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Non-weight modules over the mirror Heisenberg-Virasoro algebra

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Abstract

In this paper, we study irreducible non-weight modules over the mirror Heisenberg-Virasoro algebra \({\cal D}\), including Whittaker modules, \({\cal U}\left( {{\mathbb{C}d_0}} \right)\)-free modules and their tensor products. More precisely, we give the necessary and sufficient conditions for the Whittaker modules to be irreducible. We determine all the \({\cal D}\)-module structures on \({\cal U}\left( {{\mathbb{C}d_0}} \right)\), and find the necessary and sufficient conditions for these modules to be irreducible. At last, we determine the necessary and sufficient conditions for the tensor products of Whittaker modules and \({\cal U}\left( {{\mathbb{C}d_0}} \right)\)-free modules to be irreducible, and obtain that any two such tensor products are isomorphic if and only if the corresponding Whittaker modules and \({\cal U}\left( {{\mathbb{C}d_0}} \right)\)-free modules are isomorphic. These lead to many new irreducible non-weight modules over \({\cal D}\).

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Acknowledgements

The first author was supported by China Scholarship Council (Grant No. 201906340096) and National Natural Science Foundation of China (Grant Nos. 11771410 and 11931009). The second author was supported by National Natural Science Foundation of China (Grant No. 11801066). The third author was supported by National Natural Science Foundation of China (Grant No. 11871190) and Natural Sciences and Engineering Research Council of Canada (Grant No. 311907-2020). This work was carried out during the visit of the first author at Wilfrid Laurier University in 2019–2021. The authors thank the referees for helpful suggestions to improve the paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

    Dongfang Gao

  2. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Yao Ma

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

    Kaiming Zhao

  4. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China

    Kaiming Zhao

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  1. Dongfang Gao
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  2. Yao Ma
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Correspondence to Kaiming Zhao.

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Gao, D., Ma, Y. & Zhao, K. Non-weight modules over the mirror Heisenberg-Virasoro algebra. Sci. China Math. 65, 2243–2254 (2022). https://doi.org/10.1007/s11425-021-1939-5

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