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Arithmetic differential equations on $GL_n$, II: arithmetic Lie theory

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.