Non-weight modules over the super-BMS$_3$ algebra

In the present paper, a class of non-weight modules over the super-BMS$_3$ algebras $S^{epsilon}$ ($epsilon=0$ or $frac{1}{2}$) are constructed. These modules when regarded as $S^{0}$-modules and further restricted as modules over the Cartan subalgebra $mathfrak{h}$ are free of rank $1$, while when regarded as $S^{frac{1}{2}}$-modules and further restricted as modules over the Cartan subalgebra $mathfrak{H}$ are free of rank $2$. We determine the necessary and sufficient conditions for these modules being simple, as well as determining the necessary and sufficient conditions for two $S^{epsilon}$-modules being isomorphic. At last, we present that these modules constitute a complete classification of free $U(mathfrak{h})$-modules of rank $1$ over $S^{0}$, and also constitute a complete classification of free $U(mathfrak{H})$-modules of rank $2$ over $S^{frac{1}{2}}$.