On representations of Lie superalgebras (Q1101539)

Let \({\mathfrak g}={\mathfrak g}_ 0+{\mathfrak g}_ 1\) be a Lie superalgebra. The authors wish to classify and construct irreducible representations of \({\mathfrak g}\) by using results about representations of \({\mathfrak g}_ 0\). Thus let \(V_ 0\) be a \({\mathfrak g}_ 0\)-module and assume that there exists a bilinear map \(B: {\mathfrak g}_ 1\times {\mathfrak g}_ 1\to {\mathfrak g}\ell (V_ 0)\). Under certain canonical conditions on B they prove that there exists a representation V of \({\mathfrak g}\) such that the degree zero subspace of V is \(V_ 0\). Further this representation is irreducible if and only if \(V_ 0\) is an irreducible module for the subalgebra of \({\mathfrak g}\ell (V_ 0)\) generated by \(\{\) \({\mathfrak g}_ 0,B(x,y):\) \(x_ y\in {\mathfrak g}_ 1\}\). They also consider the case of unitary extensions of a unitary \({\mathfrak g}_ 0\)-module. Using these results they classify explicitly all representations of osp((2/1) and all unitary representations of type A.