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arXiv · 2211.08337

Hopf algebras of multiple polylogarithms, and holomorphic 1-forms

Zachary Greenberg·Dani Kaufman·Haoran Li·Christian K. Zickert
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Abstract

We associate to a multiple polylogarithm a holomorphic 1-form on the universal abelian cover of its domain. We relate the 1-forms to the symbol and variation matrix and show that the 1-forms naturally define a lift of the variation of mixed Hodge structure associated to a polylogarithm. The results are conveniently described in terms of a variant H of Goncharov's Hopf algebra of multiple polylogarithms. In particular, we show that the association of a 1-form to a multiple polylogarithm induces a map from the Chevalley-Eilenberg complex of the Lie coalgebra of indecomposables of H to the de Rham complex.

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Zachary Greenberg, Dani Kaufman, Haoran Li, Christian K. Zickert. 2022-11-15. Hopf algebras of multiple polylogarithms, and holomorphic 1-forms. https://arxiv.org/abs/2211.08337

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