The centralizer of~$\frak{sl}(0,n)$ in the generalized Witt Lie superalgebra over fields of prime characteristic

Abstract

This paper considers centralizers of the Lie superalgebra~$\frak{sl}(0,n)$ over prime characteristic fields. Using homological methods, the centralizers of the even and odd parts of ~$\frak{sl}(0,n)$ in the generalized Witt Lie superalgebra are calculated and a summary of their structural properties is provided.

Keywords

• Centralizer
• Lie superalgebra
• zero cohomology group

References

1. B. DeWitt, Supermanifolds, Second edition, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 1992.
2. P. Fayet and S. Ferrara, Supersymmetry, Phys. Rep., 32C(5) (1977), 249-334.
3. G. Hochschild and J.-P. Serre, Cohomology of Lie algebras, Ann. of Math. (2), 57 (1953), 591-603.
4. N. Jacobson, Lie Algebras, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, New York-London, 1962.
5. N Jacobson, Lie Algebras, Republication of the 1962 original, Dover Publications, New York, 1979.
6. J. Q. Liu, Q. G. Qian and K. L. Zheng, The centralizer of~$\frak{sl}(0,3)$ in the generalized Witt Lie superalgebra over fields of prime characteristic, J. Northeast Normal Univ. (Natural Science Edition), 52(1) (2020), 10-12. (Chinese)
7. J. Q. Liu, Q. G. Qian and K. L. Zheng, Centralizers of~$\frak{sl}(2,1)$ and~$\frak{sl}(1,2)$ in the generalized Witt Lie superalgebra over fields of prime characteristics, Journal of Harbin University of Science and Technology, 26(1) (2021), 144-148. (Chinese)
8. R. Mokhtari, R. Hoseini Sani and A. Chenaghlou, Supersymmetry approach to the Dirac equation in the presence of the deformed Woods-Saxon potential, Eur. Phys. J. Plus, 134 (2019), 446 (7 pp).

9. J. J. Rotman, An Introduction to Homological Algebra, 2nd ed, Universitext, Springer, New York, 2009.
10. M. Scheunert, The Theory of Lie Superalgebras. An Introduction, Lecture Notes in Math., 716, Springer, Berlin, 1979.
11. M. Scheunert and R. B. Zhang, Cohomology of Lie superalgebras and their generalizations, J. Math. Phys., 39(9) (1998), 5024-5061.
12. L. Y. Tian, Y. Hou and K. L. Zheng, The centralizer of~$\frak{gl}(0,3)$ in the generalized Witt Lie superalgebra over fields of prime characteristic, Natur. Sci. J. Harbin Normal Univ., 32(2) (2016), 5-7. (Chinese)
13. Y. Zhang, Finite-dimensional Lie superalgebras of Cartan type over fields of prime characteristic, Chinese Sci. Bull., 42(9) (1997), 720-724.
14. Q. Zhang and Y. Zhang, Derivation algebras of the modular Lie superalgebras $W$ and $S$ of Cartan-type, Acta Math. Sci. Ser. B (Engl. Ed.), 20(1) (2000), 137-144.
15. D. Mao and K. L. Zheng, Centralizer of general linear Lie superalgebra in generalized Witt Lie superalgebra, J. Jilin Univ. Sci., 60(1) (2022), 27-34. (Chinese)