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Restricted Cohomology of Restricted Lie Superalgebras

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Abstract

Suppose the ground field \(\mathbb{F}\) is an algebraically closed field characteristic of p > 2. In this paper, we investigate the restricted cohomology theory of restricted Lie superalgebras. Algebraic interpretations of low dimensional restricted cohomology of restricted Lie superalgebra are given. We show that there is a family of restricted model filiform Lie superalgebra \(L_{p,p}^\lambda\) structures parameterized by elements \(\lambda \in {\mathbb{F}^p}\). We explicitly describe both the 1-dimensional ordinary and restricted cohomology superspaces of \(L_{p,p}^\lambda\) with coefficients in the 1-dimensional trivial module and show that these superspaces are equal. We also describe the 2-dimensional ordinary and restricted cohomology superspaces of \(L_{p,p}^\lambda\) with coefficients in the 1-dimensional trivial module and show that these superspaces are unequal.

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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

  1. School of Mathematical Sciences, Heilongjiang University, Harbin, 150080, P. R. China

    Ji Xia Yuan

  2. School of Mathematical and Statistics, Northeast Normal University, Changchun, 130024, P. R. China

    Ji Xia Yuan & Liang Yun Chen

  3. Department of Mathematics, Harbin University of Science and Technology, Harbin, 150080, P. R. China

    Yan Cao

Authors
  1. Ji Xia Yuan
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  2. Liang Yun Chen
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  3. Yan Cao
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Correspondence to Liang Yun Chen.

Additional information

Supported by the National Natural Science Foundation of China (Grant Nos. 11601135, 11771069, 11801121 and 12071405), Natural Science Foundation of Heilongjiang Province of China (Grant Nos. QC2017002 and QC2018006), Fundamental Research Funds for the Colleges and Universities in Heilongjiang Province (Grant No. 2020-KYYWF-1018) and Project funded by China Postdoctoral Science Foundation (Grant No. 2018M630311)

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Yuan, J.X., Chen, L.Y. & Cao, Y. Restricted Cohomology of Restricted Lie Superalgebras. Acta. Math. Sin.-English Ser. 38, 2115–2130 (2022). https://doi.org/10.1007/s10114-022-1088-4

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  • DOI: https://doi.org/10.1007/s10114-022-1088-4

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