Distance-regular graphs and the \(q\)-tetrahedron algebra
The authors define the \(q\)-tetrahedron algebra generated by \(\{x_{ij} \mid i,j\in \mathbb Z_4, j-i=1 \text{ or } j-i=2\}\) and relations such as \(x_{ij}x_{ji}=1\) if \(j-i=2\) and certain Weyl-type and cubic \(q\)-Serre type of relations, as in semisimple Lie algebra theory. They then exhibit four \(\mathbb C\)-algebra homomorphisms from the quantum algebra \(U_q(\widehat{\mathfrak{sl}}_2)\) to the \(q\)-tetrahedron algebra. They then consider \(\Gamma\), a distance-regular graph, e.g., the Platonic solids, and assume that \(\Gamma\) is formally self-dual, i.e., the intersection numbers coincide with the Krein parameters. They impose further restrictions on the intersection numbers of \(\Gamma\). They proceed to define eight matrices indexed by the set of vertices in the graph and with complex coefficients, called \(A, A^*, B, B^*, K, K^*, \Phi, \Psi\). The main result is that there exists a \(q\)-tetrahedron module structure on the standard module \(V\) of the graph \(\Gamma\) where the action is essentially given by (products of) the above eight matrices. Using the homomorphisms, they also provide \(V\) with a \(U_q(\widehat{\mathfrak{sl}}_2)\)-module structure.
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