## Abstract

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.

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.

Original language | English |
---|---|

Journal | Arkiv foer Matematik |

Publication status | Accepted/In press - 30 Jan 2023 |