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 UU({mathfrak {h}}) is equivalent to the noetherianity of UU({mathfrak {g}}). The second part of the paper focuses on Lie subalgebras of W≥− 1 = Der(\U0001d55c[t]). In particular, we prove that certain subalgebras of W≥− 1 (denoted by L(f), where f ∈ \U0001d55c[t]) have non-noetherian universal enveloping algebras, and provide a sufficient condition for a subalgebra of W≥− 1 to have a non-noetherian universal enveloping algebra. Furthermore, we make significant progress on a classification of subalgebras of W≥− 1 by showing that any infinite-dimensional subalgebra must be contained in some L(f) in a canonical way.