Homogeneous irreducible supermanifolds and graded Lie superalgebras

A depth one grading g = g−1 g0 g1 · · · g of a finite dimensional Lie superalgebra g is called nonlinear irreducible if the isotropy representation adg0|g−1 is irreducible and g1 = (0). An example is the full prolongation of an irreducible linear Lie superalgebra g0 ⊂ gl(g−1) of finite type with non-trivial first prolongation. We prove that a complex Lie superalgebra g which admits a depth one transitive nonlinear irreducible grading is a semisimple Lie superalgebra with the socle s (Cn), where s is a simple Lie superalgebra, and we describe such gradings. The graded Lie superalgebra g defines an isotropy irreducible homogeneous supermanifold M = G/G0 where G, G0 are Lie supergroups, respectively associated with the Lie superalgebras g and g0 :=p≥0 gp.