Given a Lie group, it is often useful to have a
parametrization of the set of its
lattices. In Euclidean space ℝn, for
example, each lattice corresponds to a basis, and
any lattice is equivalent to the standard integer lattice under an automorphism
in
GL(n, ℝ). In the nilpotent case, the lattices of the
Heisenberg groups are classified,
up to automorphisms, by certain sequences of positive integers with divisibility
conditions (see [1]). In this paper we will study the
set of lattices in a class of simply
connected, solvable, but not nilpotent groups G.
The construction of G depends on
a diagonal n×n matrix Δ
with distinct non-zero eigenvalues, of trace 0; we will write
formula here