An analogue of the Kostant criterion for quadratic Lie superalgebras

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Let r be a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form, let p be a finite dimensional complex super vector space with a non-degenerate super-symmetric bilinear form, and let ν: r(p) be a Lie superalgebra homomorphism. In this paper, we give a necessary and sufficient condition for r to be a quadratic Lie superalgebra. The criterion obtained is an analogue of a constancy condition given by Kostant in the Lie algebra setting. As an application, we prove an analogue of the Parthasarathy's formula for the square of the Dirac operator attached to a pair of quadratic Lie superalgebras.