A new method for constructing Anosov Lie algebras (Q2796528)

A diffeomorphism \(f:M\to M\), where \(M\) is a closed manifold, is said to be Anosov if the tangent bundle to \(M\) has a continuous splitting into \(df\) invariant vector bundles with the property that \(df\) is contracting on one and expanding on the other. Thus a hyperbolic automorphism of a torus is Anosov. \textit{S. Smale} [Bull. Am. Math. Soc. 73, 747--817 (1967; Zbl 0202.55202)] gave the first examples of non-toral Anosov diffeomorphisms, and raised the question of which closed manifolds admit an Anosov diffeomorphism. One of several conjectures related to this question is the suggestion that every Anosov diffeomorphism is topologically conjugate to an affine infra-nilmanifold automorphism. In this paper a general approach to constructing examples on nilmanifolds is given, motivated in part by some earlier constructions. One approach is to start with an automorphism of a free nilpotent Lie algebra and then quotient out by an invariant ideal. Another is to start with a nilpotent Lie algebra over some number field with a hyperbolic automorphism and find via explicit basis calculations a rational form of the Lie algebra which is invariant under the automorphism. Here the latter method is generalized using Lie and Galois theoretic methods, leading to much shorter arguments and new examples. This also allows questions raised in work of \textit{J. Lauret} [J. Algebra 262, No. 1, 201--209 (2003; Zbl 1015.37022)] and \textit{J. Lauret} and \textit{C. E. Will} [Trans. Am. Math. Soc. 361, No. 5, 2377--2395 (2009; Zbl 1165.37009)] to be answered.