アイテム詳細

要旨

Zeta generators are derivations associated with odd Riemann zeta values that
act freely on the Lie algebra of the fundamental group of Riemann surfaces with
marked points. The genus-zero incarnation of zeta generators are Ihara
derivations of certain Lie polynomials in two generators that can be obtained
from the Drinfeld associator. We characterize a canonical choice of these
polynomials, together with their non-Lie counterparts at even degrees $w\geq
2$, through the action of the dual space of formal and motivic multizeta
values. Based on these canonical polynomials, we propose a canonical
isomorphism that maps motivic multizeta values into the $f$-alphabet. The
canonical Lie polynomials from the genus-zero setup determine canonical zeta
generators in genus one that act on the two generators of Enriquez' elliptic
associators. Up to a single contribution at fixed degree, the zeta generators
in genus one are systematically expanded in terms of Tsunogai's geometric
derivations dual to holomorphic Eisenstein series, leading to a wealth of
explicit high-order computations. Earlier ambiguities in defining the
non-geometric part of genus-one zeta generators are resolved by imposing a new
representation-theoretic condition. The tight interplay between zeta generators
in genus zero and genus one unravelled in this work connects the construction
of single-valued multiple polylogarithms on the sphere with
iterated-Eisenstein-integral representations of modular graph forms.