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# Classification of simple weight modules over the Schrödinger algebra

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Published:2017-05-23
Printed: Mar 2018
• V. V. Bavula,
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK
• T. Lu,
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK
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## Abstract

A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer $C_{\mathcal{S}}(H)$ (and some of its prime factor algebras) of the Cartan element $H$ in the universal enveloping algebra $\mathcal{S}$ of the Schrödinger (Lie) algebra. The simple $C_{\mathcal{S}}(H)$-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra $\mathcal{S}$ (over the centre). It is proved that some (prime) factor algebras of $\mathcal{S}$ and $C_{\mathcal{S}}(H)$ are tensor homological/Krull minimal.
 Keywords: weight module, simple module, centralizer, Krull dimension, global dimension, tensor homological minimal algebra, tensor Krull minimal algebra
 MSC Classifications: 17B10 - Representations, algebraic theory (weights) 17B20 - Simple, semisimple, reductive (super)algebras 17B35 - Universal enveloping (super)algebras [See also 16S30] 16E10 - Homological dimension 16P90 - Growth rate, Gelfand-Kirillov dimension 16P40 - Noetherian rings and modules 16P50 - Localization and Noetherian rings [See also 16U20]

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