Geodesic
Cohomology of Hom-Lie superalgebras and $q$-deformed Witt superalgebra
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 3, pp. 721-761.

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Hom-Lie algebra (superalgebra) structure appeared naturally in $q$-deformations, based on $\sigma $-derivations of Witt and Virasoro algebras (superalgebras). They are a twisted version of Lie algebras (superalgebras), obtained by deforming the Jacobi identity by a homomorphism. In this paper, we discuss the concept of $\alpha ^k$-derivation, a representation theory, and provide a cohomology complex of Hom-Lie superalgebras. Moreover, we study central extensions. As application, we compute derivations and the second cohomology group of a twisted ${\rm osp}(1,2)$ superalgebra and $q$-deformed Witt superalgebra.
DOI : 10.1007/s10587-013-0049-6
Classification : 17A70, 17B56, 17B68
Keywords: Hom-Lie superalgebra; derivation; cohomology; $q$-deformed superalgebra 
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Ammar, Faouzi; Makhlouf, Abdenacer; Saadaoui, Nejib. Cohomology of Hom-Lie superalgebras and $q$-deformed Witt superalgebra. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 3, pp. 721-761. doi : 10.1007/s10587-013-0049-6. http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0049-6/

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