Noncommutative solutions to the local tetrahedron equation

Abstract

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.

Keywords

• n-simplex maps
• Tetrahedron maps
• Local Zamolodchikov tetrahedron equation
• Yang–Baxter maps
• Division rings
• Conditional n-simplex maps