A Quantization of the Loday-Ronco Hopf Algebra

Abstract

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