Enveloping algebras of derivations of commutative and noncommutative algebras

2024/11/27 by Jason P. Bell, Lucas Buzaglo, Bell, Jason +1 · 1 citation
Mathematics · #16P40. Secondary: 17B65 #16W25 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Primary: 17B35 #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2411.17972

openalex publication_date 2024/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \Bbbk be a field of characteristic zero. Motivated by the fundamental question of whether it is possible for the universal enveloping algebra of an infinite-dimensional Lie algebra to be noetherian, we study Lie algebras of derivations of associative algebras. The main result of this paper is that the universal enveloping algebra of the Lie algebra of derivations of a finitely generated \Bbbk-algebra is not noetherian. This extends a result of Sierra and Walton on the Witt algebra, as well as a result of the second author on Krichever-Novikov algebras. We highlight that the result applies to derivations of both commutative and noncommutative algebras without restriction on their growth.

Cited by

• Derivations and Hochschild cohomology of quantum nilpotent algebras

Related

• Enveloping algebras of Krichever-Novikov algebras are not noetherian
• Lie algebras whose derivation algebras are simple
• Structure of Algebras of Weyl Type
• Derivations of certain algebras defined by étale descent
• Noetherian enveloping algebras of simple graded Lie algebras