On the solutions of the local Zamolodchikov tetrahedron equation

We study the solutions of the local Zamolodhcikov tetrahedron equation in the form of correspondences derived by 3× 3 matrices. We present all the associated generators of 4-simplex maps satisfying the local tetrahedron equation. Moreover, we demonstrate that, from some of our solutions, we can recover the 4-simplex extensions of Kashaev--Korepanov--Sergeev and Hirota type tetrahedron maps. Finally, we construct several novel 4-simplex maps.