Research

PhD Thesis

Title : Algebraic structures associated to double shuffle relations between multiple polylogrithm values at roots of unity

Date of defense : March 14^th, 2023.

Abstract : Racinet attached to each finite cyclic group \(G\) and group embedding \(\iota : G \to \mathbb{C}^{\times}\) a \(\mathbb{Q}\)-scheme \(\mathsf{DMR}^{\iota}\) which describes the double shuffle and regularization relations between multiple polylogarithm values at \(N\)^th roots of unity, with \(N\) the order of \(G\). He also exhibited a \(\mathbb{Q}\)-group scheme \(\mathsf{DMR}_0^G\), which Enriquez and Furusho identified with the stabilizer of a coproduct arising in Racinet’s formalism with respect to the action of the group \(\mathcal{G}\) of grouplike elements of a noncommutatives series Hopf algebra, equipped with the twisted Magnus product. We reformulate Racinet’s construction in terms of crossed product. Racinet coproduct can then be identified with a coproduct \(\widehat{\Delta}^{\mathcal{M}, \mathrm{DR}}_G\) defined on a module \(\widehat{\mathcal{M}}_G^{\mathrm{DR}}\) over an algebra \(\widehat{\mathcal{W}}_G^{\mathrm{DR}}\), which is equipped with its own coproduct \(\widehat{\Delta}^{\mathcal{W}, \mathrm{DR}}_G\). We show that there are compatible group actions of a semidirect product involving \(\mathcal{G}\) on \(\widehat{\mathcal{M}}_G^{\mathrm{DR}}\) and \(\widehat{\mathcal{W}}_G^{\mathrm{DR}}\). This yields an explicit stabilizer group scheme containing \(\mathsf{DMR}_0^G\), which we also express in the Racinet formalism. Furthermore, for \(G=\{1\}\), Enriquez and Furusho showed that a subscheme \(\mathsf{DMR}^{\iota}_{\times}\) of \(\mathsf{DMR}^{\iota}\) is a torsor of isomorphisms relating de Rham and Betti objects. In the second part of this work, we define the main ingredients for a generalization of this result to any finite cyclic group \(G\): we exhibit a module \(\widehat{\mathcal{M}}_N^{\mathrm{B}}\) over an algebra \(\widehat{\mathcal{W}}_N^{\mathrm{B}}\) (where \(N\) is the order of \(G\)) and we prove the existence of two compatible coproducts \(\widehat{\Delta}^{\mathcal{W}, \mathrm{B}}_N\) and \(\widehat{\Delta}^{\mathcal{M}, \mathrm{B}}_N\) on \(\widehat{\mathcal{W}}_N^{\mathrm{B}}\) and \(\widehat{\mathcal{M}}_N^{\mathrm{B}}\) respectively such that \(\mathsf{DMR}^{\iota}_{\times}\) is contained in the torsor of isomorphisms relating \(\widehat{\Delta}^{\mathcal{W}, \mathrm{B}}_N\) (resp. \(\widehat{\Delta}^{\mathcal{M}, \mathrm{B}}_N\)) to \(\widehat{\Delta}^{\mathcal{W}, \mathrm{DR}}_G\) (resp. \(\widehat{\Delta}^{\mathcal{M}, \mathrm{DR}}_G\)).

Publications

Preprints

• A work is in progress…

Journal articles

• K. Yaddaden, The cyclotomic double shuffle torsor in terms of Betti and de Rham coproducts. To appear in Journal of Pure and applied algebra and available at (arXiv 2304.04061).
• K. Yaddaden, Crossed product interpretation of the double shuffle Lie algebra attached to an abelian group. To appear in Publications of the Research Institute for Mathematical Sciences and available at (arXiv 2112.14140).

Presentations

Monthly seminar of the mathematics department of the university of Nagoya

Date : November 09^th, 2023

Location : Nagoya, Japan

Title : Double shuffle torsor of cyclotomic MZVs and de Rham and Betti coproducts stabilizers

Mid-term workshop of the ANR HighAGT

Date : June 02^nd, 2023

Location : Strasbourg, France

Title : Double shuffle torsor of cyclotomic multizeta values and stabilizer de Rham and Betti coproducts

Abstract : Racinet describes the double shuffle and regularization relations between multiple polylogarithm values at \(N\)^th roots of unity via a \(\mathbb{Q}\)-scheme \(\mathsf{DMR}^{\iota}\) where \(\iota : G \hookrightarrow \mathbb{C}^{\times}\) is a group embedding from a finite cyclic group \(G\) of order \(N\) to \(\mathbb{C}^{\times}\). Then, Enriquez and Furusho show, when \(G=\{1\}\), that a subscheme \(\mathsf{DMR}^{\iota}_{\times}\) is a torsor of isomorphisms between Betti and de Rham objects. In this talk, we establish a cyclotomic generalization of this result. First, we explicit the torsor structure of \(\mathsf{DMR}^{\iota}_{\times}\). Then, we introduce in this context the adequate de Rham and Betti objects : the former arise from a crossed product algebra and enables a reformulation of Racinet’s harmonic coproduct closer to the formalism introduced by Enriquez and Furusho; the latter, on the other hand, arise from a group algebra of the orbifold fundamental group \(\left(\mathbb{C}^{\times} \smallsetminus \mu_{N}\right) / \mu_{N}\), where \(\mu_{N}\) is the group of \(N\)^th roots of unity. Finally, we show the existence of a coalgebra and Hopf algebra coproduct such that \(\mathsf{DMR}^{\iota}_{\times}\) is a torsor of isomorphisms relying these Betti coproducts to their de Rham counterparts.

The “Algebra, Topology and the Grothedieck-Teichmüller group” conference

Date : August 30^th, 2022

Location : Les Diablerets, Switzerland

Title : The stablizer Lie algebra of the harmonic coproduct

Abstract : For a finite abelian group \(G\), Racinet constructed a Lie algebra \(\mathfrak{dmr}_0^G\), which for \(G=\mu_N\) describes double shuffle and regularisation relations between multiple polylogarithm values specialized to \(N^{th}\) roots of unity. Enriquez and Furusho then identified this Lie algebra with the stabilizer Lie algebra \(\mathfrak{stab}(\Delta^M)\) of a coalgebra \((M, \Delta^M)\) appearing in Racinet’s formalism. On the other hand, Racinet’s formalism provides a Hopf algebra \((\mathbb{Q}\langle\langle Y_G\rangle\rangle, \Delta_{\star}^{alg, G})\). When \(G=1\), this Hopf algebra is equipped with a Lie algebra action, which gives rise to a stabilizer Lie algebra \(\mathfrak{stab}(\Delta_{\star}^{alg,1})\), which can be shown to be equal to \(\mathfrak{stab}(\Delta^M)\). However, when \(G \neq \{1\}\), no such action exists on \((\mathbb{Q}\langle\langle Y_G\rangle\rangle, \Delta_{\star}^{alg, G})\), making a direct analogue of this construction impossible. We show how Racinet’s theory fits in a crossed product formalism, which allows for the construction of an alternative generalisation \((W, \Delta^W)\) for general \(G\) of the Hopf algebra \((\mathbb{Q}\langle\langle Y_G\rangle\rangle, \Delta_{\star}^{alg, G})\), which is moreover equipped with a Lie algebra action and, therefore, allows for the construction of a stabilizer Lie algebra \(\mathfrak{stab}(\Delta^W)\), which can be shown to contain \(\mathfrak{dmr}_0^G\).

Monthly seminar of the mathematics department of the university of Nagoya

Date : August 22^nd, 2022

Location : Web seminar

Title : Double shuffle Lie algebra and crossed product

Quantum Seminar of the Institut de Recherche en Mathématiques Avancées

Date : May 11^th, 2022

Location : Strasbourg, France

Title : Double shuffle Lie algebra and crossed product

Abstract : For any integer \(N \geq 1\), Racinet studied the scheme associated to the double shuffle and regularisation relations between multiple polylogarithm values at \(N^{th}\) roots of unity. He showed, in particular, that this scheme is equipped with a torsor structure under the action of an affine group scheme, specialisation for \(G=\mu_N\) of a group scheme \(\mathsf{DMR}_0^G\), associated to any finite abelian group \(G\). Enriquez and Furusho then identified the Lie algebra \(\mathfrak{dmr}_0^G\) of \(\mathsf{DMR}_0^G\) with the stabilizer Lie algebra of a coproduct appearing in Racinet’s formalism. We reformulate Racinet’s construction in terms of crossed product. Racinet’s coproduct is then identified to the coproduct of a coalgebra \((\mathcal{M}^G,\Delta^{\mathcal{M}}_G)\) that appears in this formalism. Furthermore, this new setting allows us to construct a Hopf algebra \((\mathcal{W}^G,\Delta^{\mathcal{W}}_G)\) under which \((\mathcal{M}^G, \Delta^{\mathcal{M}}_G)\) is a module-coalgebra, the whole structure being equipped with the ambiant Lie algebra action. This leads us to the construction of a stabilizer Lie algebra \(\Delta^{\mathcal{W}}_G\) which contains the stabilizer Lie algebra of \(\Delta^{\mathcal{M}}_G\) which we will express in terms of Racinet’s formalism.

Rotating Colloquium of the GDR Lie Theory in Algebra and Geometry

Date : March 16^th, 2022

Location : Dijon, France

Title : Double shuffle Lie algebra and crossed product

Link : https://dubouloz.perso.math.cnrs.fr/Confs/TLAG-2022-Dijon/index.html

Master Theses

Prior to my Ph. D. I did two M. Sc. resulting in two Master theses described as follows:

• 2018 : Stratification of reductive group actions on affine varieties

Supervisor: Daniel Panazolo.

Initiation to Mumford’s Geometric Invariant Theory which is caracterized by actions of reductive groups on algebraic varieties. Then give a decomposition of the algebraic variety into a finite number of partitions which allows to a better understanding of the variety. This work is based on this paper of Victoria Hoskins.

• 2017 : Métriques riemanniennes sur les groupoïdes de Lie

Supervisor : Yacine Aït Amrane.

We study the set-theoric groupoid structure and equip it with differential structure obtaining what is called a Lie groupoid. Then we consider its action on a smooth manifold. This aims to equip a Lie groupoid with a Riemannian metric with respect to the underlying metrics on the structure. This work is based on this paper of Fernandes and del Hoyo.