Quaternities, correspondences, and tetrahedron equations (Summa tetralogiae)

TL;DR

This paper introduces a generalization of tetrahedron equations using R-correspondences, explores their relation to Wronskian evolutions, and discusses underlying cohomological structures called quaternities, offering new mathematical insights.

Contribution

It proposes a novel generalization of tetrahedron equations with R-correspondences and links them to Wronskian evolutions and cohomological structures called quaternities.

Findings

Introduction of R-correspondences replacing R-matrices.

Reformulation of equations via Wronskian evolutions.

Discussion of cohomological structures called quaternities.

Abstract

The aim of this note is: (a) to propose a generalization of tetrahedron equations from \cite{S} and of their solutions. Due to appearance of a larger number of parameters the $R$-matrices from \cite{S} will be replaced by "$R$-correspondences". (b) To rephrase these equations in terms of Wronskian evolutions in the spirit of \cite{SV}. (c) To discuss some elementary structures of cohomological flavour lying behind our considerations. We call them "quaternities", or "bibitorsors"; they might be not without an independent interest.