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Register a new userClassification of simple weight modules for the Neveu-Schwarz algebra with a finite-dimensional weight space
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We show that the support of a simple weight module over the Neveu-Schwarz algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all non-trivial weight spaces of such module are infinite-dimensional. As a corollary we obtain that every simple weight module over the Neveu-Schwarz algebra, having a non-trivial finite-dimensional weight space, is a Harish-Chandra module (and hence is either a highest or lowest weight module, or else a module of the intermediate series). This result generalizes a theorem which was originally given on the Virasoro algebra.
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In this paper, the tensor product of highest weight modules with intermediate series modules over the Neveu-Schwarz algebra is studied. The weight spaces of such tensor products are all infinitely dimensional if the highest weight module is nontrivial. We find that all such tensor products are indecomposable. We give the necessary and sufficient conditions for these tensor product modules to be irreducible by using shifting technique established for the Virasoro case in [13]. The necessary and sufficient conditions for any two such tensor products to be isomorphic are also determined.
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 $mathbb C^n$.
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.
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.
In this paper, we study weight representations over the Schr{o}dinger Lie algebra $mathfrak{s}_n$ for any positive integer $n$. It turns out that the algebra $mathfrak{s}_n$ can be realized by polynomial differential operators. Using this realization, we give a complete classification of irreducible weight $mathfrak{s}_n$-modules with finite dimensional weight spaces for any $n$. All such modules can be clearly characterized by the tensor product of $mathfrak{so}_n$-modules, $mathfrak{sl}_2$-modules and modules over the Weyl algebra.
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