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 \(U({\mathfrak {h}})\) is equivalent to the noetherianity of U \(U({\mathfrak {g}})\). The second part of the paper focuses on Lie subalgebras of W≥− 1 = Der(𝕜[t]). In particular, we prove that certain subalgebras of W≥− 1 (denoted by L(f), where f ∈ 𝕜[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.
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Acknowledgements
We thank Pavel Etingof for extensive comments and useful discussion on an earlier version of the paper. We also thank Jason Bell for useful comments and discussion.
Funding
This research is part of the author’s PhD work at the University of Edinburgh, under the supervision of Dr Susan J. Sierra, and supported by the University of Edinburgh’s School of Mathematics.
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Presented by: Pramod Achar
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Buzaglo, L. Enveloping Algebras of Krichever-Novikov Algebras are not Noetherian. Algebr Represent Theor 26, 2085–2111 (2023). https://doi.org/10.1007/s10468-022-10168-9
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DOI: https://doi.org/10.1007/s10468-022-10168-9