On the rational forms of nilpotent Lie algebras and lattices in nilpotent Lie groups. (Q1419574)

Let \(L\) be a real finite-dimensional nilpotent Lie algebra and \(H\) be a rational subalgebra of \(L\). \(H\) is called a rational form for \(L\) if there exists a basis of \(H\) over \(Q\) which is also a real basis for \(L\). Rational forms for the Lie algebra of a nilpotent Lie group give rise to lattices in the group. There are examples of \(L\) which do not have any rational forms, examples where there is a unique form and examples where there are an infinite number of forms. Of the latter, a 6-dimensional example due to Malcev is studied showing when the forms are isomorphic using a simple condition on a parameter. Then a class is constructed which have a unique form. These algebras are the direct sum of a generalized Heisenberg and an abelian Lie algebra. Other results in this spirit are obtained.