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.
maink1m@cmich.edu
Cynthia E. Will
FaMAF and CIEM, Universidad Nacional de Córdoba, Haya de la Torre s/n, 5000 Córdoba, Argentina
cwill@famaf.unc.edu.ar
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
[ Fulltext-pdf (297 KB)] for subscribers only.