Singular quadratic Lie superalgebras

In this paper, we generalize some results on quadratic Lie algebras to quadratic Lie superalgebras, by applying graded Lie algebras tools. We establish a one-to-one correspondence between non-Abelian quadratic Lie superalgebra structures and nonzero even super-antisymmetric 3-forms satisfying a structure equation. An invariant number of quadratic Lie superalgebras is then defined, called the dup-number. Singular quadratic Lie superalgebras (i.e. those with nonzero dup-number) are studied. We show that their classification follows the classifications of O(m)-adjoint orbits of o(m) and Sp(2n)-adjoint orbits of sp(2n). An explicit formula for the quadratic dimension of singular quadratic Lie superalgebras is also provided. Finally, we discuss a class of 2-nilpotent quadratic Lie superalgebras associated to a particular super-antisymmetric 3-form.