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}.

Dmitri V Alekseevsky, Andrea Santi (2018). Homogeneous irreducible supermanifolds and graded Lie superalgebras. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 4, 1045-1079 [10.1093/imrn/rnw262].

Homogeneous irreducible supermanifolds and graded Lie superalgebras

Andrea Santi
2018

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 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}.
2018
Dmitri V Alekseevsky, Andrea Santi (2018). Homogeneous irreducible supermanifolds and graded Lie superalgebras. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 4, 1045-1079 [10.1093/imrn/rnw262].
Dmitri V Alekseevsky; Andrea Santi
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/626867
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