Dirac Operators on Quadratic Lie Superalgebras

Assume thatτis a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form,p is a finite dimensional complex super vector space with a nondegenerate super-symmetric bilinear form,and v:τ→osp(p)is a homomorphism of Lie superalgebras.In this paper,we give a necessary and sufficient condition forτ■p to be a quadratic Lie superalgebra.Then,we define the cubic Dirac operator D(g,τ)on g and give a formula of(D(g,τ))^(2).Finally,we get the Vogan’s conjecture for quadratic Lie superalgebras by D(g,τ).