  This paper is the first part of a series of papers aimed at improving the
classification by Formanek of the irreducible representations of Artin braid
groups of small dimension. In this paper we classify all the irreducible
complex representations $\rho$ of Artin braid group $B_n$ with the condition
$rank (\rho (\sigma_i)-1)=2$ where $\sigma_i$ are the standard generators. For
$n \geq 7$ they all belong to some one-parameter family of $n$-dimensional
representations.
