Koszul calculus for N-homogeneous algebras

Roland Berger (1)

1 AGL - Algèbre, géométrie, logique

We extend Koszul calculus defined on quadratic algebras by Berger, Lambre, Solotar (Koszul calculus, Glasg. Math. J.) 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.

Mots clés: Koszul (co)homology; Hochschild (co)homology; cup and cap products; N-Koszul algebras; N-homogeneous algebras

Domaines: Anneaux et algèbres [math.RA]

Roland Berger. Koszul calculus for N-homogeneous algebras. Journal of Algebra, 2019, 519, pp.149-189. ⟨10.1016/j.jalgebra.2018.10.031⟩. ⟨hal-01918544⟩