Alfons I. Ooms

A class of counterexamples to the Gel’fand-Kirillov conjecture

LIMBURGS UNIV CENTRUM,DEPT WNI,B-3500 DIEPENBEEK,BELGIUM.Alev, J, UNIV REIMS,UFR SCI,DEPT MATH,MOULIN HOUSSE,BP 347,F-51062 REIMS,FRANCE.

On Lie algebras with primitive envelopes, supplements

Let L be a finite dimensional Lie algebra over a field k of characteristic zero, U(L) its universal enveloping algebra and Z(D(L)) the center of the division ring of quotients of U(L). A number of conditions on L are each shown to be equivalent with the primitive of U(L). Also, a formula is given for the transcendency degree of Z(D(L)) over k.

On commutative polarizations

Let L be a finite-dimensional Lie algebra over a field k of characteristic zero and let U(L) be its enveloping algebra with quotient division ring D(L). Let P be a commutative Lie subalgebra of L. In [O2] the necessary and sufficient condition on P was given in order for D(P ) to be a maximal (commutative) subfield of D(L). In particular, this condition is satisfied if P is a commutative polarization (CP) with respect to any regular f ∈ L and the converse holds if L is ad-algebraic. The purpose of this paper is to study Lie algebras admitting these CP’s and to demonstrate their widespread occurrence. First we have the following characterisation if L is completely solvable: P is a CP of L if and only if there exists a descending chain of Lie subalgebras

The polynomiality of the Poisson center and semi-center of a Lie algebra and Dixmier's fourth problem

Abstract Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We provide necessary and also some sufficient conditions in order for its Poisson center and semi-center to be polynomial algebras over k. This occurs for instance if g is quadratic of index 2 with [ g , g ] ≠ g and also if g is nilpotent of index at most 2. The converse holds for filiform Lie algebras of type L n , Q n , R n and W n . We show how Dixmiers fourth problem for an algebraic Lie algebra g can be reduced to that of its canonical truncation g Λ . Moreover, Dixmiers statement holds for all Lie algebras of dimension at most eight. The nonsolvable ones among them possess a polynomial Poisson center and semi-center.