Ideals in the enveloping algebra of the positive Witt algebra

TL;DR

This paper investigates the structure of ideals in the universal enveloping algebra of the positive Witt algebra, revealing finite Gelfand-Kirillov dimension for certain quadratic-generated ideals and proposing broader conjectures.

Contribution

It demonstrates that quadratic-generated ideals in $U(W_+)$ lead to finite Gelfand-Kirillov dimension and satisfy the ascending chain condition, and formulates conjectures for general ideals.

Findings

Quadratic-generated ideals have finite Gelfand-Kirillov dimension.

Such ideals satisfy the ascending chain condition.

Conjectures are proposed for arbitrary ideals and verified for radical Poisson ideals.

Abstract

Let $W_+$ be the positive Witt algebra, which has a $C$-basis $\{e_n: n \in Z_{\geq 1}\}$, with Lie bracket $[ e_i, e_j] = (j-i) e_{i+j}$. We study the two-sided ideal structure of the universal enveloping algebra $U(W_+)$ of $W_+$. We show that if $I$ is a (two-sided) ideal of $U(W_+)$ generated by quadratic expressions in the $e_i$, then $U(W_+)/I$ has finite Gelfand-Kirillov dimension, and that such ideals satisfy the ascending chain condition. We conjecture that analogous facts hold for arbitrary ideals of $U(W_+)$, and verify a version of these conjectures for radical Poisson ideals of the symmetric algebra $S(W_+)$.