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Hidekazu Furusho

Publications and source records attributed to Hidekazu Furusho.

At least 19 recordsLinked to original sources
arXiv2026

A stabilizer interpretation of the Grothendieck-Teichmüller group $\mathsf{GRT}_1(\mathbf k)$

If $\mathfrak u$ and $\mathfrak v$ are Lie algebras, then the product $\mathrm{Out}(\mathfrak u)\times \mathrm{Out}(\mathfrak v)$ of their outer automorphism groups naturally acts on the set of outer Lie algebra morphisms from $\mathfrak u$ to $\mathfrak v$; the stabilizer of the outer class of a given such morphism is then a subgroup of $\mathrm{Out}(\mathfrak u)\times \mathrm{Out}(\mathfrak v)$. We show that this leads to two related interpretation of the Grothendieck-Teichmüller group $\mathsf{GRT}_1(\mathbf k)$, where $\mathfrak u,\mathfrak v$ are the Lie algebra of infinitesimal braids on the plane (resp. framed infinitesimal braids on the sphere) with 3 and 4 (resp. 4 and 5) strands: namely, it can be expressed as the joint intersection of the stabilizer groups of the outer classes of certain strand doubling morphisms $ϕ$ and $ψ$ with $\mathrm{Out}^*(\mathfrak u)\times\mathrm{Out}(\mathfrak v)$, where $\mathrm{Out}^*(\mathfrak u)$ is a subgroup of $\mathrm{Out}(\mathfrak u)$ of outer classes of inertia-preserving automorphisms of $\mathfrak u$.

math.QA↗
arXiv2026

Associators in mould theory

By developing various techniques of mould theory and establishing a quasi-involutive reformulation of Drinfeld's associator set, we introduce $\mathsf{GARI}(\mathscr{F})_{\mathsf{as}+\mathsf{bal}}$, a mould theoretic formulation of Drinfeld's associator set. We give a mould-theoretical generalization of the result that associator relations imply double shuffle relations, namely, we explain that $\mathsf{GARI}(\mathscr{F})_{\mathsf{as}+\mathsf{bal}}$ is embedded into Ecalle's set $\mathsf{GARI}(\mathscr{F})_{\mathsf{as}\ast\mathsf{is}}$ which is a mould theoretic version of Racinet's double shuffle set.

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arXiv2026

On linearised and elliptic versions of the Kashiwara-Vergne Lie algebra

The goal of this article is to define a linearized or depth-graded version $\mathfrak{lkv}$, and a closely related elliptic version $\mathfrak{krv}_{ell}$, of the Kashiwara-Vergne Lie algebra $\mathfrak{krv}$ originally constructed by Alekseev and Torossian as the space of solutions to the linearized Kashiwara-Vergne problem. We show how the elliptic Lie algebra $\mathfrak{krv}_{ell}$ is related to earlier constructions of elliptic versions $\mathfrak{grt}_{ell}$ and $\mathfrak{ds}_{ell}$ of the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}$ and the double shuffle Lie algebra $\mathfrak{ds}$. In particular we show that there is an injective Lie morphism $\mathfrak{ds}_{ell}\hookrightarrow \mathfrak{krv}_{ell}$, and an injective Lie algebra morphism $\mathfrak{krv}\rightarrow \mathfrak{krv}_{ell}$ extending the known morphisms $\mathfrak{grt}\hookrightarrow\mathfrak{grt}_{ell}$ (Enriquez section) and $\mathfrak{ds}\rightarrow\mathfrak{ds}_{ell}$ (Écalle map).

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arXiv2026

Double shuffle Lie algebra and special derivations

Racinet's double shuffle Lie algebra $\mathfrak{dmr}_0$ is a Lie subalgebra of the Lie algebra $\mathfrak{tder}$ of tangential derivations of the free Lie algebra with generators $x_0,x_1$, i.e. of derivations such that $x_1\mapsto 0$ and $x_0\mapsto [a,x_0]$ for some element $a$. We prove: (1) $\mathfrak{dmr}_0$ is contained in the Lie subalgebra $\mathfrak{sder}$ of $\mathfrak{tder}$ of special derivations, i.e. satisfying the additional condition that $x_\infty\mapsto [b,x_\infty]$ for some element $b$, where $x_\infty:=x_1-x_0$; (2) $\mathfrak{dmr}_0$ is stable under the involution of $\mathfrak{sder}$ induced by the exchange of $x_0$ and $x_\infty$. The first statement: (a) says that any element of $\mathfrak{dmr}_0$ satisfies the "senary relation" (a fact announced without proof by Ecalle in 2011); (b) implies the inclusion $\mathfrak{dmr}_0\subset \mathfrak{krv}_2$ (which was proved by Schneps in 2012 only conditionally to the truth of (1)). We also derive the analogues of statements (a) and (b) respective to Racinet's ``double shuffle schemes'' $\mathsf{DMR}_μ(\mathbf k)$ and to the Betti double shuffle group $\mathsf{DMR}^{\mathrm{B}}(\mathbf k)$ introduced in our earlier work.

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arXiv2025

$p$-adic hypergeometric function related with $p$-adic multiple polylogarithms

This paper introduces a $p$-adic analogue of Gauss's hypergeometric function, constructed via a method that is distinct from distinct from Dwork's approach. The idea of our construction is motivated by the Ohno-Zagier formula, which is elucidated through the relationship between the hypergeometric differential equation and the Knizhnik-Zamolodchikov (KZ) equation. We develop a rigorous framework for the residue-wise analytic prolongation of our $p$-adic hypergeometric function by exploring its relationship with $p$-adic multiple polylogarithms. Through a detailed analysis of its local behavior near the point $1$, we show a $p$-adic version of Gauss hypergeometric theorem for the function.

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arXiv2025

An alternative $\mathbb{Q}$-form of the cyclotomic double shuffle Lie algebra

We present an alternative $\mathbb{Q}$-form for Racinet's cyclotomic double shuffle Lie algebra, inspired by the double shuffle relations among congruent multiple zeta values studied by Yuan and Zhao. Our main result establishes an invariance characterization theorem, demonstrating how these two $\mathbb{Q}$-forms can be reconstructed from each other under Galois action.

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arXiv2024

Notes on Kashiwara-Vergne and double shuffle Lie algebras

We explain the current situation of the relationship between the Kashiwara-Vergne Lie algebra $\mathfrak{krv}$ and the double shuffle Lie algebra $\mathfrak{dmr}$. We also show the validity of Ecalle's senary relation for small depths.

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arXiv2022

The $\ell$-adic hypergeometric function and associators

We introduce an $\ell$-adic analogue of Gauss's hypergeometric function arising from the Galois action on the fundamental torsor of the projective line minus three points. Its definition is motivated by a relation between the KZ-equation and the hypergeometric differential equation in the complex case. We show two basic properties, analogues of Gauss's hypergeometric theorem and of Euler's transformation formula for our $\ell$-adic function. We prove them by detecting a connection of a certain two-by-two matrix specialization of even unitary associators with the associated gamma function, which extends the result of Ohno and Zagier.

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arXiv2022

Kashiwara-Vergne and dihedral bigraded Lie algebras in mould theory

We introduce the Kashiwara-Vergne bigraded Lie algebra associated with a finite abelian group and give its mould theoretic reformulation. By using the mould theory, we show that it includes Goncharov's dihedral Lie algebra, which generalizes the result of Raphael and Schneps.

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arXiv2022

The stabilizer bitorsors of the module and algebra harmonic coproducts are equal

In earlier work, we constructed a pair of "Betti" and "de Rham" Hopf algebras and a pair of module-coalgebras over this pair, as well as the bitorsors related to both structures (which will be called the "module" and "algebra" stabilizer bitorsors). We showed that Racinet's torsor constructed out of the double shuffle and regularization relations between multiple zeta values is essentially equal to the "module" stabilizer bitorsor, and that the latter is contained in the "algebra" stabilizer bitorsor. In this paper, we show the equality of the "algebra" and "module" stabilizer bitorsors. We reduce the proof to showing the equality of the associated "algebra" and "module" graded Lie algebras. The argument for showing this equality involves the relation of the "algebra" Lie algebra with the kernel of a linear map, the expression of this linear map as a composition of three linear maps, the relation of one of them with the "module" Lie algebra and the computation of the kernel of the other one by discrete topology arguments.

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arXiv2022

The Betti side of the double shuffle theory. II. Double shuffle relations for associators

We derive from the compatibility of associators with the module harmonic coproduct, obtained in Part I of the series, the inclusion of the torsor of associators into that of double shuffle relations, which completes one of the aims of this series. We define two stabilizer torsors using the module and algebra harmonic coproducts from Part I. We show that the double shuffle torsor can be described using the module stabilizer torsor, and that the latter torsor is contained in the algebra stabilizer torsor.

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arXiv2022

The Betti side of the double shuffle theory. III. Bitorsor structures

In the two first parts of the series, we constructed stabilizer subtorsors of a `twisted Magnus' torsor, studied their relations with the associator and double shuffle torsors, and explained their `de Rham' nature. In this paper, we make the associated bitorsor structures explicit and explain the `Betti' nature of the corresponding right torsors; we thereby complete one aim of the series. We study the discrete and pro-p versions of the `Betti' group of the double shuffle bitorsor.

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arXiv2022

Analytic continuation of multiple polylogarithms in positive characteristic

Our aim of this paper is to propose a method of analytic continuation of Carlitz multiple (star) polylogarithms to the whole space by using Artin-Schreier equation and present a treatment of their branches by introducing the notion of monodromy modules. As applications of this method, we obtain (1) a method of continuation of the logarithms of higher tensor powers of Carlitz module, (2) the orthogonal property (Chang-Mishiba functional relations), (3) a branch independency of the Eulerian property.

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arXiv2022

The pentagon equation and the confluence relations

We show an equivalence of Drinfeld's pentagon equation and Hirose-Sato's confluence relations. As a corollary, we obtain a pentagon-free presentation of the Grothendieck-Teichmüller group $GRT_1$ and associators.

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arXiv2021

The Betti side of the double shuffle theory. I. The harmonic coproducts

This paper is the first in a series which aims at: (a) giving a proof that the associator relations between multizeta values imply the double shuffle and regularization (DSR) ones, alternative to that of the second-named author's 2010 paper; (b) enhancing Racinet's construction of a torsor structure over the Q-scheme of DSR relations to an explicit bitorsor structure. In this paper, we revisit Racinet's original DSR formalism, whose main character is an algebra coproduct, called the harmonic coproduct, and we introduce a variant which is a module coproduct; we explain the `de Rham' nature of this formalism and construct a `Betti' counterpart of it; we show how both formalisms can be interpreted in terms of geometry, following the ideas of Deligne and Terasoma's unfinished 2005 preprint; we use Bar-Natan's interpretation of associators as functors from the category of parenthesized braids to that of chord diagrams to show that any associator relates the Betti and de Rham geometric objects, both in the `algebraic' and in the `module' setups; we derive that any associator relates the Betti and de Rham algebra coproducts, as well as their module counterparts. These results will be used in the next parts of the series.

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