A Quantization of the Loday-Ronco Hopf Algebra

TL;DR

This paper introduces a quantized algebraic structure extending the Loday-Ronco Hopf algebra, connecting it to Topological Recursion solutions across arbitrary genus, and explores its algebraic and cohomological properties.

Contribution

It presents a new quantized Hopf algebra $k[Y^ obreak^ ext{infty}]_h$ related to Topological Recursion, extending previous work to include higher genus solutions.

Findings

$k[Y^ obreak^ ext{infty}]_h$ is a Hopf algebra quantization.

Solution space $ obreak ext{ extcal{A}}^h_{ ext{TopRec}}$ is a subalgebra of a quotient algebra.

Discussion on cohomology of $ obreak extcal{A}^h_{ ext{TopRec}}$ in low degree.

Abstract

We propose a quantization algebra of the Loday-Ronco Hopf algebra $k[Y^\infty]$, based on the Topological Recursion formula of Eynard and Orantin. We have shown in previous works that the Loday-Ronco Hopf algebra of planar binary trees is a space of solutions for the genus 0 version of Topological Recursion, and that an extension of the Loday Ronco Hopf algebra as to include some new graphs with loops is the correct setting to find a solution space for arbitrary genus. Here we show that this new algebra $k[Y^\infty]_h$ is still a Hopf algebra that can be seen in some sense to be made precise in the text as a quantization of the Hopf algebra of planar binary trees, and that the solution space of Topological Recursion $\mathcal{A}^h_{\text{TopRec}}$ is a subalgebra of a quotient algebra $\mathcal{A}_{\text{Reg}}^h$ obtained from $k[Y^\infty]_h$ that nevertheless doesn't inherit the Hopf algebra structure. We end the paper with a discussion on the cohomology of $\mathcal{A}^h_{\text{TopRec}}$ in low degree.