Journal of Lie Theory, Vol. 9, No. 2, pp. 305-320 (1999)

Relations between invertibility of Casimir operators and semisimplicity of quadratic Lie superalgebras

Hedi Benamor and Sad Benayadi

Universite de Metz, Departement de Mathematiques, Ile du Saulcy, 57045 METZ Cedex 01, France, benamor@poncelet.univ-metz.fr, benayadi@poncelet.univ-metz.fr

Abstract: We choose a definition of semisimplicity for Lie superalgebras. As a consequence of this definition: semisimple Lie superalgebras are rigid. We study relation between semisimplicity and invertibility of Casimir operator in the case of a quadratic Lie superalgebra $\scriptstyle({\frak g} = {\frak g}_{\bar 0} \oplus {\frak g}_{\bar 1}, B)$. In particular, we show that if the representation of the Lie algebra $\scriptstyle{\frak g}_{\bar 0}$ on $\scriptstyle{\frak g}_{\bar 1}$ is completely reducible then semisimplicity is equivalent to the invertibility of Casimir operator.

Keywords: semisimple Lie superalgebras, Casimir operators, invertibility, quadratic Lie superalgebras

Classification (MSC91): 17B20

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