Cohomologies of $n$-simplex relations

TL;DR

This paper develops a cohomology theory for set-theoretic n-simplex relations, extending known algebraic structures like quandle and Yang-Baxter cohomologies, with a focus on the tetrahedron case and explicit solutions.

Contribution

It introduces a new cohomology framework for n-simplex relations, generalizing permutation solutions to quantum n-simplex equations and providing explicit examples.

Findings

Constructed a cohomology theory for set-theoretic n-simplex relations.

Generalized permutation-type solutions to quantum n-simplex equations.

Presented explicit solutions involving nontrivial cocycles.

Abstract

A theory of (co)homologies related to set-theoretic $n$-simplex relations is constructed in analogy with the known quandle and Yang--Baxter (co)homologies, with emphasis made on the tetrahedron case. In particular, this permits us to generalize Hietarinta's idea of "permutation-type" solutions to the quantum (or "tensor") $n$-simplex equations. Explicit examples of solutions to the tetrahedron equation involving nontrivial cocycles are presented.