Irreducible Representations of Braid Groups of corank two

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.