Noncommutative Carrollian Geometry Foundations Extend to Almost Commutative Frameworks Via Lie-Rinehart Pairs

Carroll manifolds provide a natural geometric language for describing physics in extreme relativistic scenarios, and researchers are now extending these ideas into the realm of noncommutative geometry. Andrew James Bruce, from the Institute of Mathematics, Polish Academy of Sciences, and colleagues demonstrate how Carrollian geometry translates to this more abstract mathematical setting, where standard algebraic rules no longer strictly apply. The team establishes a connection between Carrollian geometry and Lie-Rinehart pairs, revealing analogous principles within noncommutative frameworks. By constructing Carrollian structures on examples such as the extended plane and a noncommutative torus, they lay the foundations for a rigorous mathematical investigation of noncommutative Carrollian geometry, potentially offering new insights into the behaviour of physical systems at the most fundamental levels.

Carrollian Geometry via ρ-Lie-Rinehart Pairs

This work presents a comprehensive exploration of Carrollian geometry, extending its principles to the realm of almost commutative geometry using ρ-Lie-Rinehart pairs. Scientists demonstrate that core tenets of Carrollian geometry, traditionally defined for smooth spaces, hold analogous validity within the more general framework of almost commutative algebras. This generalization utilizes ρ-Lie-Rinehart pairs, which provide a powerful tool for studying symmetries and geometric structures in this noncommutative setting. The research establishes a foundation for rigorously investigating Carrollian geometry beyond classical descriptions, opening new avenues for exploring spacetime structures at a fundamental level.

The core idea involves extending concepts of Carrollian geometry, a non-Lorentzian spacetime where light cones collapse to a point, to the realm of almost commutative geometry. ρ-Lie-Rinehart pairs act as the fundamental building blocks, enabling the definition of geometric structures in this generalized setting. Almost commutative geometry, developed by Alain Connes, provides tools to define differential geometry and topology on spaces that are not necessarily traditional manifolds. This work positions Carrollian spacetimes as a limit of Lorentzian spacetimes, similar to how Galilean spacetimes relate to Newtonian spacetimes, highlighting its relevance to fundamental physics.

Noncommutative Carrollian Geometry and ρ-Lie-Rinehart Pairs

This work establishes a rigorous framework for noncommutative Carrollian geometry, building upon the foundations of almost commutative algebras and Lie-Rinehart pairs. Scientists demonstrate that tenets of Carrollian geometry, traditionally defined for smooth manifolds, have analogous statements within the more general framework of almost commutative algebras. This generalization utilizes ρ-Lie-Rinehart pairs, which provide a powerful tool for studying symmetries and geometric structures in this noncommutative setting. The research introduces a refined version of Lie-Rinehart pairs, termed ρ-Lie-Rinehart pairs, where graded structures, Lie brackets, and anchor maps are all compatible, allowing for the definition of connections and curvature tensors in this noncommutative setting.

Scientists demonstrate that connections on these ρ-Lie-Rinehart pairs generalize core elements of Lie algebroid connections, providing a crucial link between classical and noncommutative geometry. They define Carrollian ρ-Lie-Rinehart pairs as graded almost commutative generalizations of Carrollian manifolds, establishing their core properties and laying the groundwork for studying spacetime at the quantum level. To illustrate this framework, the team equipped Manin’s extended quantum plane and a noncommutative 2-torus with chosen Carrollian structures and compatible connections.

Noncommutative Carrollian Geometry Established and Validated

This work establishes a framework for extending Carrollian geometry, which describes the ultra-relativistic limit of physics, into the realm of noncommutative geometry. Scientists successfully generalised Carrollian Lie algebroids, geometric structures central to this ultra-relativistic framework, to the setting of almost commutative geometry, where algebraic elements do not strictly commute. This extension was achieved through the introduction of specific pairings, demonstrating analogous principles hold in this more general, noncommutative world. The research establishes a foundation for rigorously investigating Carrollian geometry beyond classical descriptions, opening new avenues for exploring spacetime structures at a fundamental level.

The team validated this approach by constructing Carrollian structures on two distinct examples, the extended plane and the noncommutative torus, providing concrete instances of this extended geometry. This development opens avenues for rigorous investigation of noncommutative Carrollian geometry using the tools of almost commutative geometry, potentially offering new insights into the nature of spacetime at extreme scales. Future work may focus on applying these techniques to more complex examples and exploring potential physical interpretations of the resulting geometric structures.