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

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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