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Foundations of Noncommutative Carrollian Geometry via Lie-Rinehart Pairs

Andrew James Bruce

TL;DR

This work addresses the problem of formulating Carrollian geometry in noncommutative settings by developing the theory on almost commutative algebras using ρ-Lie-Rinehart pairs as algebraic analogues of Lie algebroids. The authors define Carrollian ρ-Lie-Rinehart pairs with a degenerate metric whose kernel is a free cyclic submodule, and they introduce Carroll distributions as the image of the anchor, proving their involutivity and foliated structure. They construct metric-compatible ρ-connections, derive curvature and torsion as ρ-tensors, and provide a Koszul-type formula in the non-degenerate case, establishing a path to Levi-Civita-type results in this framework. Two explicit toy models—the extended quantum plane and the noncommutative 2-torus—demonstrate Carrollian structures and compatible connections, offering a rigorous platform for noncommutative Carrollian physics and potential links to flat-space holography and condensed-matter applications.

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.

Foundations of Noncommutative Carrollian Geometry via Lie-Rinehart Pairs

TL;DR

This work addresses the problem of formulating Carrollian geometry in noncommutative settings by developing the theory on almost commutative algebras using ρ-Lie-Rinehart pairs as algebraic analogues of Lie algebroids. The authors define Carrollian ρ-Lie-Rinehart pairs with a degenerate metric whose kernel is a free cyclic submodule, and they introduce Carroll distributions as the image of the anchor, proving their involutivity and foliated structure. They construct metric-compatible ρ-connections, derive curvature and torsion as ρ-tensors, and provide a Koszul-type formula in the non-degenerate case, establishing a path to Levi-Civita-type results in this framework. Two explicit toy models—the extended quantum plane and the noncommutative 2-torus—demonstrate Carrollian structures and compatible connections, offering a rigorous platform for noncommutative Carrollian physics and potential links to flat-space holography and condensed-matter applications.

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 -commutative geometry, (also known as almost commutative geometry), where the underlying algebras commute up to a numerical factor. Via -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 -torus with Carrollian structures. This opens up the rigorous study of noncommutative Carrollian geometry via almost commutative geometry.
Paper Structure (9 sections, 8 theorems, 69 equations)

This paper contains 9 sections, 8 theorems, 69 equations.

Table of Contents

  1. Introduction and Background
  2. Introduction
  3. Almost Commutative Algebras and Related Notions
  4. rho-Lie-Rinehart Pairs and Carrollian Structures
  5. rho-Lie-Rinehart Pairs
  6. Connections on rho-Lie-Rinehart Pairs
  7. Carrollian Structures on rho-Lie-Rinehart Pairs
  8. Toy Examples
  9. Concluding Remarks

Key Result

Proposition 2.10

Let $(\mathcal{A}, \mathfrak{g})$ be a $\rho$-Lie-Rinehart pair with anchor $\mathsf{a} : \mathfrak{g} \rightarrow \rho\mathop{\mathrm{Der}}\nolimits(\mathcal{A})$. Then $(\mathcal{A},\mathfrak{g}_{\mathcal{A}})$ is a $\rho$-Lie-Rinehart pair whose anchor is the zero map.

Theorems & Definitions (54)

  • Definition 1.1
  • Example 1.2
  • Example 1.3
  • Definition 1.4
  • Remark 1.5
  • Example 1.6
  • Example 1.7
  • Definition 1.8
  • Example 1.9: Vector fields on supermanifolds
  • Definition 2.1
  • ...and 44 more