Let $W$ be the Witt algebra of vector fields on the punctured complex plane, and let $\text{Vir}$ be the Virasoro algebra, the unique nontrivial central extension of $W$. $\text{Vir}$ is an infinite-dimensional Lie algebra that is ubiquitous in representation theory and important in physics. The symmetric algebras of $\text{Vir}$ and $W$ are polynomial rings in infinitely many variables which also have a Poisson structure induced from the bracket on the Lie algebras. Poisson ideals of $\text{Sym(Vir)}$ and $\text{Sym(W)}$ approximate two-sided ideals in the corresponding universal enveloping algebras and give evidence for the two-sided structure of the enveloping algebras.

We study prime Poisson ideals in these symmetric algebras, focussing on understanding Poisson primitive ideals, which are the Poisson cores of maximal ideals of $\text{Sym(Vir)}$ and of $\text{Sym(W)}$. Surprisingly, these give a way to do finite-dimensional algebraic geometry with these infinite polynomial rings! We give a complete classification of maximal ideals of $\text{Sym(W)}$ which have nontrivial Poisson cores. We then lift this classification to $\text{Sym(Vir)}$, and use it to show that if $q$ is a nonzero complex number, then $(z-q)$ is a maximal Poisson ideal in $\text{Sym(Vir)}$.

This is joint work with Alexey Petukhov.