Non-weight modules over the algebra HW(b)

For the parameter $b\in C$, let $\mathrm{HW}\left(b\right)$ be the semidirect product of the Witt algebra and the loop Heisenberg Lie algebra. In this paper, we study some non-weight modules over $\mathrm{HW}\left(b\right)$, specifically focusing on restricted modules, $U\left(C{L}_0\right)$-free modules of rank 1 and the tensor product of both. We prove that these three classes of non-weight $\mathrm{HW}\left(b\right)$-modules are pairwise non-isomorphic. Finally, we transform some tensor product modules over $\mathrm{HW}\left(b\right)$ into induced modules from modules of its certain subalgebras for the case $b\ne \pm 1$.