A Quantization of the Loday-Ronco Hopf Algebra

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.

Figures (6)

• Figure 2.1: A planar binary tree of order 3
• Figure 2.2: Planar binary tree with levels that is the image of $\mathbf{(132)}$
• Figure 2.3: The identity and the generator in $k[Y^\infty]$
• Figure 2.4: $\mathbf{(1)}\ast\mathbf{(1)}\ast\mathbf{(1)}=\mathbf{(123)}+\mathbf{(321)}+\mathbf{(312)}+\mathbf{(132)}+\mathbf{(231)}+\mathbf{(213)}$ computed in $k[S^\infty]$. Note that in $k[Y^\infty]$ the fourth and the fifth trees are the same.
• Figure 4.1: Word $\vcenter{}\ast_h \vcenter{}\ast_h \vcenter{}$ that corresponds to the correlation function $W_1^2(p)$.

Theorems & Definitions (27)

• Example 1: The Airy spectral curve
• Definition 1
• Example 2
• Proposition 1
• Theorem 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Example 3