Noncommutative solutions to the local tetrahedron equation

We study the solutions of the local Zamolodchikov tetrahedron equation on noncommutative groups and division rings in the form of correspondences derived from 3 × 3 matrices with free noncommutative variables. The complete set of generators for 4-simplex maps that adhere to the local tetrahedron equation is presented. We study the difference in classification between commutative and noncommutative cases. Additionally, we introduce a procedure for obtaining novel 4-simplex maps associated with known tetrahedron maps. Also, we introduce the “conditional $n$-simplex maps” and study the case of 4-simplex maps via examples. Lastly, several new 4-simplex maps on noncommutative groups are constructed.