A Quantization of the Loday-Ronco Hopf Algebra
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866911761135828992 |
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| author | Esteves, João N. |
| author_facet | Esteves, João N. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_09680 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A Quantization of the Loday-Ronco Hopf Algebra Esteves, João N. Quantum Algebra Mathematical Physics Combinatorics 05C10, 05C25, 16T05 (Primary) 81R10, 81R50, 81T32 (Secondary) 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. |
| title | A Quantization of the Loday-Ronco Hopf Algebra |
| topic | Quantum Algebra Mathematical Physics Combinatorics 05C10, 05C25, 16T05 (Primary) 81R10, 81R50, 81T32 (Secondary) |
| url | https://arxiv.org/abs/2109.09680 |