2021/12/16 by Lucas Buzaglo, Buzaglo, Lucas · 1 citation
Mathematics · #16P40 #16S30 #16W25 #17B65 #17B66 (Primary) #17B68 (Secondary) #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2112.08828
openalex publication_date 2021/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathfrakh is a Lie subalgebra of \mathfrakg of finite codimension, then the noetherianity of U(\mathfrakh) is equivalent to the noetherianity of U(\mathfrakg). The second part of the paper focuses on Lie subalgebras of W≥ -1 = Der(\Bbbk[t]). In particular, we prove that certain subalgebras of W≥ -1 (denoted by L(f), where f ∈ \Bbbk[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.