[Paper Review] Algebraic structures on typed decorated rooted trees

Loïc Foissy|arXiv (Cornell University)|Nov 19, 2018

Advanced Topics in Algebra15 references9 citations

TL;DR

This paper introduces and studies algebraic structures on typed, decorated rooted trees, generalizing key constructions in combinatorial Hopf algebras. It establishes that the space of such trees forms a multiple pre-Lie algebra, constructs associated Hopf algebras via Guin-Oudom and Connes-Kreimer-type functors, and proves that the Bruned-Hairer-Zambotti renormalization framework arises as a subquotient of these structures, unifying and extending prior results in the field.

ABSTRACT

Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation processon stochastic PDEs. We here study the algebraic structures on these objects: multiple prelie algebrasand 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 algebraof typed and decorated trees is isomorphic to a Connes-Kreimer Hopf algebra of non--typed and decoratedtrees (the set of decorations of vertices being bigger), through a contraction process,and finally obtain the Bruned-Hairer-Zambotti construction as a subquotient.

Motivation & Objective

• To develop a universal algebraic framework for typed decorated rooted trees used in stochastic PDE renormalization.
• To generalize pre-Lie algebra and Hopf algebra constructions (e.g., Grossman-Larson, Connes-Kreimer) to the typed, decorated setting.
• To prove that the Bruned-Hairer-Zambotti renormalization construction is a subquotient of this framework.
• To establish isomorphisms between Hopf algebras for different parameter choices, showing structural robustness.

Proposed method

• Define grafting operations ‚t for each type t, forming a T-multiple pre-Lie algebra on typed, D-decorated trees.
• Use the Guin-Oudom construction to lift the pre-Lie product to a Hopf algebra structure on the symmetric algebra of forests.
• Construct dual Hopf algebras via a cointeraction bialgebra framework, generalizing Calaque-Ebrahimi-Fard-Manchon.
• Introduce a contraction-extraction coproduct δ to model the Bruned-Hairer-Zambotti construction.
• Prove that any Connes-Kreimer Hopf algebra of typed trees is isomorphic to one of non-typed trees with a larger decoration set via a contraction process.
• Use a nonassociative permutative coproduct and Livernet’s rigidity theorem to establish isomorphism invariance under parameter variation.

Experimental results

Research questions

• RQ1 Can the algebraic structures of typed decorated rooted trees be systematically generalized from non-typed cases?
• RQ2 How do multiple pre-Lie algebras and their operads relate to the combinatorics of typed trees?
• RQ3 Is the Bruned-Hairer-Zambotti renormalization framework a natural subquotient of a more general algebraic construction?
• RQ4 Are Hopf algebras constructed from different parameter choices (λ) isomorphic, and what does this imply for renormalization invariance?
• RQ5 Can the Connes-Kreimer Hopf algebra on typed trees be reduced to one on non-typed trees via a contraction process?

Key findings

• The space of T-typed, D-decorated rooted trees forms a free T-multiple pre-Lie algebra, generalizing Chapoton-Livernet’s result.
• The operad of T-multiple pre-Lie algebras is combinatorially described by T-typed, indexed trees.
• The Hopf algebra HGLλD,T via the Guin-Oudom construction generalizes the Grossman-Larson algebra to the typed case.
• The dual Hopf algebra HCKλD,T generalizes the Connes-Kreimer algebra and satisfies a Hochschild cohomological universal property.
• For any nonzero λ, the Hopf algebras HGLλD,T and HCKλD,T are isomorphic to those with any other nonzero parameter, showing structural invariance.
• The Bruned-Hairer-Zambotti bialgebra construction is realized as a subquotient of the constructed Hopf algebra framework, unifying the renormalization process.