A Tannakian interpretation of the elliptic infinitesimal braid Lie algebras

Benjamin Enriquez; Pavel Etingof

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arXiv · 1709.03430

A Tannakian interpretation of the elliptic infinitesimal braid Lie algebras

Benjamin Enriquez·Pavel Etingof

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Abstract

Let $n\geq 1$. The pro-unipotent completion of the pure braid group of $n$ points on a genus 1 surface has been shown to be isomorphic to an explicit pro-unipotent group with graded Lie algebra using two types of tools: (a) minimal models (Bezrukavnikov), (b) the choice of a complex structure on the genus 1 surface, making it into an elliptic curve $E$, and an appropriate flat connection on the configuration space of $n$ points in $E$ (joint work of the authors with D. Calaque). Following a suggestion by P. Deligne, we give an interpretation of this isomorphism in the framework of the Riemann-Hilbert correspondence, using the total space $E^\#$ of an affine line bundle over $E$, which identifies with the moduli space of line bundles over $E$ equipped with a flat connection.

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Benjamin Enriquez, Pavel Etingof. 2017-09-11. A Tannakian interpretation of the elliptic infinitesimal braid Lie algebras. https://arxiv.org/abs/1709.03430

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