Homogeneous irreducible supermanifolds and graded Lie superalgebras

Abstract

Abstract. 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 ad g 0 | g −1 is irreducible and g 1 = (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⊗Λ(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/G0 where G, G0 are Lie supergroups respectively associated with the Lie superalgebras g and g0 := p≥0 g p .