Foundations of noncommutative Carrollian geometry via Lie–Rinehart pairs

Source abstract

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 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.