We consider when the symmetric algebra of an infinite-dimensional Lie
algebra, equipped with the natural Poisson bracket, satisfies the ascending
chain condition (ACC) on Poisson ideals. We define a combinatorial condition on
a graded Lie algebra which we call Dicksonian because it is related to
Dickson's lemma on finite subsets of Nk. Our main result is:
Theorem. If g is a Dicksonian graded Lie algebra over a field of
characteristic zero, then the symmetric algebra S(g) satisfies the
ACC on radical Poisson ideals.
As an application, we establish this ACC for the symmetric algebra of any
graded simple Lie algebra of polynomial growth, and for the symmetric algebra
of the Virasoro algebra. We also derive some consequences connected to the
Poisson primitive spectrum of finitely Poisson-generated algebras.Comment: 29 pages; comments welcome; v2 minor changes to introduction,
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