Category image over a deformation of the symplectic oscillator algebra

We discuss the representation theory of Hf, which is a deformation of the symplectic oscillator algebra image, where image is the ((2n+1)-dimensional) Heisenberg algebra. We first look at a more general algebra with a triangular decomposition. Assuming the PBW theorem, and one other hypothesis, we show that the BGG category image is abelian, finite length, and self-dual. We decompose image as a direct sum of blocks image, and show that each block is a highest weight category. In the second part, we focus on the case Hf for n=1, where we prove all these assumptions, as well as the PBW theorem.