A Quantization of the Loday-Ronco Hopf Algebra

We propose a quantization algebra of the Loday-Ronco Hopf algebra \(k[Y^\infty ]\) $k[Y^{\infty }]$ , 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 \(k[Y^\infty ]_h\) $k{[Y^{\infty }]}_h$ 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 \(\mathcal {A}^h_{\text {TopRec}}\) ${\mathcal{A}}_{\text{TopRec}}^h$ is a subalgebra of a quotient algebra \(\mathcal {A}_{\text {Reg}}^h\) ${\mathcal{A}}_{\text{Reg}}^h$ obtained from \(k[Y^\infty ]_h\) $k{[Y^{\infty }]}_h$ that nevertheless doesn’t inherit the Hopf algebra structure. We end the paper with a discussion on the cohomology of \(\mathcal {A}^h_{\text {TopRec}}\) ${\mathcal{A}}_{\text{TopRec}}^h$ in low degree.