Global and Arithmetic Hom-Lie Algebras

Hom-Lie algebras are non-associative, non-commutative algebras generalizing Lie algebras by twisting the Jacobi identity by a homomorphism. The main examples are algebras of twisted derivations (i.e., linear maps with a generalized Leibniz rule). Such generalized derivations appear in all parts of number theory, so hom-Lie algebras appear to have a natural role to play in many number-theoretical studies. In this paper we also give an alternative construction of hom-Lie algebras which is useful for constructing hom-Lie algebras from number fields and A-motives, and vice versa. As a possible interesting application we show that hom-Lie algebras can be used to construct (non-trivial) field extensions of a given number field.