Journal of Lie Theory, Vol. 10, No. 1, pp. 207-212 (2000)

Fitting decomposition of Casimir operators of quadratic Lie superalgebras.

Hedi Benamor and Saïd Benayadi

Université de Metz,
Département de Mathématiques
Ile du Saulcy, 57045 METZ Cedex 01

Abstract: Semisimple Lie superalgebras are Lie superalgebras with non-degenerate Killing forms. In this paper we show that a quadratic Lie superalgebra $({\frak g}, B)$ is semisimple if and only if its Casimir operator, $\Omega_B$, associated to $B$ is invertible. This result anables us to characterize quadratic Lie superalgebras with nilpotent Casimir operators: $\Omega_B$ is nilpotent if and only if ${\frak g}$ is a Lie superalgebra without any nonzero semisimple ideal.

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