi_F\otimes\un:{\mathfrak F}_{\Lambda}\to\cl B.(\cl H.)$ will be the action of ${\mathfrak F}_{\Lambda}$ on $\cl H.$. The quantum connection is given by the set of operators $$\{{\widehat}{T}_{\Phi_{ij}(\ell)}^{(\ell)}\,\mid\,\ell\in \Lambda^1,\;i,j=1,\ldots, k\}\quad
 \hbox{where}\quad {\widehat}{T}_f^{(\ell)}:=\un\otimes\big(\un\otimes\cdots\otimes\un\otimes T^{(\ell)}_f\otimes\un\otimes\cdots\otimes\un\big)$$ and $T^{(\ell)}_f$ is the multiplication operator on the $\ell^{\rm th}$ factor, hence ${\widehat}{T}_{\Phi_{ij}(\ell)}^{(\ell)}$ acts as the identity on all the other factors of $\cl H.$. Likewise, for the gauge momenta we take $${\widehat}{P}^{(\ell)}_{\Psi(\ell)}:=\un\otimes\big(\un\otimes\cdots\otimes\un\otimes P^{(\ell)}_{\Psi(\ell)}\otimes\un\otimes\cdots\otimes\un\big),\quad
\ell\in \Lambda^1$$ where $P^{(\ell)}_X$ is the $P_X$ operator on 