Zheng, Keli; Zhang, Yongzheng On \((\alpha, \beta, \gamma)\)-derivations of Lie superalgebras. (English) Zbl 1370.17023 Int. J. Geom. Methods Mod. Phys. 10, No. 10, Article ID 1350050, 18 p. (2013). 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. Cited in 6 Documents MSC: 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras 17B05 Structure theory for Lie algebras and superalgebras Keywords:Lie superalgebras; generalized derivations; \((\alpha, \beta, \gamma)\)-derivations PDFBibTeX XMLCite \textit{K. Zheng} and \textit{Y. Zhang}, Int. J. Geom. Methods Mod. Phys. 10, No. 10, Article ID 1350050, 18 p. (2013; Zbl 1370.17023) Full Text: DOI References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.