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On \((\alpha, \beta, \gamma)\)-derivations of Lie superalgebras. (English) Zbl 1370.17023

Summary: This paper is primarily concerned with \((\alpha, \beta, \gamma)\)-derivations of finite-dimensional Lie superalgebras over the field of complex numbers. Some properties of \((\alpha, \beta, \gamma)\)-derivations of the Lie superalgebras are obtained. In particular, two examples for \((\alpha, \beta, \gamma)\)-derivations of low-dimensional non-simple Lie superalgebras are presented and the super-spaces of \((\alpha, \beta, \gamma)\)-derivations for simple Lie superalgebras are determined. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie superalgebras. A special case for the generalization of 1-cocycles with respect to the adjoint representation is exactly \((\alpha, \beta, \gamma)\)-derivations. Furthermore, two-dimensional twisted cocycles of the adjoint representation are investigated in detail.

MSC:

17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
17B05 Structure theory for Lie algebras and superalgebras
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[1] DOI: 10.1016/0370-1573(77)90066-7 · doi:10.1016/0370-1573(77)90066-7
[2] DOI: 10.1103/RevModPhys.47.573 · Zbl 0557.17004 · doi:10.1103/RevModPhys.47.573
[3] DOI: 10.1007/978-1-4757-1910-9 · doi:10.1007/978-1-4757-1910-9
[4] DOI: 10.1016/0001-8708(77)90017-2 · Zbl 0366.17012 · doi:10.1016/0001-8708(77)90017-2
[5] DOI: 10.1007/BFb0070929 · doi:10.1007/BFb0070929
[6] DOI: 10.1081/AGB-120021887 · Zbl 1125.17313 · doi:10.1081/AGB-120021887
[7] DOI: 10.1080/00927870903236228 · Zbl 1211.17006 · doi:10.1080/00927870903236228
[8] DOI: 10.1007/s11202-009-0049-9 · Zbl 1221.17019 · doi:10.1007/s11202-009-0049-9
[9] DOI: 10.1016/j.jalgebra.2010.09.032 · Zbl 1218.17011 · doi:10.1016/j.jalgebra.2010.09.032
[10] DOI: 10.1023/A:1021172031215 · doi:10.1023/A:1021172031215
[11] DOI: 10.4134/JKMS.2010.47.3.495 · Zbl 1191.16039 · doi:10.4134/JKMS.2010.47.3.495
[12] Maksa G., Proc. Amer. Math. Soc. 81 pp 406– (1981)
[13] DOI: 10.1016/j.jalgebra.2008.09.007 · Zbl 1218.17010 · doi:10.1016/j.jalgebra.2008.09.007
[14] DOI: 10.1006/jabr.1999.8250 · Zbl 0961.17010 · doi:10.1006/jabr.1999.8250
[15] DOI: 10.1016/j.geomphys.2007.10.005 · Zbl 1162.17014 · doi:10.1016/j.geomphys.2007.10.005
[16] DOI: 10.1016/j.laa.2008.11.003 · Zbl 1215.17013 · doi:10.1016/j.laa.2008.11.003
[17] DOI: 10.1142/S0219887810004737 · Zbl 1217.39034 · doi:10.1142/S0219887810004737
[18] DOI: 10.1063/1.532508 · Zbl 0928.17023 · doi:10.1063/1.532508
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