[Paper Review] Planar Shuffle Product, Co-Addition and the non-associative Exponential

Lothar Gerritzen|ArXiv.org|Feb 17, 2005

Advanced Combinatorial Mathematics3 references3 citations

TL;DR

This paper introduces a planar shuffle product for finite planar reduced rooted trees and establishes its duality with the co-addition coproduct in non-associative Hopf algebras. It proves that the generic non-associative exponential series $\mathrm{EXP}$ satisfies $\hat{\Delta}(\mathrm{EXP}) = \mathrm{EXP} \hat{\otimes} \mathrm{EXP}$, leading to quadratic relations for its coefficients involving planar shuffle products, thus extending non-associative analogues of classical exponential structures.

ABSTRACT

In this note we introduce the concept of a shuffle product $\sq$ for planar tree polynomials and give a formula to compute the planar shuffle product $S \ \sq T$ of two finite planar reduced rooted trees $S, T.$ It is shown that $\sq$ is dual to the co-addition $Δ$ which leads to a formula for the coefficients of $Δ(f).$ It is also proved that $Δ(EXP) = EXP {\hat \otimes} EXP$ where $EXP$ is the generic planar tree exponential series, see [G]. Systems of quadratic relations for the coefficients of EXP are derived.

Motivation & Objective

• To define a planar shuffle product $\sqcup\!\sqcup$ for finite planar reduced rooted trees and establish its duality with the co-addition coproduct $\Delta$.
• To derive a formula for the coefficients of $\Delta(f)$ in terms of coefficients of the planar shuffle product $V \sqcup\!\sqcup W$.
• To prove that the generic non-associative exponential series $\mathrm{EXP}$ satisfies $\hat{\Delta}(\mathrm{EXP}) = \mathrm{EXP} \hat{\otimes} \mathrm{EXP}$ under the formal power series completion.
• To derive systems of quadratic relations for the coefficients of $\mathrm{EXP}$ using the planar shuffle product.
• To extend the canonical projection from associative to non-associative settings using the derived algebraic structure.

Proposed method

• Define the planar shuffle product $\sqcup\!\sqcup$ as the dual operation to the co-addition $\Delta$, constructed via contraction of planar trees onto leaf sets.
• Use the reduction of planar rooted trees to define reduced trees $\mathrm{Red}(T)$, ensuring $ar_T(a) \neq 1$ for all vertices $a$, to maintain algebraic consistency.
• Establish that the co-addition $\Delta$ is a continuous $K$-algebra homomorphism on the formal power series completion $K\{\{x\}\}_{\infty}$, extending from the polynomial algebra.
• Prove $\hat{\Delta}(\mathrm{EXP}) = \mathrm{EXP} \hat{\otimes} \mathrm{EXP}$ via a functional equation argument using $K$-automorphisms $\varphi_k$ and evaluation maps $\pi_k$ to $\mathbb{Q}$.
• Derive quadratic relations for coefficients $a(T)$ of $\mathrm{EXP}$ by expressing $a(V) \cdot a(W)$ as a sum over $c_T(V \sqcup\!\sqcup W) \cdot a(T)$, linking coefficients to shuffle products.
• Use recursive formulas for coefficients: $a(T) = \frac{{q \choose m}}{q^n - q} \cdot a(T_1) \cdots a(T_m)$ for $T = T_1 \cdot \cdots \cdot T_m$ with $m = \mathrm{ar}(T)$, $n = \deg(T)$.

Experimental results

Research questions

• RQ1 How can a planar shuffle product be defined for finite planar reduced rooted trees, and what are its algebraic properties?
• RQ2 What is the precise duality relationship between the planar shuffle product and the co-addition coproduct $\Delta$?
• RQ3 Does the generic non-associative exponential series $\mathrm{EXP}$ satisfy $\hat{\Delta}(\mathrm{EXP}) = \mathrm{EXP} \hat{\otimes} \mathrm{EXP}$ in the formal power series completion?
• RQ4 What systems of quadratic relations emerge for the coefficients of $\mathrm{EXP}$, and how are they expressed via the planar shuffle product?
• RQ5 Can the canonical projection from associative to non-associative settings be extended using the derived structure?

Key findings

• The planar shuffle product $\sqcup\!\sqcup$ is shown to be commutative and associative, and its coefficients are used to compute the coefficients of $\Delta(f)$ for any $f$ in the algebra.
• The co-addition $\Delta$ is extended to a continuous coproduct $\hat{\Delta}$ on the formal power series completion $K\{\{x\}\}_{\infty}$, and it satisfies $\hat{\Delta}(\mathrm{EXP}) = \mathrm{EXP} \hat{\otimes} \mathrm{EXP}$.
• The coefficients $a(T)$ of the non-associative exponential series $\mathrm{EXP}$ satisfy the quadratic relation $a(V) \cdot a(W) = \sum_T c_T(V \sqcup\!\sqcup W) \cdot a(T)$, where $c_T(V \sqcup\!\sqcup W)$ are the coefficients of the shuffle product.
• For $m \geq 3$, the shuffle product $x \sqcup\!\sqcup x^m$ is computed explicitly as $ (m+1)x^{m+1} + 2\sum_{i=1}^m x^{2} \cdot x^{m-1} \text{ (with }x\text{ in }i\text{-th position)} + x \cdot x^m + x^m \cdot x $, and the coefficient relations are verified to hold.
• The recursive formula $a(T) = \frac{{q \choose m}}{q^n - q} \cdot a(T_1) \cdots a(T_m)$ for $T = T_1 \cdot \cdots \cdot T_m$ is used to compute specific coefficients, and these are shown to satisfy the derived quadratic identities.
• The verification in Example 5.3 confirms that the right-hand side of the quadratic relation equals $a(x) \cdot a(x^m) = a(x^m)$, thus validating the general formula.