Ideals in the enveloping algebra of the positive Witt algebra

Abstract

Let $W_{+}$ be the positive Witt algebra, which has a $C$-basis $\{e_n:n\in Z_{\ge 1}\}$, 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 $S(W_{+})$.

arXiv comment22 pages; v2 extensive revisions to Section 4 to improve readability of proofs. To appear in Algebras and Representation Theory