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.
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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.