In their seminal work [1] on the
fields of fractions of the enveloping algebra of an
algebraic Lie algebra, Gel'fand and Kirillov formulate the following
conjecture.
Assume that [gfr ] is a finite-dimensional algebraic Lie algebra over a
field of characteristic
zero. Then D([gfr ]) is a Weyl skew-field over a purely transcendental
extension of the
base field.
They showed that neither the conjecture nor its negation holds for all
non-algebraic algebras. In [2], A. Joseph gave
a particularly easy non-algebraic
counterexample devised by L. Makar-Limanov: this is a non-algebraic 5-dimensional
solvable Lie algebra, providing a counterexample despite the fact that
the centre is
one-dimensional. Besides, he raised a question of generalization of this
method for
any completely solvable Lie algebra.
On the other hand, consider [Ascr ](V, δ, Γ),
the McConnell algebra for the triple
(V, δ, Γ) as defined in [4, 14.8.4]
and
below. McConnell in [3] described the completely
prime quotients of the enveloping algebra of a solvable Lie algebra in
terms of
[Ascr ](V, δ, Γ), and found a complete set of invariants
to separate
them. In [2], A. Joseph
raised the question whether the fields of fractions of these McConnell
algebras remain
non-isomorphic. The purpose of this note is to extend the work of L. Makar-Limanov
reported in [2, Section 6], and so provide
an integer-valued invariant which, for
McConnell algebras defined over ℤ, says precisely when this skew-field
is isomorphic
to a Weyl skew-field: this number has simply to be positive. This result
therefore gives
a large supply of skew-fields which ‘resemble’ a Weyl skew-field
very nearly, but
nevertheless are not isomorphic to it.