4-tensor product of the $H_{j_{qi}}$ above, implemented by $A_v$. The extension of the topological data from $SU(2)^5_q$ to $SU(N)^r_q$ is presented in the next section.

Fermionic Cycles and Gauge Symmetry Unification {#sec:unification}
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The natural symmetry for the quantum tetrahedron, as discussed in Section \[qt\], is the spatial rotations $SO(3)$ in 3D, implemented by the covering group $SU(2)$, and hence its fifth-root-of-unity deformation. But the generalization of $SU(2)=A_{1}$ in the Weyl-Cartan or Chevalley basis to the higher dimensional simple Lie algebras, eq. (\[eq:SU3algebrarelation\]), indicates that we can incorporate the so called internal symmetry or charge space in the same way. The simplest way is to consider the symmetry $SU(2)\times G$, with $G$ one of the higher dimensional simple Lie algebras [@Bianchi:2010bn]. Because of the importance of $A_{1}$ the natural first candidate is $G=A_{2}=SU(3)$. Following this, we will also define an extension of the topological data from $SU(2)^5_q$ to $SU(N)^r_q$. To get onto multi-dimensional Fibonacci-like anyonic representations and fulfill the quantizing postulate eq. (\[eq:commutationJplusminos-1-1\]), we will further restrict $r$ to be a multiple of $5$ larger than $N$. Examples useful for gauge unification are given by: $SU(3)^5_q$, $SU(4)^5_q$, $SU(5)^{10}_q$, $SU(8)^{10}_q$ and $SU(9)^{10}_q$. This allows us to address the higher dimensional $A_{4}$, 