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Register a new userSimple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields
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Let ${mathcal W}_n$ be the Lie algebra of polynomial vector fields. We classify simple weight ${mathcal W}_n$-modules $M$ with finite weight multiplicities. We prove that every such nontrivial module $M$ is either a tensor module or the unique simple submodule in a tensor module associated with the de Rham complex on $mathbb C^n$.
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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.
In this paper, a family of non-weight modules over Lie superalgebras $S(q)$ of Block type are studied. Free $U(eta)$-modules of rank $1$ over Ramond-Block algebras and free $U(mathfrak{h})$-modules of rank $2$ over Neveu-Schwarz-Block algebras are constructed and classified. Moreover, the sufficient and necessary conditions for such modules to be simple are presented, and their isomorphism classes are also determined. The results cover some existing results.
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 $S^{frac{1}{2}}$.
We construct a class of non-weight modules over the twisted $N=2$ superconformal algebra $T$. Let $mathfrak{h}=C L_0oplusC G_0$ be the Cartan subalgebra of $T$, and let $mathfrak{t}=C L_0$ be the Cartan subalgebra of even part $T_{bar 0}$. These modules over $T$ when restricted to the $mathfrak{h}$ are free of rank $1$ or when restricted to the $mathfrak{t}$ are free of rank $2$. We provide the sufficient and necessary conditions for those modules being simple, as well as giving the sufficient and necessary conditions for two $T$-modules being isomorphic. We also compute the action of an automorphism on them. Moreover, based on the weighting functor introduced in cite{N2}, a class of intermediate series modules $A_sigma$ are obtained. As a byproduct, we give a sufficient condition for two $T$-modules are not isomorphic.
We classify all simple bounded highest weight modules of a basic classical Lie superalgebra $mathfrak g$. In particular, our classification leads to the classification of the simple weight modules with finite weight multiplicities over all classical Lie superalgebras. We also obtain some character formulas of strongly typical bounded highest weight modules of $mathfrak g$.
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