Bulletin of the Korean Mathematical Society (대한수학회보)

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A NOTE ON THE FINITE-DIMENSIONAL ODD CONTACT SUPERALGEBRA OVER A FIELD OF PRIME CHARACTERISTIC

  • Received : 2020.07.28
  • Accepted : 2021.07.23
  • Published : 2021.09.30
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Abstract

This paper aims to analyze the PTG module for the finite-dimensional odd Contact superalgebra over a field of prime characteristic by using the method of Hu and Shen's mixed product realization. The general acting law in odd Contact superalgebra is obtained. In addition, the structure and irreducibility of graded module for odd Contact superalgebra are discussed.

Keywords

Acknowledgement

This work was financially supported by NSFC 11501274 and LNECFC L2015203.

References

  1. Y. Chen and W. Liu, Finite-dimensional odd contact superalgebras over a field of prime characteristic, J. Lie Theory 21 (2011), no. 3, 729-754.
  2. R. Farnsteiner, Extension functors of modular Lie algebras, Math. Ann. 288 (1990), no. 4, 713-730. https://doi.org/10.1007/BF01444560
  3. R. Farnsteiner and H. Strade, Shapiro's lemma and its consequences in the cohomology theory of modular Lie algebras, Math. Z. 206 (1991), no. 1, 153-168. https://doi.org/10.1007/BF02571333
  4. J. Fu, Q. Zhang, and C. Jiang, The Cartan-type modular Lie superalgebra KO, Comm. Algebra 34 (2006), no. 1, 107-128. https://doi.org/10.1080/00927870500346065
  5. R. R. Holmes, Simple restricted modules for the restricted contact Lie algebras, Proc. Amer. Math. Soc. 116 (1992), no. 2, 329-337. https://doi.org/10.2307/2159737
  6. N. H. Hu, The graded modules for the graded contact Cartan algebras, Comm. Algebra 22 (1994), no. 11, 4475-4497. https://doi.org/10.1080/00927879408825082
  7. Q. Mu, L. Ren, and Y. Zhang, Ad-nilpotent elements, isomorphisms, and the Weisfeiler filtration of infinite-dimensional modular odd contact superalgebras, Comm. Algebra 39 (2011), no. 10, 3581-3593. https://doi.org/10.1080/00927872.2010.502162
  8. G. Y. Shen, Graded modules of graded Lie algebras of Cartan type. I. Mixed products of modules, Sci. Sinica Ser. A 29 (1986), no. 6, 570-581.
  9. G. Y. Shen, Graded modules of graded Lie algebras of Cartan type. II. Positive and negative graded modules, Sci. Sinica Ser. A 29 (1986), no. 10, 1009-1019.
  10. G. Y. Shen, Graded modules of graded Lie algebras of Cartan type. III. Irreducible modules, Chinese Ann. Math. Ser. B 9 (1988), no. 4, 404-417.
  11. B. Shu and Y.-F. Yao, Character formulas for restricted simple modules of the special superalgebras, Math. Nachr. 285 (2012), no. 8-9, 1107-1116. https://doi.org/10.1002/mana.201000064
  12. B. Shu and C. Zhang, Representations of the restricted Cartan type Lie superalgebra W(m, n, 1), Algebr. Represent. Theory 14 (2011), no. 3, 463-481. https://doi.org/10.1007/s10468-009-9198-6
  13. H. Strade and R. Farnsteiner, Modular Lie algebras and their representations, Monographs and Textbooks in Pure and Applied Mathematics, 116, Marcel Dekker, Inc., New York, 1988.
  14. Y. Su and R. B. Zhang, Generalised Verma modules for the orthosympletic Lie superalgebra ospk|2, J. Algebra 357 (2012), 94-115. https://doi.org/10.1016/j.jalgebra.2012.01.026
  15. Y.-F. Yao and B. Shu, Restricted representations of Lie superalgebras of Hamiltonian type, Algebr. Represent. Theory 16 (2013), no. 3, 615-632. https://doi.org/10.1007/s10468-011-9322-2
  16. J. Yuan, W. Liu, and W. Bai, Associative forms and second cohomologies of Lie superalgebras HO and KO, J. Lie Theory 23 (2013), no. 1, 203-215.
  17. Y. Z. Zhang and W. D. Liu, Modular Lie Superalgebras, Science Press, Beijing, 2004.
  18. K. Zheng and Y. Zhang, Some properties of generalized reduced Verma modules over Z-graded modular Lie superalgebras, Czechoslovak Math. J. 67(142) (2017), no. 3, 699-713. https://doi.org/10.21136/CMJ.2017.0050-16