or $SU(2)\times SU(2)$ in 4d, the respective covering groups of the Lorentz group and the 4D rotation group $SO(4)$. The variables are $SU(2)$ group elements $U_e$ associated to edges $e$ in the bulk of $\triangle$\* or $U_l$ associated to links $l$ on the boundary, and algebra elements $L_f=L^i_fJ_i$ associated to faces $f$ of $\triangle$\* or $L_l=L^i_l J_i$ associated to links on the boundary. $J_i$ are the $SU(2)$ generators, $J_i=\frac{i}{2}\sigma_{i}$, where $\sigma_{i}$ are the usual Pauli matrices. The quantization involves operators $U_l$ and $L_l$ on the boundary realizing a commutation relation on a Hilbert space, $$\left[U_{l},L_{l'}^{i}\right]=i\beta\delta_{ll'}U_{l}J^{i}\label{eq:commutationUL}$$ and $$\left[L_{l}^{i},L_{l'}^{j}\right]=i\beta\delta_{ll'}\varepsilon_{k}^{ij}L_{l}^{k},\label{eq:commutationsu2algebra}$$ where $\beta$ is a constant related to the gravitational constant, $\delta_{ll'}$ is the Kronecker-delta and $\varepsilon_{k}^{ij}$ is the totally antisymmetric Levi-Civita symbol. States are wavefunctions $\Psi(U_{l})$ or $\Psi(L_{l})$ of $L$ group/algebra elements on $L$ links of the boundary graph modulo the $SU(2)$ gauge symmetry implemented on the nodes. The Hilbert space is the space of square integrable functions of these coordinates: $$\mathcal{H}_{\Gamma}=L_{2}\left[SU(2)^{L}/SU(2)^{N}\right]_{\Gamma}\label{eq:hilbertspaceLQG}$$ where $\Gamma$ is the boundary graph. The gauge invariant states must satisfy $\Psi(U_{l})=\Psi\left(\lambda_{s_{l}}U_{l}\lambda_{t_{l}}^{-1}\right)$ for every node $n$ of the boundary graph. One major realization of loop quantum gravity is that the kinematics are the same in 3D and 4D $-$ the 