Foundations of Noncommutative Carrollian Geometry via Lie-Rinehart Pairs

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 $ρ$-commutative geometry, (also known as almost commutative geometry), where the underlying algebras commute up to a numerical factor. Via $ρ$-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.

💡 Research Summary

The paper develops a systematic framework for Carrollian geometry in the setting of almost‑commutative (ρ‑commutative) algebras, thereby providing a bridge between ultra‑relativistic physics and non‑commutative geometry. After motivating the need for non‑commutative structures in quantum gravity, the authors recall the definition of a G‑graded algebra equipped with a bicharacter ρ:G×G→K× that controls the graded commutation rule xy = ρ(|x|,|y|) yx. They introduce ρ‑derivations, which satisfy a twisted Leibniz rule, and show that the space ρDer(A) of such derivations forms a ρ‑Lie algebra and a left A‑module.

With these tools, the central algebraic object is defined: a ρ‑Lie‑Rinehart pair (A, g). Here A is a ρ‑commutative algebra, g is a ρ‑Lie algebra that is a left G‑graded A‑module, and there exists an anchor map a:g→ρDer(A) satisfying two compatibility conditions: (2.1a) a(