On the compatibility of the Betti harmonic coproduct with cyclotomic filtrations
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| Format: | Preprint |
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2025
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| author | Enriquez, Benjamin Yaddaden, Khalef |
| author_facet | Enriquez, Benjamin Yaddaden, Khalef |
| contents | In a previous paper, the second author introduced a Betti counterpart of $N$-cyclotomic double shuffle theory for any $N \geq 1$. The construction is based on the group algebra of the free group $F_2$, endowed with a filtration relative to a morphism $F_2 \to μ_N$ (where $μ_N$ is the group of $N$-th roots of unity). One of the main results therein is the construction of a complete Hopf algebra coproduct $\widehatΔ^{\mathcal{W}, \mathrm{B}}_N$ on the relative completion of a specific subalgebra $\mathcal{W}^\mathrm{B}$ of the group algebra of $F_2$. However, an explicit formula for this coproduct is missing. In this paper, we show that the discrete Betti harmonic coproduct $Δ^{\mathcal{W}, \mathrm{B}}$ defined in \cite{EF1} for the classical case ($N=1$) by the first author and Furusho remains compatible with the filtration structure on $\mathcal{W}^\mathrm{B}$ induced by the relative completion for arbitrary $N$. This compatibility suggests that the completion corresponding to $Δ^{\mathcal{W}, \mathrm{B}}$ is a candidate for an explicit realization of $\widehatΔ^{\mathcal{W}, \mathrm{B}}_N$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12208 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the compatibility of the Betti harmonic coproduct with cyclotomic filtrations Enriquez, Benjamin Yaddaden, Khalef Algebraic Topology Number Theory Quantum Algebra Rings and Algebras Primary 11M32, 16W70, 16T10. Secondary 16T05 In a previous paper, the second author introduced a Betti counterpart of $N$-cyclotomic double shuffle theory for any $N \geq 1$. The construction is based on the group algebra of the free group $F_2$, endowed with a filtration relative to a morphism $F_2 \to μ_N$ (where $μ_N$ is the group of $N$-th roots of unity). One of the main results therein is the construction of a complete Hopf algebra coproduct $\widehatΔ^{\mathcal{W}, \mathrm{B}}_N$ on the relative completion of a specific subalgebra $\mathcal{W}^\mathrm{B}$ of the group algebra of $F_2$. However, an explicit formula for this coproduct is missing. In this paper, we show that the discrete Betti harmonic coproduct $Δ^{\mathcal{W}, \mathrm{B}}$ defined in \cite{EF1} for the classical case ($N=1$) by the first author and Furusho remains compatible with the filtration structure on $\mathcal{W}^\mathrm{B}$ induced by the relative completion for arbitrary $N$. This compatibility suggests that the completion corresponding to $Δ^{\mathcal{W}, \mathrm{B}}$ is a candidate for an explicit realization of $\widehatΔ^{\mathcal{W}, \mathrm{B}}_N$. |
| title | On the compatibility of the Betti harmonic coproduct with cyclotomic filtrations |
| topic | Algebraic Topology Number Theory Quantum Algebra Rings and Algebras Primary 11M32, 16W70, 16T10. Secondary 16T05 |
| url | https://arxiv.org/abs/2508.12208 |