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Special Issue SIGMA (Symetry, Integrability and Geometry: Methods and Applications) Year : 2021

Algebraic structures on typed decorated rooted trees

Abstract

Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation process on stochastic PDEs. We here study the algebraic structures on these objects: multiple prelie 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.
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Dates and versions

hal-01924416 , version 1 (15-11-2018)
hal-01924416 , version 2 (01-04-2021)
hal-01924416 , version 3 (21-09-2021)

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Loïc Foissy. Algebraic structures on typed decorated rooted trees. SIGMA (Symetry, Integrability and Geometry: Methods and Applications), 17 (086), pp.28, 2021, Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday, ⟨10.3842/SIGMA.2021.086⟩. ⟨hal-01924416v3⟩
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