Algebraic Structures on Typed Decorated Rooted Trees

Abstract

Typically decorated trees are used by Bruned, Hairer, and Zambotti to describe a renormalisation process on stochastic PDEs. We here study the algebraic structures on these objects: multiple pre-Lie algebras and related operads (generalizing a result by Chapoton and Livernet), noncommutative and cocommutative Hopf algebras (generalizing Grossman and Larson's construction), commutative and noncocommutative Hopf algebras (generalizing Connes and Kreimer's construction), bialgebras in cointeraction (generalizing Calaque, Ebrahimi-Fard, and Manchon's result). We also define families of morphisms and, in particular, we prove that any Connes-Kreimer Hopf algebra of typed and decorated trees is isomorphic to a Connes-Kreimer Hopf algebra of non-typed and decorated trees (the set of decorations of vertices being bigger), through a contraction process, and finally obtain the Bruned-Hairer-Zambotti construction as a subquotient.