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\title{Explicit formula for the Drinfeld associator in low degrees}
\author{S.\,Duzhin}
\date{}

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\begin{abstract}
I am reporting just a result of my computer calculations,
giving an explicit formula for the logarithm of the
Knizhnik--Zamolodchikov Drinfeld associator expanded over the Lyndon basis
of the free Lie algebra in two variables up to degree 12. This formula might
turn useful in low-degree computations, checking conjectures etc.
I made the corresponding calculation in 2004; to the best of my knowledge,
no explicit expansion of the KZ associator to degrees higher than 4 is given
in the existing literature (with the exception of my own joint paper
\cite{CD} where degree 5 is also written up). Therefore I decided to upload
it on the arXiv.
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\section{Introduction}}
The Knozhnik--Zamolodchikov Drinfeld associator is a remarkable series
in two non-commuting variables with coefficients belonging to the algebra of
multiple zeta values (MZV). See \cite{KnZa,Dr1,Dr2,EG,Koh4,Oht,CDB 
LM2,LM4} about the associator and \cite{Hoff,Zag} about the multiple
zeta values. 
The associator is a group-like element, therefore, its logarithm
is a well-defined element of the completed free Lie algebra in two
generators over the ring of MZV's. We will use the Lyndon basis
of the free Lie algebra in our formulas; as the Lyndon words are in 1-to-1
correspondence with irreducible polynomials over the field $\F_2$, the 
logarithm of the associator can be described as a very special mapping 
from the set of irreducible polynomials over $\Z_2$ into $\cal{Z}$.

\section{Expansion of $\Phi$)

\section{Expansion of $\log\Phi$)


\begin{thebibliography}{JM}
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