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Journal of Lie Theory 25 (2015), No. 3, 857--873
Copyright Heldermann Verlag 2015

Characterization of 9-Dimensional Anosov Lie Algebras

Meera Mainkar
Dept. of Mathematics, Pearce Hall, Central Michigan University, Mt. Pleasant, MI 48859, U.S.A.

Cynthia E. Will
FaMAF and CIEM, Universidad Nacional de Córdoba, Haya de la Torre s/n, 5000 Córdoba, Argentina

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.

Keywords: Anosov Lie algebras, nilmanifolds, nilpotent Lie algebras, hyperbolic automorphisms.

MSC: 22E25; 37D20, 20F34

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