Under a center hypothesis, every derivation of a uniparameter quantum nilpotent algebra decomposes uniquely as an inner derivation plus a homogeneous derivation, and HH1 of U_q^+(g) is the free module over the center with basis the rank-many diagonal derivations D_i.
Enveloping algebras of derivations of commutative and noncommutative algebras
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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.
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Derivations and Hochschild cohomology of quantum nilpotent algebras
Under a center hypothesis, every derivation of a uniparameter quantum nilpotent algebra decomposes uniquely as an inner derivation plus a homogeneous derivation, and HH1 of U_q^+(g) is the free module over the center with basis the rank-many diagonal derivations D_i.