N-Cartan calculus

TL;DR

This paper explores the extension of N-Koszul calculus for N-symmetric algebras, proposing a conjecture that it generalizes Cartan calculus to noncommutative manifolds for N>2, potentially advancing noncommutative geometry.

Contribution

It introduces the N-Cartan calculus for N-symmetric algebras and conjectures its extension to manifolds, generalizing classical Cartan calculus.

Findings

N=2 case recovers classical Cartan calculus

Proposes conjecture for N>2 extending to noncommutative manifolds

Lays groundwork for new noncommutative differential geometry

Abstract

We examine the N-Koszul calculus for the N-symmetric algebras. The case N=2 corresponds to the Elie Cartan calculus. We conjecture that, as in the case N=2, the N-Cartan calculus extends to manifolds when N>2, which would provide a new type of noncommutative differential geometry.