Socle and some invariants of quadratic Lie superalgebras

We construct some new invariants of the quadratic Lie superalgebras. These invariants are closely related to the socle and the decomposability of quadratic Lie superalgebras. Next, we establish some relations between these invariants. We use these relations in order to characterize the simple Lie algebras and the basic classical Lie superalgebras among the quadratic Lie superalgebras with completely reducible action of the even part on the odd part and to discuss the problem of characterization of quadratic Lie superalgebras having a unique (up to a constant) quadratic structure. We give a characterization of the socle of a quadratic Lie superalgebra. Several examples are included to show that the situations in the super case change drastically. Lower and upper bounds of dimension of the vector space of even supersymmetric invariant bilinear forms on a quadratic Lie superalgebra are obtained. Finally, we give converses of Koszulʹs theorems.