Koszul calculus for N-homogeneous algebras

Abstract

We extend Koszul calculus defined on quadratic algebras by Berger et al. (2018) [9] to N-homogeneous algebras for any N≥2, quadratic algebras corresponding to N=2. We emphasize that N-homogeneous algebras are considered in full generality, with no Koszulity assumption. Koszul cup and cap products are introduced and are reduced to usual cup and cap products if N=2, but if N>2, they are defined by very specific expressions. These specific expressions are compatible with the Koszul differentials and provide associative products on classes. There is no associativity in general on chains–cochains, suggesting that Koszul cochains should constitute an A∞-algebra, acting as an A∞-bimodule on Koszul chains.