Enveloping algebras of Krichever-Novikov algebras are not noetherian

Abstract

This work is part of the overarching question of whether it is possible for the universal enveloping algebra of an infinite-dimensional Lie algebra to be noetherian. The main result of this paper is that the universal enveloping algebra of any Krichever-Novikov algebra is not noetherian, extending a result of Sierra and Walton on the Witt (or classical Krichever-Novikov) algebra. As a subsidiary result, which may be of independent interest, we show that if $\mathfrak{h}$ is a Lie subalgebra of $\mathfrak{g}$ of finite codimension, then the noetherianity of $U(\mathfrak{h})$ is equivalent to the noetherianity of $U(\mathfrak{g})$. The second part of the paper focuses on Lie subalgebras of $W_{\geq -1} = \operatorname{Der}(\Bbbk[t])$. In particular, we prove that certain subalgebras of $W_{\geq -1}$ (denoted by $L(f)$, where $f \in \Bbbk[t]$) have non-noetherian universal enveloping algebras, and provide a sufficient condition for a subalgebra of $W_{\geq -1}$ to have a non-noetherian universal enveloping algebra. Furthermore, we make significant progress on a classification of subalgebras of $W_{\geq -1}$ by showing that any infinite-dimensional subalgebra must be contained in some $L(f)$ in a canonical way.