Homogeneous irreducible supermanifolds and graded Lie superalgebras

A depth one grading g=g^{−1}+g^{0}+g^{1}+...+g^{l} of a finite dimensional Lie superalgebra g is called nonlinear irreducible if the isotropy representation is irreducible and g^{1}\neq (0). An example is the full prolongation of an irreducible linear Lie superalgebra g^{0}⊂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⊗\Lambda(C^n), 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/G_{0} where G, G_{0} are Lie supergroups,respectively associated with the Lie superalgebras g and g_{0}:= +_{p≥0} g^{p}.