Simple $\mathcal{W}⋉\widehat{H_4}$-modules from tensor products

This paper investigates simple modules of the semi-direct product algebra $\mathcal{W}⋉\widehat{H_4}$, where $\mathcal{W}$ is the Witt algebra and $\widehat{H_4}$ is the loop Diamond algebra. We first use simple modules over the Weyl algebra to construct a family of simple $\mathcal{W}⋉\widehat{H_4}$-modules. Then, we classify simple $\mathcal{W}⋉\widehat{H_4}$-modules that are free $U(\mathbb{C}{L}_0\oplus \mathbb{C}{a}_0)$-modules of rank 1. Finally, we give a necessary and sufficient condition for finitely many simple $U(\mathbb{C}{L}_0\oplus \mathbb{C}{a}_0)$-free modules to be simple, and then determine their isomorphism classes.