Acta Mathematica Sinica, English Series

, Volume 34, Issue 10, pp 1578–1588 | Cite as

Irreducible Wakimoto-like Modules for the Lie Superalgebra D(2, 1;α)

  • Jin Cheng
  • Zi Ting ZengEmail author


By using the idea of Wakimoto’s free field, we construct a class of representations for the Lie superalgebra D(2, 1; α) on the tensor product of a polynomial algebra and an exterior algebra involving one parameter λ. Then we obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter λ satisfies \((\lambda + m)(\lambda - \frac{1+\alpha}{\alpha}m) \neq 0\) for any m ∈ ℤ+.


Lie superalgebra representation Wakimoto’s free field 

MR(2010) Subject Classification

17B10 17B25 


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We thank Dr. Yong Jie Wang for some useful discussion. We thank the referees for their time and comments.


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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsShandong Normal UniversityJi’nanP. R. China
  2. 2.School of Mathematical ScienceBeijing Normal UniversityBeijingP. R. China

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