of the underlying scale $l_{p}$. This representation theory is as well understood as that of $SU(2)$. There are 3 independent Casimir invariants for each face $C_{aq}=L_{a}^{+}L_{a}^{-}+\left[h_{a}\right]_{q}\left[h_{a}+1\right]_{q}$, $X_{a}=(L_{a}^{+})^{5}$ and $Y_{a}=(L_{a}^{-})^{5}$. When $X_{a}$ and $Y_{a}$ have zero eigenvalues for all vectors in the linear vector space carrying the irreducible representation, the representations have classical analogs, and can be constructed by specializing from the general values of $q$ to the specific root of unity. In this case the spin quantum numbers that label the representations of $SU(2)_{q}$ are constrained to $j_{a}\leq\frac{3}{2}$, and are related to the Casimir $C_{aq}$ of eigenvalues $[j_{a}]_{q}[(j_{a}+1)]_{q}$. In this way the area spectrum interpretation $$A_{a}=\sqrt{[j_{a}]_{q}[(j_{a}+1)]_{q}}\label{eq:quantumarea}$$ is not immediate and a scale should emerge in an appropriate limit.

The vertex amplitude is naturally given by the specialization of eq. (\[eq:transitionamplitudedelta1\]) to $SU(2)^5_q$ symmetry defined by the Hilbert space of the tensor product of 4 representations at the vertex $H_{j_{q1}}\times H_{j_{q2}}\times H_{j_{q3}}\times H_{j_{q4}}$, and it is a function of the representations on the 6 boundary faces defined by this vertex $$W_{\triangle_{1}}^{5}(j_{lv})=(-1)^{j_{v}}\left\{ \begin{array}{ccc}
j_{1} & j_{2} & j_{3}\\
j_{4} & j_{5} & j_{6}
\end{array}\right\} _{5},\label{eq:transitionamplitudedelta1-5}$$ with $j_{lv}=j_{1},...,j_{6}$ and the index 5 representing the $SU(2)^5_q$ deformation. The q-deformed 6-j symbols, $\left\{ 6j\right\} _{5}$, 