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arXiv · 2606.23863

The Goncharov Lie coalgebra of a field

Alexander Kupers·Daniil Rudenko·Ismael Sierra
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Abstract

This paper relates algebraic $K$-theory of fields to polylogarithms via general linear groups. We introduce the Goncharov Lie coalgebra, defined in terms of the $E_\infty$-homology of general linear groups. Using Steinberg modules, we find a presentation, compute its Lie cobracket, and construct motivic and Hodge realisations. Combining these results with the Rognes rank spectral sequence, we give symbolic descriptions of the rationalisation of the algebraic $K$-theory of fields beyond the cases studied by Matsumoto-Milnor and Bloch-Suslin: we express $K^{(3)}_4(F)$ and the indecomposable part of $K^{(3)}_5(F)$ in terms of Goncharov's polylogarithmic complex of weight 3.

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Alexander Kupers, Daniil Rudenko, Ismael Sierra. 2026-06-22. The Goncharov Lie coalgebra of a field. https://arxiv.org/abs/2606.23863

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