The Goncharov Lie coalgebra of a field

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 $E_{\mathrm{\infty }}$-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 $K_4^{(3)}(F)$ and the indecomposable part of $K_5^{(3)}(F)$ in terms of Goncharov’s polylogarithmic complex of weight 3.