The Fay relations satisfied by the elliptic associator (Q6584672)

The Kronecker function satisfies a well known equation often called the Fay relation. The elliptic associator \(\mathbf{A}_\tau\) is a power series in two non-commutative variables that is defined as an iterated integral of the Kronecker function. Since the Kronecker function satisfies a Fay relation, it is natural to consider analogous Fay relations satisfied by the elliptic associator. Broedel, Matthes, and Schlotterer [\textit{J. Broedel} et al., J. Phys. A, Math. Theor. 49, No. 15, Article ID 155203, 49 p. (2016; Zbl 1354.81045)] previously pursued this, and up to non-explicit correction terms, determined these Fay relations. This paper uses mould theory to provide an alternative route to obtaining Fay relations. An advantage to this route is the ability to determine the correction term explicitly. On the other hand, using mould theory in this way requires working over the ring generated by the coefficients of \(\mathbf{A}_\tau\) modulo the ideal generated by \(2\pi i\). This, however, leads to \(\mathbf{A}_\tau\) being zero. To remedy this, the author instead turns to studying a power series \(\mathbf{A}_\tau^{1/2\pi i}\), which has coefficients in the ring generated by the coefficients of \(\mathbf{A}_\tau\), but whose structure after reducing the coefficients modular \(2\pi i\), denoted \(\bar{A}_\tau\), is non-trivial. The main results of the paper surround establishing various (corrected) Fay relations satisfied by different elliptic generating series and elliptic associators. Details are provided on how to use these results to compute Fay relations for \(\bar{A}_\tau\), including the correction terms.\N\NTo develop the theory and results presented in this paper, the author weaves various areas of mathematics together. Notably, the author forges connections between Fay relations, mould theory, and the Kashiwara-Vergne Lie algebra. Indeed, the first main theorem of this paper (Theorem A) provides an isomorphism between the elliptic Kashiwara-Vergne Lie algebra and a certain space of moulds satisfying a specific family of Fay relations. Meanwhile, there are different types of reduced elliptic generating series and reduced elliptic associators of interest, and the second main theorem (Theorem B) details the Fay relations satisfied by reductions called the `reduced Lie-Like elliptic generating series' and `reduced Lie-like elliptic associator.' The other cases are discussed in Section 3.3, where additional details are found on how to use Theorem B to compute explicit Fay relations for the `reduced group-like elliptic generating series' and also the reduced elliptic associator \(\bar{A}_\tau\).