Infinitesimal deformations of the Lie superalgebra Ln,m

In this paper, we continue the study of the infinitesimal deformations of the Lie superalgebra Ln,mLn,m that we have started in [M. Bordemann, J.R. Gómez, Yu. Khakimdjanov, R.M. Navarro, Some deformations of nilpotent Lie superalgebras, J. Geom. Phys. 57 (2007) 1391–1403]. These deformations allow us to obtain all filiform Lie superalgebras. In [M. Bordemann, J.R. Gómez, Yu. Khakimdjanov, R.M. Navarro, Some deformations of nilpotent Lie superalgebras, J. Geom. Phys. 57 (2007) 1391–1403], we gave a method that allows us to determine the dimension of the space of deformations of type Hom(S2(L1n,m),L0n,m) and we calculated a basis of the aforementioned space of deformations for n≥2m−1n≥2m−1. In this paper, we conclude the study by developing a method to calculate a basis of the space of deformations Hom(S2(L1n,m),L0n,m) for the rest of possibilities n<2m−1n<2m−1. We particularize for even nn and also give an algorithm for computing a cocycle basis for the given concrete dimensions nn and mm.