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Cartan structure equations and Levi-Civita connection in noncommutative geometry

Paolo Aschieri

TL;DR

The paper develops a comprehensive braided noncommutative Riemannian geometry for algebras $A$ with a triangular Hopf action, allowing metrics that are not central and extending Cartan calculus to left and right module connections. It proves a general Koszul formula yielding existence and uniqueness of Levi-Civita connections for arbitrary braided-symmetric metrics, and it establishes Cartan structure equations and Bianchi identities in this setting. The framework unifies form-based and vector-field formulations of curvature and torsion, and applies duality to relate connections on modules and their duals. Examples include the noncommutative torus and the tensor square of Sweedler's Hopf algebra, illustrating explicit Levi-Civita connections, Ricci curvature, and Einstein conditions in braided geometries, including Drinfeld-twist deformations and cotriangular algebras.

Abstract

We study the differential and Riemannian geometry of algebras $A$ endowed with an action of a triangular Hopf algebra $H$ and noncommutativity compatible with the associated braiding. The modules of one forms and of braided derivations are modules in a compact closed category of $H$-equivariant $A$-bimodules, whose internal morphisms correspond to tensor fields. Vector fields and forms approaches to curvature and torsion are proven to be equivalent by extending the Cartan calculus to left (right) $A$-module (not necessarily $A$-bimodule) connections. The Cartan structure equations and the Bianchi identities are derived. Existence and uniqueness of the Levi-Civita connection for arbitrary pseudo-Riemannian metrics is proven via a Koszul formula. The general theory includes Drinfeld twists of commutative geometries and also cotriangular Hopf algebras. It is illustrated with the example of the tensor square of Sweedler Hopf algebra which becomes a noncommutative Einstein manifold via a non-central metric.

Cartan structure equations and Levi-Civita connection in noncommutative geometry

TL;DR

The paper develops a comprehensive braided noncommutative Riemannian geometry for algebras with a triangular Hopf action, allowing metrics that are not central and extending Cartan calculus to left and right module connections. It proves a general Koszul formula yielding existence and uniqueness of Levi-Civita connections for arbitrary braided-symmetric metrics, and it establishes Cartan structure equations and Bianchi identities in this setting. The framework unifies form-based and vector-field formulations of curvature and torsion, and applies duality to relate connections on modules and their duals. Examples include the noncommutative torus and the tensor square of Sweedler's Hopf algebra, illustrating explicit Levi-Civita connections, Ricci curvature, and Einstein conditions in braided geometries, including Drinfeld-twist deformations and cotriangular algebras.

Abstract

We study the differential and Riemannian geometry of algebras endowed with an action of a triangular Hopf algebra and noncommutativity compatible with the associated braiding. The modules of one forms and of braided derivations are modules in a compact closed category of -equivariant -bimodules, whose internal morphisms correspond to tensor fields. Vector fields and forms approaches to curvature and torsion are proven to be equivalent by extending the Cartan calculus to left (right) -module (not necessarily -bimodule) connections. The Cartan structure equations and the Bianchi identities are derived. Existence and uniqueness of the Levi-Civita connection for arbitrary pseudo-Riemannian metrics is proven via a Koszul formula. The general theory includes Drinfeld twists of commutative geometries and also cotriangular Hopf algebras. It is illustrated with the example of the tensor square of Sweedler Hopf algebra which becomes a noncommutative Einstein manifold via a non-central metric.

Paper Structure

This paper contains 31 sections, 31 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Hopf algebras, braidings and representations
  3. Closed monoidal categories
  4. Modules over a Hopf Algebra
  5. Modules over a triangular Hopf Algebra
  6. Algebras and bimodules
  7. Graded algebras and bimodules
  8. Finitely generated projective modules and their duals
  9. Compact closed categories
  10. Examples
  11. Differential and Cartan Calculus
  12. Braided derivations and differential calculus
  13. Cartan Calculus
  14. Examples
  15. Right connections, left connections, curvature and torsion
  16. ...and 16 more sections

Key Result

Lemma 2.1

( Dual Basis Lemma). Let $A$ be an algebra. A left $A$-module $\Gamma$ is finitely generated and projective if and only if there exists a family of elements $\lbrace s_i\in \Gamma:i=1,\dots,n\rbrace$ and left $A$-linear maps $\lbrace {}^{*\!}s^i \in{}^*\Gamma :={_A}\mathrm{Hom}(\Gamma,A):i=1,\dots,n A right $A$-module $\Sigma$ is finitely generated and projective if and only if there exists a fami

Theorems & Definitions (76)

  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8
  • proof
  • ...and 66 more