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On finite-dimensional representations of finite W-superalgebras

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Abstract

We first formulate and prove a version of Premet’s conjecture for finite W-superalgebras associated with basic Lie superalgebras. As in the case of W-algebras, Premet’s conjecture is very close to giving a classification of finite-dimensional simple modules of finite W-superalgebras. In the case of basic type I Lie superalgebras, we classify the finite-dimensional simple supermodules with the integral central character and give an algorithm to compute their characters based on the \({\mathfrak{g}_{\bar 0}}\)-rough structure of \(\mathfrak{g}\)-modules.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11801113) and Research Institute for Mathematical Sciences (RIMS), an International Joint Usage/Research Center Located in Kyoto University. This work was motivated by communications with Tomuyuki Arakawa and a part of it was written during the author’s visit to Tomuyuki Arakawa at Research Institute for Mathematical Sciences. The author is indebted much to him for fruitful and helpful discussion. He thanks Hao Chang for the tremendous help in improving the language. He also thanks for the helpful communications from Bin Shu and comments from Yang Zeng. Finally, the author thanks the referees for their numerous helpful comments.

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  1. College of Mathematical Sciences, Harbin Engineering University, Harbin, 150001, China

    Husileng Xiao

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  1. Husileng Xiao
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Xiao, H. On finite-dimensional representations of finite W-superalgebras. Sci. China Math. 66, 1737–1750 (2023). https://doi.org/10.1007/s11425-021-2048-x

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  • DOI: https://doi.org/10.1007/s11425-021-2048-x

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