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Quaternities, correspondences, and tetrahedron equations (Summa tetralogiae)

Gleb Koshevoy, Vadim Schechtman, Alexander Varchenko

TL;DR

The paper generalizes tetrahedron equations through the introduction of $R$-correspondences, reframing them in terms of $Wr$-based evolutions to reveal underlying quaternities (bibitorsors) with a cohomological flavor. It builds a categorical framework via ${ rak A}(n,2)$ and ${ rak B}(n,2)$ and establishes the equivalence of tetrahedron and sonnet formulations in the $ ext{Corr}$ setting, highlighting how compositions of correspondences encode higher-dimensional integrability. Central to the approach is the Wronskian calculus, which links normalized flag data, transition maps, and evolutions through the theorem $Wr(A_i(a)u)=A_i(a)Wr(u)$, enabling a concrete realization of Sergeev-type identities as Wronskian evolutions. The work provides explicit constructions of $R$-correspondences (and reductions to $R$-matrices in special cases), proves tetrahedron/sonnet equalities in representative scenarios (including Lusztig-type flips), and develops a rich algebraic-geometry toolkit (quaternity calculus, torsor actions) that may illuminate deeper cohomological structures behind integrable systems.

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.

Quaternities, correspondences, and tetrahedron equations (Summa tetralogiae)

TL;DR

The paper generalizes tetrahedron equations through the introduction of -correspondences, reframing them in terms of -based evolutions to reveal underlying quaternities (bibitorsors) with a cohomological flavor. It builds a categorical framework via and and establishes the equivalence of tetrahedron and sonnet formulations in the setting, highlighting how compositions of correspondences encode higher-dimensional integrability. Central to the approach is the Wronskian calculus, which links normalized flag data, transition maps, and evolutions through the theorem , enabling a concrete realization of Sergeev-type identities as Wronskian evolutions. The work provides explicit constructions of -correspondences (and reductions to -matrices in special cases), proves tetrahedron/sonnet equalities in representative scenarios (including Lusztig-type flips), and develops a rich algebraic-geometry toolkit (quaternity calculus, torsor actions) that may illuminate deeper cohomological structures behind integrable systems.

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 -matrices from \cite{S} will be replaced by "-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.
Paper Structure (126 sections, 524 equations)

This paper contains 126 sections, 524 equations.

Table of Contents

  1. Introduction
  2. •
  3. Sonnet equations.
  4. Category of rational correspondences.
  5. Categories ${\cal B}(n,2)$.
  6. •
  7. Evolutions along reduced decompositions.
  8. $c$-upper triangular version.
  9. Wronskian evolutions.
  10. •
  11. •
  12. Acknowledgements.
  13. Berenstein - Zelevinsky transitions
  14. Flags and bases.
  15. Adding up the diagonal.
  16. ...and 111 more sections