Socle and some invariants of quadratic Lie superalgebras
Let \(\mathfrak g\) be a Lie superalgebra with a nondegenerate bilinear function \(B(x,y)\) such that \(B(x,y)=(-1)^{ab}B(y,x)\) if \(x\in {\mathfrak g}_a, y\in {\mathfrak g}_b\); \(B([x,y],z)=B(x,[y,z])\) for all \(x,y,z\in \mathfrak g\); \(B(x,y)=0\) if \(x\in {\mathfrak g}_0, y\in {\mathfrak g}_1\). In the first sections of the paper minimal graded ideals and the socle of \(\mathfrak g\) are considered. These results are applied to a characterization of simple Lie superalgebras. In the third section a problem of uniqueness of the function \(B\) is considered. In the final section some properties of new invariants related to the socle are discussed and a converse of Koszul's theorem is established.
scientific article; zbMATH DE number 1903595