  Motivated by the search of a concept of linearity in the theory of arithmetic
differential equations we introduce here an arithmetic analogue of Lie algebras
and a concept of skew arithmetic differential cocycles. We will then construct
such skew cocycles, based on certain remarkable lifts of Frobenius for the
classical groups $GL_n, SL_n, SO_n, Sp_{n}$. The theory for $GL_n$, especially
on the Galois side, will be further developed in a sequel to this paper.
