#### Vol. 3, No. 1, 2009

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### Sinan Ünver

Vol. 3 (2009), No. 1, 1–34
##### Abstract

Let $k$ be a field of characteristic zero, and let $k{\left[\epsilon \right]}_{n}:=k\left[\epsilon \right]∕\left({\epsilon }^{n}\right)$. We construct an additive dilogarithm ${Li}_{2,n}:{B}_{2}\left(k{\left[\epsilon \right]}_{n}\right)\to {k}^{\oplus \left(n-1\right)}$, where ${B}_{2}$ is the Bloch group which is crucial in studying weight two motivic cohomology. We use this construction to show that the Bloch complex of $k{\left[\epsilon \right]}_{n}$ has cohomology groups expressed in terms of the K-groups ${K}_{\left(\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}\right)}\left(k{\left[\epsilon \right]}_{n}\right)$ as expected. Finally we compare this construction to the construction of the additive dilogarithm by Bloch and Esnault defined on the complex ${T}_{n}ℚ\left(2\right)\left(k\right)$.

##### Keywords
polylogarithms, additive polylogarithms, mixed Tate motives, Hilbert's 3rd problem
Primary: 11G55