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 Ln, Qn, Rn and Wn.We show how Dixmier's fourth problem for an algebraic Lie algebra g can be reduced to that of its canonical truncation gΛ. Moreover, Dixmier's statement holds for all Lie algebras of dimension at most eight. The nonsolvable ones among them possess a polynomial Poisson center and semi-center.
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