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Register a new userIrreducible weight modules over the Schr{o}dinger Lie algebra in $(n+1)$ dimensional space-time
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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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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, the property and the classification the simple Whittaker modules for the schr{o}dinger algebra are studied. A quasi-central element plays an important role in the study of Whittaker modules of level zero. For the Whittaker modules of nonzero level, our arguments use the Casimir element of semisimple Lie algebra $sl_2$ and the description of simple modules over conformal Galilei algebras by R. L{u}, V. Mazorchuk and K. Zhao.
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 $mathbb C^n$.
The rank $n$ symplectic oscillator Lie algebra $mathfrak{g}_n$ is the semidirect product of the symplectic Lie algebra $mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite dimensional weight spaces over $mathfrak{g}_n$. When $dot z eq 0$, it is shown that there is an equivalence between the full subcategory $mathcal{O}_{mathfrak{g}_n}[dot z]$ of the BGG category $mathcal{O}_{mathfrak{g}_n}$ for $mathfrak{g}_n$ and the BGG category $mathcal{O}_{mathfrak{sp}_{2n}}$ for $mathfrak{sp}_{2n}$. Then using the technique of localization and the structure of generalized highest weight modules, we also give the classification of simple weight modules over $mathfrak{g}_n$ with finite-dimensional weight spaces.
In this paper, we classify all indecomposable Harish-Chandra modules of the intermediate series over the twisted Heisenberg-Virasoro algebra. Meanwhile, some bosonic modules are also studied.
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