N-Cartan calculus
We examine the N-Koszul calculus as defined by the author (Koszul calculus for N-homogeneous algebras, arXiv:1610.01035) in the example of the N-symmetric algebras. The case N = 2 is fundamental in differential geometry since this case corresponds to the Elie Cartan calculus. Therefore, it is natural to conjecture the existence of an N-Cartan calculus on manifolds when N > 2, which would provide a new prototype of noncommutative differential geometry. The goal of this paper is to express in terms of wedge product the defining formulas of the N-Koszul calculus for N-symmetric algebras. We hope that these formulas will be useful for performing an extension to manifolds.