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Register a new userAutomorphisms and Verma modules for Generalized Schr{o}dinger-Virasoro algebras
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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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Let $mathfrak g(G,lambda)$ denote the deformed generalized Heisenberg-Virasoro algebra related to a complex parameter $lambda eq-1$ and an additive subgroup $G$ of $mathbb C$. For a total order on $G$ that is compatible with addition, a Verma module over $mathfrak g(G,lambda)$ is defined. In this paper, we completely determine the irreducibility of these Verma modules.
We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra $hat{mathfrak{g}}$, from $mathfrak{g}$-module homomorphisms. When $mathfrak{g}=mathfrak{sl}_2$, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to $hat{mathfrak{g}}$, and we identify obstructions to the construction arising from the possible non-existence of series solutions having a prescribed form.
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.
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.
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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