Geometry of the Trilogarithm and the Motivic Lie Algebra of a Field

We express explicitly the Aomoto trilogarithm by classical trilogarithms and investigate the algebro-geometric structures lying behind it: different realizations of the weight three motivic complexes. Applying these results, we describe the motivic structure of the Grassmannian tetralogarithm function and determine the structure of the motivic Lie coalgebra in degrees ≤ 4. Using this we give an explicit construction of the Bore1 regulator map$$r4\;:{K_7}\left( \mathbb{C} \right) \to \mathbb{R}$$which together with Borel’s theorem leads to results about ζ _F (4).