# Braided Cartan Calculi and Submanifold Algebras

## TL;DR

This paper develops a noncommutative Cartan calculus within braided commutative algebras, extending classical geometric tools to noncommutative settings and exploring their deformation and substructure properties.

## Contribution

It introduces a braided Cartan calculus applicable to all braided commutative algebras, including new notions of covariant derivatives and metrics, and demonstrates their behavior under twist deformations.

## Key findings

- Constructed a braided Lie derivative, insertion, and de Rham differential.
- Proved existence and uniqueness of an equivariant Levi-Civita covariant derivative.
- Showed that braided Cartan calculus projects consistently to 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.