Elliptic associators

Details Elliptic associators Elliptic associators

2010-03-04

arxiv.org/abs/1003.1012v2

Mathematics

Quantum Algebra

54 pages

Scientific paper

We study the action of the arithmetic fundamental group of the moduli space of elliptic curves on completions of braid groups in genus 1. This action factors through an explicit profinite group GT_ell, which is an analogue of the Grothendieck-Teichmueller group and may be viewed as the group of universal automorphisms of elliptic structures over braided monoidal categories. We relate the action of GT_ell on braid groups in genus 1 to that of a prounipotent version GT_ell(-) of GT_ell. We prove that the Lie algebra of GT_ell(-) is isomorphic to a graded Lie algebra, using a scheme of elliptic associators. This enables us to compute the Zariski closure of the image of the mapping class group in genus one B_3 in the automorphism groups of the prounipotent completions of the braid groups in genus 1. We study algebraic relations between multiple zeta values (MZVs) and iterated Mellin transforms of Eisenstein series, derived from the action of B_3 on a particular elliptic associator.

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