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Keywords:
modular Lie superalgebra; restricted Lie superalgebra; filtration
Summary:
In this paper, we continue to investigate some properties of the family $\Gamma $ of finite-dimensional simple modular Lie superalgebras which were constructed by X. N. Xu, Y. Z. Zhang, L. Y. Chen (2010). For each algebra in the family, a filtration is defined and proved to be invariant under the automorphism group. Then an intrinsic property is proved by the invariance of the filtration; that is, the integer parameters in the definition of Lie superalgebras $\Gamma $ are intrinsic. Thereby, we classify these Lie superalgebras in the sense of isomorphism. Finally, we study the associative forms and Killing forms of these Lie superalgebras and determine which superalgebras in the family are restrictable.
References:
[1] Block, R. E., Wilson, R. L.: Classification of the restricted simple Lie algebras. J. Algebra 114 (1988), 115-259. DOI 10.1016/0021-8693(88)90216-5 | MR 0931904
[2] Bouarroudj, S., Grozman, P., Leites, D.: Classification of finite dimensional modular Lie superalgebras with indecomposable Cartan matrix. SIGMA, Symmetry Integrability Geom. Methods Appl. (electronic only) 5 Paper 060, 63 pages (2009). MR 2529187 | Zbl 1220.17010
[3] Martin, A. J. Calderón, Delgado, J. M. Sánchez: On the structure of graded Lie superalgebras. Mod. Phys. Lett. A 27 (2012), 1250142, 18 pages. DOI 10.1142/S0217732312501428 | MR 2966788
[4] Chen, L. Y., Meng, D. J., Zhang, Y. Z.: The Frattini subalgebra of restricted Lie superalgebras. Acta Math. Sin., Engl. Ser. 22 (2006), 1343-1356. DOI 10.1007/s10114-005-0670-x | MR 2251395 | Zbl 1127.17020
[5] Draper, C., Elduque, A., González, C. Martín: Fine gradings on exceptional simple Lie superalgebras. Int. J. Math. 22 (2011), 1823-1855. DOI 10.1142/S0129167X11007392 | MR 2872534
[6] Fei, Q. Y.: On new simple Lie algebras of Shen Guangyu. Chin. Ann. Math., Ser. B 10 (1989), 448-457. MR 1038379 | Zbl 0695.17004
[7] Kac, V. G.: The classification of the restricted simple Lie algebras over a field with non-zero characteristic. Math. USSR, Izv. 4 (1970), 391-413. DOI 10.1070/IM1970v004n02ABEH000912 | MR 0276286
[8] Kac, V. G.: A description of filtered Lie algebras with which graded Lie algebras of Cartan type are associated. Math. USSR, Izv. 8 (1975), 801-835 \kern 3sp translated from Izv. Akad. Nauk SSSR Ser. Mat. 8 (1975), 800-834 Russian. MR 0369452
[9] Kac, V. G.: Lie superalgebras. Adv. Math. 26 (1977), 8-96. DOI 10.1016/0001-8708(77)90017-2 | MR 0486011 | Zbl 0367.17007
[10] Kac, V. G.: Classification of infinite-dimensional simple linearly compact Lie superalgebras. Adv. Math. (1998), 139 1-55. MR 1652530 | Zbl 0929.17026
[11] Kochetkov, Y., Leites, D.: Simple Lie algebras in characteristic 2 recovered from superalgebras and on the notion of a simple finite group. Algebra, Proc. Int. Conf. Memory A. I. Mal'cev, Novosibirsk/USSR 1989, Contemp. Math. 131 (1992), 59-67. MR 1175822 | Zbl 0765.17006
[12] Leites, D.: Towards classification of simple finite dimensional modular Lie superalgebras. J. Prime Res. Math. 3 (2007), 101-110. MR 2397769 | Zbl 1172.17011
[13] Liu, W. D., Zhang, Y. Z., Wang, X. L.: The derivation algebra of the Cartan-type Lie superalgebra $HO$. J. Algebra 273 (2004), 176-205. DOI 10.1016/j.jalgebra.2003.10.019 | MR 2032456 | Zbl 1162.17308
[14] Liu, W. D., Zhang, Y. Z.: Automorphism groups of restricted Cartan-type Lie superalgebras. Commun. Algebra 34 (2006), 3767-3784. DOI 10.1080/00927870600862615 | MR 2262384 | Zbl 1193.17010
[15] Petrogradskiĭ, V. M.: Identities in the enveloping algebras of modular Lie superalgebras. J. Algebra 145 (1992), 1-21. DOI 10.1016/0021-8693(92)90173-J | MR 1144655
[16] Scheunert, M.: The Theory of Lie Superalgebras. An Introduction. Lecture Notes in Mathematics 716 Springer, Berlin (1979). MR 0537441 | Zbl 0407.17001
[17] Shen, G. Y.: An intrinsic property of the Lie algebra $K(m,n)$. Chin. Ann. Math. 2 (1981), 105-115. Zbl 0498.17009
[18] Shen, G. Y.: New simple Lie algebras of characteristic $p$. Chin. Ann. Math., Ser. B 4 (1983), 329-346. MR 0742032 | Zbl 0507.17007
[19] Strade, H.: The classification of the simple modular Lie algebras. IV: Determining the associated graded algebra. Ann. Math. (2) 138 (1993), 1-59. MR 1230926 | Zbl 0790.17011
[20] Strade, H., Farnsteiner, R.: Modular Lie Algebras and Their Representations. Monographs and Textbooks in Pure and Applied Mathematics 116 Marcel Dekker, New York (1988). MR 0929682 | Zbl 0648.17003
[21] Strade, H., Wilson, R. L.: Classification of simple Lie algebras over algebraically closed fields of prime characteristic. Bull. Am. Math. Soc., New Ser. 24 (1991), 357-362. DOI 10.1090/S0273-0979-1991-16033-7 | MR 1071032 | Zbl 0725.17023
[22] Wang, Y., Zhang, Y. Z.: A new definition of restricted Lie superalgebras. Chinese Kexue Tongbao 44 (1999), 807-813. MR 1733605
[23] Wang, Y., Zhang, Y. Z.: The associative forms of the graded Cartan type Lie superalgebras. Adv. Math., Beijing 29 (2000), 65-70. MR 1769128 | Zbl 1009.17015
[24] Wang, W. Q., Zhao, L.: Representations of Lie superalgebras in prime characteristic. I. Proc. Lond. Math. Soc. 99 (2009), 145-167. DOI 10.1112/plms/pdn057 | MR 2520353 | Zbl 1176.17013
[25] Wang, X. L., Liu, W. D.: Filtered Lie superalgebras of odd Hamiltonian type $HO$. English, Chinese summary Adv. Math., Beijing 36 (2007), 710-720. MR 2417896
[26] Wilson, R. L.: A structural characterization of the simple Lie algebras of generalized Cartan type over fields of prime characteristic. J. Algebra 40 (1976), 418-465. DOI 10.1016/0021-8693(76)90206-4 | MR 0412239 | Zbl 0355.17012
[27] Xu, X. N., Zhang, Y. Z., Chen, L. Y.: The finite-dimensional modular Lie superalgebra $\Gamma$. Algebra Colloq. 17 (2010), 525-540. MR 2660443 | Zbl 1203.17009
[28] Xu, X. N., Chen, L. Y., Zhang, Y. Z.: On the modular Lie superalgebra $\Omega$. J. Pure Appl. Algebra 215 (2011), 1093-1101. DOI 10.1016/j.jpaa.2010.07.014 | MR 2747241
[29] Zhang, Y. Z.: Finite-dimensional Lie superalgebras of Cartan type over fields of prime characteristic. Chin. Sci. Bull. 42 (1997), 720-724. DOI 10.1007/BF03186962 | MR 1460613 | Zbl 0886.17022
[30] Zhang, Y. Z., Nan, J. Z.: Finite-dimensional Lie superalgebras $W(m,n, t)$ and $S(m,n, t)$ of Cartan type. Adv. Math., Beijing 27 (1998), 240-246. MR 1651296
[31] Zhang, Y. Z., Fu, H. C.: Finite-dimensional Hamiltonian Lie superalgebra. Commun. Algebra 30 (2002), 2651-2673. DOI 10.1081/AGB-120003981 | MR 1908231 | Zbl 1021.17017
[32] Zhang, Y. Z., Liu, W. D.: Modular Lie superalgebras. Chinese Science Press Beijing (2004). MR 2100474
[33] Zhang, Y. Z., Zhang, Q. C.: Finite-dimensional modular Lie superalgebra $\Omega$. J. Algebra 321 (2009), 3601-3619. DOI 10.1016/j.jalgebra.2009.01.038 | MR 2517804 | Zbl 1203.17010
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