Generalized prelie and permutative algebras

Loïc Foissy (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

We study generalizations of pre-Lie algebras, where the free objects are based on rooted trees which edges are typed, instead of usual rooted trees, and with generalized pre-Lie products formed by graftings. Working with a discrete set of types, we show how to obtain such objects when this set is given an associative commutative product and a second product making it a commutative extended semigroup. Working with a vector space of types, these two products are replaced by a bilinear map Φ which satisfies a braid equation and a commutation relation. Examples of such structures are defined on sets, semigroups, or groups. These constructions define a family of operads PreLie Φ which generalize the operad of pre-Lie algebras PreLie. For any embedding from PreLie into PreLie φ , we construct a family of pairs of cointeracting bialgebras, based on typed and decorated trees: the first coproduct is given by an extraction and contraction process, the types being modified by the action of Φ; the second coproduct is given by admissible cuts, in the Connes and Kreimer's way, with again types modified by the action of Φ. We also study the Koszul dual of PreLie Φ , which gives generalizations of permutative algebras.

Mots clés: Pre-Lie algebras; permutative algebras; typed rooted trees; interacting bialgebras

Domaines: Anneaux et algèbres [math.RA]

Loïc Foissy. Generalized prelie and permutative algebras. 2021. ⟨hal-03187482⟩