Braided Cartan Calculi and Submanifold Algebras

Abstract

We construct a noncommutative Cartan calculus on any braided commutative algebra and study its applications in noncommutative geometry. The braided Lie derivative, insertion and de Rham differential are introduced and related via graded braided commutators, also incorporating the braided Schouten-Nijenhuis bracket. The resulting braided Cartan calculus generalizes the Cartan calculus on smooth manifolds and the twisted Cartan calculus. While it is a necessity of derivation based Cartan calculi on noncommutative algebras to employ central bimodules our approach allows to consider bimodules over the full underlying algebra. Furthermore, equivariant covariant derivatives and metrics on braided commutative algebras are discussed. In particular, we prove the existence and uniqueness of an equivariant Levi-Civita covariant derivative for any fixed non-degenerate equivariant metric. Operating in a symmetric braided monoidal category we argue that Drinfel'd twist deformation corresponds to gauge equivalences of braided Cartan calculi. The notions of equivariant covariant derivative and metric are compatible with the Drinfel'd functor as well. Moreover, we project braided Cartan calculi to submanifold algebras and prove that this process commutes with twist deformation.

26 pages

References in corpus (23)

• Noncommutative Geometry and Gravity
• Connections on central bimodules
• Twisting all the way: from Classical Mechanics to Quantum Fields
• Noncommutative Gravity Solutions
• Noncommutative differential calculus, homotopy BV algebras and formality conjectures
• Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms
• Connections on Lie algebroids and on derivation-based noncommutative geometry
• Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature
• Submanifolds and Quotient Manifolds in Noncommutative Geometry
• On second quantization on noncommutative spaces with twisted symmetries
• Remarks on simple interpolation between Jordanian twists
• Drinfel'd Twist and q-Deforming Maps for Lie Group Covariant Heisenberg Algebras
• Cartan Calculus: Differential Geometry for Quantum Groups
• Fuzzy circle and new fuzzy sphere through confining potentials and energy cutoffs
• On Noncommutative Levi-Civita Connections
• Twisted submanifolds of R^n
• Obstructions for Twist Star Products
• Noncommutative calculus and operads
• Twist star products and Morita equivalence
• A Universal Construction of Universal Deformation Formulas, Drinfel'd Twists and their Positivity
• On the Notion of Noncommutative Submanifold
• From Quantum Planes to Quantum Groups and back; Cartan Calculus
• Braided Commutative Geometry and Drinfel'd Twist Deformations

Cited by in corpus (12)

• Gauge theories on quantum spaces
• Braided $L_{\infty }$-Algebras, Braided Field Theory and Noncommutative Gravity
• Noncommutative gauge and gravity theories and geometric Seiberg-Witten map
• Cartan structure equations and Levi-Civita connection in noncommutative geometry
• Twisted submanifolds of R^n
• Covariant connections on bicovariant differential calculus
• On the Notion of Noncommutative Submanifold
• Twisted Quadrics and Algebraic Submanifolds in ${\mathbb{R}}^n$
• Braided Commutative Geometry and Drinfel'd Twist Deformations
• Gauge theory models on $\kappa$-Minkowski space: Results and prospects
• Twisted geometry for submanifolds of ${\mathbb{R}}^n$
• Quantum Calculi: differential forms and vector fields in noncommutative geometry