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 g is a finite-dimensional algebraic Lie algebra over a field of characteristic zero. Then D(g) 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 A(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 A(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. 1991 Mathematics Subject Classification 17B35.