Nonassociative Riemannian Geometry by Twisting

TL;DR

This paper extends the cochain twist framework to Riemannian geometry, revealing hidden nonassociativity in quantum spaces, exemplified by twisting the $S^7$ algebra into a nonassociative hyperbolic geometry.

Contribution

It introduces a method to incorporate nonassociativity into Riemannian geometry via twisting, expanding the geometric toolkit for quantum spaces.

Findings

Demonstrates nonassociative structures in twisted quantum geometries

Provides explicit examples including a nonassociative hyperbolic geometry

Extends the twist framework to connections and Riemannian structures

Abstract

Many quantum groups and quantum spaces of interest can be obtained by cochain (but not cocycle) twist from their corresponding classical object. This failure of the cocycle condition implies a hidden nonassociativity in the noncommutative geometry already known to be visible at the level of differential forms. We extend the cochain twist framework to connections and Riemannian structures and provide examples including twist of the $S^7$ coordinate algebra to a nonassociative hyperbolic geometry in the same category as that of the octonions.