Journal of Lie Theory

The classification of all real and rational Anosov Lie algebras up to dimension 8 was given by J. Lauret and C. E. Will [Nilmanifolds of dimension ≤ 8 admitting Anosov diffeomorphisms, Trans. Amer. Math. Soc. 361 (2009) 2377--2395]. In this paper we study 9-dimensional Anosov Lie algebras by using the properties of very special algebraic numbers and Lie algebra classification tools. We prove that there exists a unique, up to isomorphism, complex 3-step Anosov Lie algebra of dimension 9. In the 2-step case, we prove that a 2-step 9-dimensional Anosov Lie algebra with no abelian factor must have a 3-dimensional derived algebra and we characterize these Lie algebras in terms of their Pfaffian forms. Among these Lie algebras, we exhibit a family of infinitely many complex non-isomorphic Anosov Lie algebras.