Rational forms of nilpotent Lie algebras and Anosov diffeomorphisms (Q956664)

Let \(N\) be a simply connected nilpotent Lie group and assume that \(N\) admits a lattice \(\Gamma\) (i.e. a cocompact discrete subgroup). Then the compact quotient \(N/\Gamma\) is called a nilmanifold. Then it is interesting to study for such a nilmanifold the dynamics of the automorphisms of \(N\) stabilizing \(\Gamma\) or the geometry, if one equips \(N\) with a left invariant Riemannian metric, the complex structure, the symplectic structure, etc. The dynamical and geometric properties of the nilmanifold often only depend on the commensurability class of the lattice \(\Gamma\). In this connection the following problem appears: (\(\ast\)) Find all rational forms up to isomorphism of a given real nilpotent Lie algebra \(\mathfrak{n}\). Not much is known on this question, and a complete answer seems quite difficult to obtain in explicit examples, even in the low-dimensional or 2-step nilpotent cases. In a paper of \textit{F. Grunewald} and \textit{D. Segal} [``Reflections on the classification of torsion-free nilpotent groups'', Group theory, Essays for Philip Hall, 121--158 (1984; Zbl 0563.20033)], the set of isomorphism classes of rational forms on \(\mathfrak{n}\) is described by using Galois cohomology of the group \(Gal(\overline{\mathbb{Q}}/\mathbb{Q})\) with values in \(Aut(\mathfrak{n)}\). The problem can also be described in terms of rational points in the orbit space of an algebraic variety [\textit{P. Eberlein}, ``Geometry of 2-step nilpotent Lie groups'', in: Modern dynamical systems and applications. Dedicated to Anatole Katok on his 60th birthday. Cambridge: Cambridge University Press. 67--101 (2004; Zbl 1154.22009); \textit{F. J. Grunewald, D. Segal} and \textit{L. S. Sterling}, Math. Z. 179, 219--235 (1982; Zbl 0497.20026)]. A diffeomorphism \(f\) of a compact differentiable manifold \(M\) is called \textit{Anosov} if the most perfect kind of global hyperbolic behavior for a dynamical system holds: the tangent bundle \(TM\) admits a continuous invariant splitting \(TM=E^{+}\oplus E^{-}\) such that \(df\) expands \(E^{+}\) and contracts \(E^{-}\) exponentially. Simple examples are obtained from the following algebraic construction. Let \(\varphi\) be a hyperbolic automorphism of \(N\) (i.e. all the eigenvalues of its derivative have an absolute value different from 1) such that \(\varphi(\Gamma)=\Gamma\) for some lattice \(\Gamma\) of \(N\). Then \(\varphi\) defines an Anosov diffeomorphism on the nilmanifold \(M=N/\Gamma\), which is called Anosov automorphism. A rational form of \(\mathfrak{n}\) is a rational subspace \(\mathfrak{n}^\mathbb{Q}\) of \(\mathfrak{n}\) such that \(\mathfrak{n}^\mathbb{Q}\otimes \mathbb{R}=\mathfrak{n}\) and \([X,Y]\in \mathfrak{n}^\mathbb{Q}\) for all \(X,Y\in \mathfrak{n}^\mathbb{Q}\). Two rational forms \(\mathfrak{n}_1^\mathbb{Q},\mathfrak{n}_2^\mathbb{Q}\) of \(\mathfrak{n}\) are said to be isomorphic if there exists \(A\in Aut(\mathfrak{n})\) such that \(A\mathfrak{n}_1^\mathbb{Q}=\mathfrak{n}_2^\mathbb{Q}\), or equivalently, if they are isomorphic as Lie algebras over \(\mathbb{Q}\). Not every real nilpotent Lie algebra admits a rational form. By a result due to Malcev, the existence of a rational form of \(\mathfrak{n}\) is equivalent to the corresponding Lie group \(N\) admitting a lattice [\textit{M. S. Raghunathan}, Discrete subgroups of Lie groups. Ergebn. Math. 68, Springer Verlag (1972; Zbl 0254.22005)]. Another difference from the semisimple case is that sometimes \(n\) has only one rational form up to isomorphism. For a 2-step nilpotent Lie algebra \(\mathfrak{n}\) with 2-dimensional center, Grunewald, Segal and Sterling (op. cit.) gave an answer to \((\ast)\) in terms of isomorphism classes of binary forms. Such a binary form is the Pfaffian form of \(\mathfrak{n}\), which is a homogeneous polynomial of degree \(m\) in \(k\) variables attached to any 2-step nilpotent Lie algebra \(n\) of dimension \(2m+k\) and \(din[n,n]=k\). The projective equivalence class of this form is an isomorphism invariant of \(n\) [\textit{J. Scheuneman}, J. Algebra 7, 152--159 (1967; Zbl 0207.34202)]. In the present paper the author shows how one can apply Pfaffian forms, results of Grunewald, Segal and Sterling, and the so-called Scheuneman duality, to solve problem \((\ast)\) in some cases. The author computes explicitly the set of isomorphism classes of rational forms for the 2-step nilpotent Lie algebras \(\mathfrak{h}_3\oplus \mathfrak{h}_3,\mathfrak{g}, \mathfrak{h}_3\oplus \mathfrak{h}_5\) and \(\mathfrak{h}\). The author also considers the 3-step nilpotent algebra \(\mathfrak{l}_4\oplus \mathfrak{l}_4\), to which the above techniques do not apply.