AN INVARIANT FOR FRACTION FIELDS OF McCONNELL ALGEBRAS

Abstract

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.