"Nonconstant cohomology" of Hietarinta's two-color solutions to four-simplex equation

TL;DR

This paper introduces nonconstant cohomologies for solutions of the set-theoretical four-simplex equation, leading to new nonconstant quantum solutions and tetrahedron equation solutions with specific properties.

Contribution

It defines nonconstant cohomologies for FSE solutions and demonstrates their existence for Hietarinta's solutions, enabling the construction of novel quantum and tetrahedron equation solutions.

Findings

Large spaces of nonconstant cohomologies exist for Hietarinta's solutions

New quantum solutions with non-negative matrix elements are constructed

Solutions are not reducible to permutations, even with cocycle multipliers

Abstract

"Nonconstant cohomologies" are introduced for solutions of set-theoretical four-simplex equation (FSE). While usual cohomologies lead to solutions of constant quantum FSE, our "nonconstant cohomologies" lead to solutions of nonconstant quantum FSE. Computer calculations are presented showing that large spaces of such cohomologies exist for all Hietarinta's two-color linear solutions to set-theoretical FSE. After taking a partial trace of the corresponding quantum operators, combined with one additional trick, this leads to solutions of tetrahedron equation, including those with non-negative matrix elements, and not reducible to a permutation, even with cocycle multipliers.