The Poisson spectrum of the symmetric algebra of the Virasoro algebra

TL;DR

This paper investigates the Poisson ideal structure of symmetric algebras related to the Virasoro and Witt algebras, classifies prime and primitive ideals, and explores their geometric and algebraic properties.

Contribution

It classifies Poisson ideals of symmetric algebras of Virasoro and Witt algebras, constructs primitive ideals via symplectic leaves, and proves a structure theorem for finite-codimension subalgebras.

Findings

Only local functions vanish on nontrivial Poisson ideals.

Constructed Poisson primitive ideals from algebraic symplectic leaves.

Any finite-codimension subalgebra of Vir contains the central element z.

Abstract

Let $W = \mathbb{C}[t,t^{-1}]\partial_t$ be the Witt algebra of algebraic vector fields on $\mathbb{C}^\times$ and let $Vir$ be the Virasoro algebra, the unique nontrivial central extension of $W$. In this paper, we study the Poisson ideal structure of the symmetric algebras of $Vir$ and $W$, as well as several related Lie algebras. We classify prime Poisson ideals and Poisson primitive ideals of $S(Vir)$ and $S(W)$. In particular, we show that the only functions in $W^*$ which vanish on a nontrivial Poisson ideal (that is, the only maximal ideals of $S(W)$ with a nontrivial Poisson core) are given by linear combinations of derivatives at a finite set of points; we call such functions local. Given a local function $\chi\in W^*$, we construct the associated Poisson primitive ideal through computing the algebraic symplectic leaf of $\chi$, which gives a notion of coadjoint orbit in our setting. As an application, we prove a structure theorem for subalgebras of $Vir$ of finite codimension and show in particular that any such subalgebra of $Vir$ contains the central element $z$, substantially generalising a result of Ondrus and Wiesner on subalgebras of codimension 1. As a consequence, we deduce that $S(Vir)/(z-\lambda)$ is Poisson simple if and only if $\lambda \neq 0$.