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THE CATEGORY OF WEIGHT MODULES FOR SYMPLECTIC OSCILLATOR LIE ALGEBRAS

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Abstract

The rank n symplectic oscillator Lie algebra 𝔤n is the semidirect product of the symplectic Lie algebra 𝔰𝔭2n and the Heisenberg algebra Hn. In this paper, we first study weight modules with finite-dimensional weight spaces over 𝔤n. When the central charge \( \dot{z} \) ≠ 0, it is shown that there is an equivalence between the full subcategory 𝒪𝔤n \( \left[\dot{z}\right] \) of the BGG category 𝒪𝔤n for 𝔤n and the BGG category 𝒪𝔰𝔭2n for 𝔰𝔭2n. Then using the technique of localization and the structure of generalized highest weight modules, we give the classification of simple weight modules over 𝔤n with finite-dimensional weight spaces. As a byproduct we also determine all simple 𝔤n-modules (not necessarily weight modules) that have a simple Hn-submodule.

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Authors and Affiliations

  1. School of Mathematics and Statistics Institute of Contemporary Mathematics, Henan University, Kaifeng, 475004, China

    GENQIANG LIU

  2. Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, N2L 3C5, Canada

    KAIMING ZHAO

  3. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050016, Hebei, China

    KAIMING ZHAO

Authors
  1. GENQIANG LIU
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  2. KAIMING ZHAO
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Correspondence to KAIMING ZHAO.

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Genqiang Liu is supported by NSFC (11771122) and NSF of Henan Province (202300410046).

Kaiming Zhao is supported by NSFC (11871190) and NSERC (311907-2015).

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LIU, G., ZHAO, K. THE CATEGORY OF WEIGHT MODULES FOR SYMPLECTIC OSCILLATOR LIE ALGEBRAS. Transformation Groups 27, 1025–1044 (2022). https://doi.org/10.1007/s00031-021-09639-y

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