IVe present a reduction of'the adjoint representation of the Lie superalgebra . s i ( % , I ) and astudy of the quotient algebra B(,,,, = LI/U(C -c ) + L L (~ -kc).ivhere c. I; are two complex numbers. lrnder some additional conditions, we prove that every irreducible infinite dimensional reprew~tation of B(,,k, is faithful. and that B1,,h) is a primitive algebra. LVe give espliciily a set of generators of primitive degenerate ideal of infinite codimension. Essentialiv \ve prove that any minimal primitive ideal of U ( s l J P . 1 ) ) is generated, as a 2-hided ideal, by its intersection with the algebra, of eo-invariants.
We generalize to the case of Lie superalgebras the classical symplectic double extension of symplectic Lie algebras introduced in [2]. We use this concept to give an inductive description of nilpotent homogeneoussymplectic Lie superalgebras. Several examples are included to show the existence of homogeneous quadratic symplectic Lie superalgebras other than evenquadratic even-symplectic considered in [6]. We study the structures of even (resp. odd)-quadratic odd (resp. even)-symplectic Lie superalgebras and oddquadratic 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 [6]. Finally, we generalize to the case of homogeneous quadratic symplectic Lie superargebras some relations between even-quadratic even-symplectic Lie superalgebras and Manin superalgebras established in [6].
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