Pentagon Relations in Direct Sums and Grassmann Algebras

TL;DR

This paper constructs extensive families of orthogonal operators satisfying pentagon relations in direct sums of vector spaces, leading to new pentagon relations in Grassmann algebras and generalizing exotic Reidemeister torsions.

Contribution

It introduces a novel construction of orthogonal operators obeying pentagon relations in higher-dimensional vector spaces and Grassmann algebras, expanding the mathematical framework of Reidemeister torsions.

Findings

Established pentagon relations in Grassmann algebras.

Generalized exotic Reidemeister torsions.

Developed families of orthogonal operators satisfying pentagon relations.

Abstract

We construct vast families of orthogonal operators obeying pentagon relation in a direct sum of three n-dimensional vector spaces. As a consequence, we obtain pentagon relations in Grassmann algebras, making a far reaching generalization of exotic Reidemeister torsions.