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# Communications in Number Theory and Physics

## Volume 10 (2016)

### Number 1

### A relative basis for mixed Tate motives over the projective line minus three points

Pages: 87 – 131

DOI: http://dx.doi.org/10.4310/CNTP.2016.v10.n1.a4

#### Author

#### Abstract

In a previous work, the author built two families of distinguished algebraic cycles in Bloch–Kriz cubical cycle complex over the projective line minus three points.

The goal of this paper is to show how these cycles induce well-defined elements in the $\mathrm{H^0}$ of the bar construction of the cycle complex and thus generate comodules over this $\mathrm{H^0}$, that is a mixed Tate motives over the projective line minus three points.

In addition, it is shown that out of the two families only one is needed at the bar construction level. As a consequence, the author obtains that one of the family gives a basis of the tannakian Lie coalgebra of mixed Tate motives over $\mathbb{P}^1 \setminus 1 \{ 0, 1, \infty \}$ relatively to the tannakian Lie coalgebra of mixed Tate motives over $\mathrm{Spec}(\mathbb{Q})$. This in turns provides a new formula for Goncharov motivic co-product, which should really be thought as a coaction. This new presentation is explicitly controlled by the structure coefficients of Ihara’s action by special derivation on the free Lie algebra on two generators.

#### 2010 Mathematics Subject Classification

Primary 14C25. Secondary 05C05, 18D50, 19E15.