LIE SUPERALGEBRAS WITH SOME HOMOGENEOUS STRUCTURES

We generalize to the case of Lie superalgebras the classical symplectic double extension of symplectic Lie algebras introduced in [A. Aubert, Structures affines et pseudo-metriques invariantes a gauche sur des groupes de Lie, These, Universite Montpellier II (1996)]. We use this concept to give an inductive description of nilpotent homogeneous-symplectic Lie superalgebras. Several examples are included to show the existence of homogeneous quadratic symplectic Lie superalgebras other than even-quadratic even-symplectic considered in [E. Barreiro and S. Benayadi, Quadratic symplectic Lie superalgebras and Lie bi-superalgebras, J. Algebra 321(2) (2009) 582-608]. We study the structures of even (respectively, odd)-quadratic odd (respectively, even)-symplectic Lie superalgebras and odd-quadratic odd-symplectic Lie superalgebras and we give its inductive descriptions in terms of quadratic generalized double extensions and odd quadratic generalized double extensions. This study complete the inductive descriptions of homogeneous quadratic symplectic Lie superalgebras started in [E. Barreiro and S. Benayadi, Quadratic symplectic Lie superalgebras and Lie bi-superalgebras, J. Algebra 321(2) (2009) 582-608]. Finally, we generalize to the case of homogeneous quadratic symplectic Lie superalgebras some relations between even-quadratic even-symplectic Lie superalgebras and Manin superalgebras established in [E. Barreiro and S. Benayadi, Quadratic symplectic Lie superalgebras and Lie bi-superalgebras, J. Algebra 321(2) (2009) 582-608].