Foundations of Noncommutative Carrollian Geometry via Lie-Rinehart Pairs

TL;DR

This paper develops a framework for noncommutative Carrollian geometry using Lie-Rinehart pairs, extending classical Carrollian structures to almost commutative algebras, and provides explicit examples on quantum planes and noncommutative tori.

Contribution

It introduces a noncommutative version of Carrollian geometry via $ ho$-Lie-Rinehart pairs, bridging classical and noncommutative geometric structures.

Findings

Established Carrollian structures on quantum plane and noncommutative torus.

Extended classical Carrollian geometry concepts to $ ho$-commutative algebras.

Provided explicit examples demonstrating noncommutative Carrollian geometry.

Abstract

Carrollian manifolds offer an intrinsic geometric framework for the physics in the ultra-relativistic limit. The recently introduced Carrollian Lie algebroids are generalised to the setting of $\rho$-commutative geometry, (also known as almost commutative geometry), where the underlying algebras commute up to a numerical factor. Via $\rho$-Lie-Rinehart pairs, it is shown that the foundational tenets of Carrollian geometry have analogous statements in the almost commutative world. We explicitly build two toy examples: we equip the extended quantum plane and the noncommutative $2$-torus with Carrollian structures. This opens up the rigorous study of noncommutative Carrollian geometry via almost commutative geometry.