Is $A_1$ of type $B_2$

Abstract: By a theorem of Dixmier, primitive quotients of enveloping algebras of finite-dimensional complex nilpotent Lie algebras are isomorphic to Weyl algebras. In view of this result, it is natural to consider simple quotients of positive parts of quantized enveloping algebras (and more generally of uniparameter Quantum Nilpotent Algebras) as quantum analogues of Weyl algebras. In this note, we study the Lie algebra of derivations of the simple quotients of $U_q^+(B_2)$ of Gelfand-Kirillov dimension 2. For a specific family of such simple quotients, we prove that all derivations are inner (as in the case of Weyl algebras) whereas all other such algebras are quantum Generalized Weyl Algebras over a commutative Laurent polynomial algebra in one variable and have a first Hochschild cohomology group of dimension 1.