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Register a new userNon-weight modules over the super-BMS$_3$ algebra
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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 $S^{frac{1}{2}}$.
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In the present paper, using the technique of localization, we determine the center of the quantum Schr{o}dinger algebra $S_q$ and classify simple modules with finite-dimensional weight spaces over $S_q$, when $q$ is not a root of unity. It turns out that there are four classes of such modules: dense $U_q(mathfrak{sl}_2)$-modules, highest weight modules, lowest weight modules, and twisted modules of highest weight modules.
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