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Journal Articles Israel Journal of Mathematics Year : 2024

The Fay relations satisfied by the elliptic associator

Abstract

We recall the construction by B. Enriquez of the elliptic associator A τ , a power series in two noncommutative variables a, b defined as an iterated integral of the Kronecker function, and turn our attention to a family of Fay relations satisfied by A τ , derived from the original well-known Fay relation satisfied by the Kronecker function. The Fay relations of A τ were studied by Broedel, Matthes and Schlotterer, and determined up to non-explicit correction terms that arise from the necessity of regularizing the nonconvergent integral. In this article, we study a reduced version Āτ of the elliptic associator mod 2πi. We recall a different construction of Āτ in three steps, due to Matthes, Lochak and the author: first one defines the reduced elliptic generating series Ēτ which comes from the reduced Drinfel'd associator Φ KZ and whose coefficients generate the same ring R as those of Āτ ; then one defines Ψ to be the automorphism of the free associative ring R a, b defined by Ψ(a) = Ēτ and Ψ([a, b]) = [a, b]; finally one shows that the reduced elliptic associator Āτ is equal to Ψ ad(b) e ad(b) −1 (a). Using this construction and mould theory and working with Lie-like versions of the elliptic generating series and associator, we prove the following results: first, a mould satisfies the Fay relations if and only if a closely related mould satisfies the well-known "swap circneutrality" relations defining the elliptic Kashiwara-Vergne Lie algebra krv ell , second, the reduced elliptic generating series satisfies a family of Fay relations with extremely simple correction terms coming directly from those of the Drinfel'd associator, and third, the correction terms for the Fay relations satisfied by the reduced elliptic associator can be deduced explicitly from these.
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Dates and versions

hal-03854618 , version 1 (15-11-2022)
hal-03854618 , version 2 (01-11-2023)

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Leila Schneps. The Fay relations satisfied by the elliptic associator. Israel Journal of Mathematics, inPress. ⟨hal-03854618v2⟩
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