Anosov Lie algebras and algebraic units in number fields

We study nilmanifolds admitting Anosov automorphisms by applying elementary properties of algebraic units in number fields to the associated Anosov Lie algebras. We identify obstructions to the existence of Anosov Lie algebras. The case of 13-dimensional Anosov Lie algebras is worked out as an illustration of the technique. Also, we recapture the following known results: (1) Every 7-dimensional Anosov nilmanifold is toral, and (2) every 8-dimensional Anosov Lie algebra with 3 or 5-dimensional derived algebra contains an abelian factor.