Heisenberg double of some quantum group and its representations

Let $\mathfrak{g},\mathfrak{h}$ be two indecomposable solvable Lie algebras with dimension $5$ and special given structure constants. Their quantized universal enveloping algebras are Hopf algebras, and they are the Hopf duals of each other. The Heisenberg double $\mathcal{D}_{q}$ is the smash product of these two Hopf algebras. In this paper, we study the weight modules, the prime spectrum and the automorphism group of the Heisenberg double $\mathcal{D}_{q}$.