use the irreducible action of the structure group $G$ on $\C^k$ to define the function $\Phi_{ij}(\ell)\in C(G)$ by $$\label{Def-QuConn}
\Phi_{ij}(\ell)(g):=(e_i,ge_j),\quad g\in G,$$ where $\{ e_i\mid
i =1,\ldots,k\}$ is an orthonormal basis of $\C^k$. Then the matrix components of the quantum connection are taken to be the operators $T_{\Phi_{ij}(\ell)}$, which we will see transform correctly w.r.t. gauge transformations. As the $\Phi_{ij}(\ell)$ are matrix elements of elements of $G$, there are obvious relations between them which reflect the structure of $G$. The C\*-algebra generated by the operators $\{ T_{\Phi_{ij}(\ell)}\,\mid\, i,\,j=1,\ldots,k\}$ is $T\s C(G).$.

To define gauge momentum operators, we first assign to each link an element of $\mathfrak{g}$, i.e. we choose a map $\Psi:\Lambda^1\to \mathfrak{g}.$ Given such a $\Psi$, take for the associated quantum gauge momentum at $\ell$ the operator $P\s\Psi(\ell).:C^\infty(G)\to C^\infty(G)$. The generalized canonical commutation relations are $$\label{genCCR}
\big[P\s\Psi(\ell).,\,T\s\Phi_{ij}(\ell).\big]=i\sum_mT\s\Psi(\ell)_{im}\Phi_{mj}(\ell).\quad\hbox{on}\quad C^\infty(G),$$ where $\Psi(\ell)_{im}:=(e_i,\Psi(\ell)e_m).$

To obtain the G–electrical fields at $\ell$, choose a basis ${\{Y_r\mid r=1,\ldots,{\rm dim}( \mathfrak{g})\}\subset \mathfrak{g}}$, then substitute for $\Psi$ the constant map $\Psi(\ell)=Y_r$ and set $E_r(\ell):=P_{Y_r}$. In the case that $G=SU(3)$, these are the colour electrical fields, and one takes the basis $\{Y_r\}$ to be the traceless selfadjoint Gell–Mann matrices satisfying ${\rm Tr}(Y_rY_s)=\delta_{rs}$. We then define $$E_{ij}(\ell):=\sum_r(Y_r)_{ij}E_r(\ell)=\sum_r(Y_r)_{ij}P\s{Y_r}.$$ and for 