On Deformations of the Filiform Lie Superalgebra L_n,m

Abstract

Much work was done for filiform Lie algebras defined by Vergne [Vergne, M. (1970). Cohomologie des algèbres de Lie nilpotentes. Apllication à l'etude de la variété des algèbres de Lie nilpotentes. Bull. Soc. Math. France 98:81–116]. An interesting fact is that these algebras are obtained by deformations of the filiform Lie algebra L _n,m . This was used for classification in Gómez et al. [Gómez, J. R., Jimenéz-Merchán, A., Khakimdjanov, Y. (1998). Low-dimensional filiform Lie algebras. J. Pure Appl. Algebras 130:133–158]. Like filiform Lie algebras, filiform Lie superalgebras are obtained by nilpotent deformations of the Lie superalgebra L _n,m . In this paper, we recall this fact and we study even cocycles of the superalgebra L _n,m which give these nilpotent deformations. A family of independent bilinear maps will help us to describe these cocycles. At the end an evaluation of the dimension of the space is established. The description of these cocycles can help us to get some classifications which were done by Gilg [Gilg, M. (2000). Super-algèbres de Lie Nilpotentes. Thèse, Université de Haute-Alsace; Gilg, M. (2001). Low-dimensional filiform Lie superalgebras. Revista Matemática Complutense XIV(2):463–478].

Key Words:

• Super Lie algebras
• Cohomology
• Deformation
• Filiform

2000 Mathematics Subject Classification:

• 17B30
• 17B56

Acknowledgment

I should like to thank Y. Khakimdjanov and M. Goze for support and numerous discussions.

Notes

^#Communicated by J. Alev.