May 13, 2024

Double shuffle torsor of cyclotomic multiple zeta values and stabilizers of de Rham and Betti coproducts

Khalef Yaddaden, Nagoya University, Japan RIMS Kyoto room 110 + Zoom · JP:15:30 · FR:08:30

Racinet described 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 proved, when \(N=1\), 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 (\(N\geq 1\)). 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.

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