A POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS OF INFINITE-DIMENSIONAL LIE ALGEBRAS

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 over an algebraically closed
field of characteristic zero, 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.