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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.
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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, 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.
In this paper, conjugate-linear anti-involutions and unitary Harish-Chandra modules over the Schr{o}dinger-Virasoro algebra are studied. It is proved that there are only two classes conjugate-linear anti-involutions over the Schr{o}dinger-Virasoro algebra. The main result of this paper is that a unitary Harish-Chandra module over the Schr{o}dinger-Virasoro algebra is simply a unitary Harish-Chandra module over the Virasoro algebra.
Let $mathcal{L}$ be the derivation Lie algebra of ${mathbb C}[t_1^{pm 1},t_2^{pm 1}]$. Given a triangle decomposition $mathcal{L} =mathcal{L}^{+}oplusmathfrak{h}oplusmathcal{L}^{-}$, we define a nonsingular Lie algebra homomorphism $psi:mathcal{L}^{+}rightarrowmathbb{C}$ and the universal Whittaker $mathcal{L}$-module $W_{psi}$ of type $psi$. We obtain all Whittaker vectors and submodules of $W_{psi}$, and all simple Whittaker $mathcal{L}$-modules of type $psi$.
Let $mathbb{F}$ be a field of characteristic 0, $G$ an additive subgroup of $mathbb{F}$, $alphain mathbb{F}$ satisfying $alpha otin G, 2alphain G$. We define a class of infinite-dimensional Lie algebras which are called generalized Schr{o}dinger-Virasoro algebras and use $mathfrak{gsv}[G,alpha]$ to denote the one corresponding to $G$ and $alpha$. In this paper the automorphism group and irreducibility of Verma modules for $mathfrak{gsv}[G,alpha]$ are completely determined.
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