Characterization of 9-dimensional Anosov Lie Algebras

Abstract

The classification of all real and rational Anosov Lie algebras up to dimension 8 is given by Lauret and Will. 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.