Contracting Arborification in Hopf Algebras

• Contracting arborification is a Hopf-algebraic process that uniquely maps decorated rooted trees to non-commutative words, encoding combinatorial grafting and admissible cuts.
• It leverages universal mapping properties and Hochschild 1-cocycles to intertwine structures like the stuffle and shuffle products, linking series and integral representations of multiple zeta values.
• The formulation extends to semigroup-valued and planar settings, resolving noncommutativity issues and offering valuable insights for applications in renormalization and normal-form expansions of PDEs.

Contracting arborification is a Hopf-algebraic construction that sends decorated rooted trees or forests to non-commutative words while preserving the combinatorics of grafting, admissible cuts, and the relevant product–coproduct structures. In the multiple zeta value setting, it is the unique Hopf-algebra map from the Butcher–Connes–Kreimer algebra of series-type decorated rooted trees to the stuffle algebra $\mathbb{k}\langle\mathbb{N}\rangle$; in a more general semigroup-valued formulation, it is a morphism from the Hopf algebra of decorated rooted forests to Hoffman's quasi-shuffle Hopf algebra. In Fan’s formulation, contracting arborification is the series-side map $a_Y$, and the central problem is to construct a natural tree-level map relating it to simple arborification on the integral side so that the expected square commutes (Fan, 28 Aug 2025, Ebrahimi-Fard et al., 2016).

1. Algebraic setting

Let $Y=\{y_n\mid n\in\mathbb{N}_{>0}\}$. A $Y$-decorated rooted tree is a rooted tree whose vertices $v$ carry labels $\delta(v)=y_{k_v}$. The corresponding Hopf algebra $H_{BCK}^Y$ is the Connes–Kreimer, or Butcher–Connes–Kreimer, Hopf algebra generated by all nonempty $Y$-decorated rooted trees, with product given by disjoint union and coproduct

$$\Delta(T)=T\otimes 1+1\otimes T+\sum_{c\ \mathrm{proper\ cut}} P_c(T)\otimes R_c(T),$$

where $P_c(T)$ is the pruned forest above the cut and $a_Y$0 is the remaining rooted tree. This makes $a_Y$1 into a graded connected commutative Hopf algebra (Fan, 28 Aug 2025).

The target algebra on the series side is $a_Y$2, the free non-commutative polynomial algebra on the alphabet $a_Y$3, endowed with the stuffle, or quasi-shuffle, product and the deconcatenation coproduct. For words $a_Y$4 and letters $a_Y$5, $a_Y$6,

$a_Y$7

with $a_Y$8. Equipped with this product and coproduct, $a_Y$9 is again a graded connected commutative Hopf algebra. Ebrahimi-Fard, Fauvet, and Manchon formulate the same structure for a commutative semigroup $Y=\{y_n\mid n\in\mathbb{N}_{>0}\}$0, writing $Y=\{y_n\mid n\in\mathbb{N}_{>0}\}$1 with Hoffman's quasi-shuffle product $Y=\{y_n\mid n\in\mathbb{N}_{>0}\}$2 and deconcatenation coproduct $Y=\{y_n\mid n\in\mathbb{N}_{>0}\}$3 (Fan, 28 Aug 2025, Ebrahimi-Fard et al., 2016).

2. Defining morphism and universality

For each generator $Y=\{y_n\mid n\in\mathbb{N}_{>0}\}$4, Fan defines a linear operator $Y=\{y_n\mid n\in\mathbb{N}_{>0}\}$5 on $Y=\{y_n\mid n\in\mathbb{N}_{>0}\}$6 by

$Y=\{y_n\mid n\in\mathbb{N}_{>0}\}$7

This operator is a Hochschild $Y=\{y_n\mid n\in\mathbb{N}_{>0}\}$8-cocycle for the deconcatenation coproduct:

$Y=\{y_n\mid n\in\mathbb{N}_{>0}\}$9

By the universal property of the Connes–Kreimer algebra, due to Foissy–Manchon, there is a unique Hopf-algebra map

$Y$0

such that

$Y$1

for every $Y$2, where $Y$3 grafts a forest onto a new root labeled $Y$4. This unique map is the contracting arborification (Fan, 28 Aug 2025).

The same universal mechanism appears in broader Hopf-algebraic treatments. Fauvet and Menous describe the Connes–Kreimer algebra as the initial object in the category of connected graded bialgebras equipped with a Hochschild $Y$5-cocycle, which forces the existence and uniqueness of arborification-type morphisms. Ebrahimi-Fard, Fauvet, and Manchon state the semigroup-valued analogue as a unique Hopf-algebra morphism

$Y$6

characterized by

$Y$7

with $Y$8 (Fauvet et al., 2012, Ebrahimi-Fard et al., 2016).

3. Explicit combinatorics and relation to multiple zeta values

The recursive definition immediately gives $Y$9 and

$v$0

The nontrivial content appears when rooted-tree factors interact under the product in $v$1. Hoffman–Manchon’s inductive formula states that for roots labeled $v$2 and $v$3 grafted onto forests $v$4,

$v$5

This is exactly the usual stuffle rule (Fan, 28 Aug 2025).

That rule encodes the series definition of multiple zeta values,

$v$6

with a ladder tree $v$7 interpreted as the word $v$8 in $v$9. In the more general mould-theoretic presentation, contracting arborification is described as a refinement of ordinary arborification in which one contracts branches of a tree into blocks and sums over all such admissible contractions in one go, without first expanding into words. Ebrahimi-Fard, Fauvet, and Manchon give an equivalent combinatorial description as a sum over linear extensions with contractions, allowing immediate contraction of comparable vertices by adding their decorations in $\delta(v)=y_{k_v}$0 (Fan, 28 Aug 2025, Fauvet et al., 2012, Ebrahimi-Fard et al., 2016).

4. Series-side and integral-side arborifications

The integral-side alphabet is $\delta(v)=y_{k_v}$1. The algebra $\delta(v)=y_{k_v}$2, endowed with the shuffle product and deconcatenation coproduct, carries the integral representation of multiple zeta values. Fan defines a natural map

$\delta(v)=y_{k_v}$3

on words by

$\delta(v)=y_{k_v}$4

This is an algebra map from the stuffle algebra $\delta(v)=y_{k_v}$5 to the shuffle algebra $\delta(v)=y_{k_v}$6. In parallel, the simple arborification

$\delta(v)=y_{k_v}$7

is uniquely determined by

$\delta(v)=y_{k_v}$8

The desired relation is a Hopf-algebra map $\delta(v)=y_{k_v}$9 making the square commute:

$H_{BCK}^Y$0

This is the tree-level bridge between the series and integral arborifications of multiple zeta values (Fan, 28 Aug 2025).

The obvious candidate due to Manchon,

$H_{BCK}^Y$1

does make the diagram commute, but it ignores tree geometry, since $H_{BCK}^Y$2 is the ladder-tree section. Clavier proposed a more natural forest-preserving map $H_{BCK}^Y$3, but the simplest two-branch tree shows that $H_{BCK}^Y$4 in general. Fan’s exposition isolates this failure on the tree with root $H_{BCK}^Y$5 and two children $H_{BCK}^Y$6, where $H_{BCK}^Y$7, while $H_{BCK}^Y$8 fails by one extra term (Fan, 28 Aug 2025).

5. Planar lifting, error terms, and the corrected map

Fan resolves the noncommutativity problem by passing to planar rooted trees. The strategy has four steps. First, one replaces $H_{BCK}^Y$9 and $Y$0 by their planar analogues $Y$1 and $Y$2; the maps $Y$3 lift to $Y$4, and $Y$5 lifts to $Y$6. Second, for each planar tree $Y$7 one defines a finite error-sum $Y$8 by comparing the expansions $Y$9 and $$\Delta(T)=T\otimes 1+1\otimes T+\sum_{c\ \mathrm{proper\ cut}} P_c(T)\otimes R_c(T),$$0 via the shuffle/stuffle relations; one checks that $$\Delta(T)=T\otimes 1+1\otimes T+\sum_{c\ \mathrm{proper\ cut}} P_c(T)\otimes R_c(T),$$1 exactly when $$\Delta(T)=T\otimes 1+1\otimes T+\sum_{c\ \mathrm{proper\ cut}} P_c(T)\otimes R_c(T),$$2 is a ladder tree. Third, these corrections are assembled into a single linear operator $$\Delta(T)=T\otimes 1+1\otimes T+\sum_{c\ \mathrm{proper\ cut}} P_c(T)\otimes R_c(T),$$3 on $$\Delta(T)=T\otimes 1+1\otimes T+\sum_{c\ \mathrm{proper\ cut}} P_c(T)\otimes R_c(T),$$4, defined recursively by grafting the errors along a process-tree $$\Delta(T)=T\otimes 1+1\otimes T+\sum_{c\ \mathrm{proper\ cut}} P_c(T)\otimes R_c(T),$$5:

$$\Delta(T)=T\otimes 1+1\otimes T+\sum_{c\ \mathrm{proper\ cut}} P_c(T)\otimes R_c(T),$$6

Fourth, one sets

$$\Delta(T)=T\otimes 1+1\otimes T+\sum_{c\ \mathrm{proper\ cut}} P_c(T)\otimes R_c(T),$$7

and finally

$$\Delta(T)=T\otimes 1+1\otimes T+\sum_{c\ \mathrm{proper\ cut}} P_c(T)\otimes R_c(T),$$8

These are the core constructions of the paper (Fan, 28 Aug 2025).

The resulting theorem is that

$$\Delta(T)=T\otimes 1+1\otimes T+\sum_{c\ \mathrm{proper\ cut}} P_c(T)\otimes R_c(T),$$9

on planar trees, proved by induction on the number of leaves. After projection, one obtains

$P_c(T)$0

This gives the desired commutative square that Manchon asked for. A plausible implication is that the planar lift is not an auxiliary convenience but the mechanism that localizes precisely the error terms needed to preserve tree geometry while restoring commutativity (Fan, 28 Aug 2025).

6. Broader Hopf-algebraic context, applications, and terminology

In Ecalle’s mould–comould formalism, contracting arborification is tied to character factorization and duality. Fauvet and Menous describe ordinary arborification as a factorization of characters involving the shuffle or quasishuffle Hopf algebras, and contracting arborification as the refinement in which admissible contractions are incorporated directly. They also formulate the dual picture through the explicit duality between the decorated Connes–Kreimer and Grossman–Larson algebras, where coarborification is adjoint to the arborification map. In their account, contracting arborification yields explicit closed-form expressions for arborified linearization moulds, and under Brjuno or Siegel-type small-divisor conditions the corresponding expansion converges, recovering analytic linearization results of Poincaré–Siegel–Brjuno type (Fauvet et al., 2012).

Ebrahimi-Fard, Fauvet, and Manchon place the same construction inside a comodule-bialgebra structure for word-series substitution and mould composition. They introduce an internal coproduct $P_c(T)$1 on $P_c(T)$2, prove that $P_c(T)$3 is a right comodule-Hopf algebra over $P_c(T)$4, and show that contracting arborification intertwines the internal coproducts:

$P_c(T)$5

In dual terms, the diamond composition of arborescent moulds agrees with that of ordinary moulds after arborification (Ebrahimi-Fard et al., 2016).

Related applications appear in normal-form expansions for dispersive PDEs. Bruned and Schratz use the arborification map from the Butcher–Connes–Kreimer Hopf algebra to the shuffle Hopf algebra as the combinatorial backbone of repeated integrations by parts, obtaining explicit B-series-style expressions for terms in the normal-form iteration for periodic cubic NLS. Their toy example for the tree $P_c(T)$6 gives

$P_c(T)$7

illustrating the systematic packaging of shuffle identities by a Hopf morphism (Bruned, 2024).

A terminological ambiguity occasionally appears outside this algebraic literature. One exposition of Tree-Contractibility uses “contracting arborification” for the problem of deciding whether an undirected graph can be made acyclic by at most $P_c(T)$8 edge contractions; that problem has a deterministic $P_c(T)$9-time algorithm and no polynomial kernel unless $a_Y$00 (Heggernes et al., 2011). In the Hopf-algebraic literature, however, contracting arborification denotes the morphism on decorated rooted forests and its associated compatibility with shuffle, stuffle, or quasi-shuffle structures (Fan, 28 Aug 2025, Ebrahimi-Fard et al., 2016).