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Communicated by Vera Serganova MSC: primary 17B30, 17B70, 81R05 secondary 15A63, 17B81 Keywords: Solvable Nilpotent Lie (super)algebras Heisenberg Lie (super)algebras ad-Invariant supersymmetric bilinear formsFinite-dimensional complex Lie superalgebras of Heisenberg type obtained from a given Z 2 -homogeneous supersymplectic form defined on a vector superspace, are classified up to isomorphism. Those arising from even supersymplectic forms, have an ordinary Heisenberg Lie algebra as its underlying even subspace, whereas those arising from odd supersymplectic forms get based on abelian Lie algebras. The question of whether this sort of Heisenberg Lie superalgebras do or do not support a given invariant supergeometric structure is addressed, and it is found that none of them do. It is proved, however, that 1-dimensional extensions by appropriate Z 2 -homogeneous derivations do. Such 'appropriate' derivations are characterized, and the invariant supergeometric structures carried by the extensions they define are fully described. Furthermore, necessary and sufficient conditions are obtained in order that any two 1-dimensional extensions by Z 2 -homogeneous derivations be isomorphic; also, necessary and sufficient conditions are obtained in order that any two extensions carrying invariant supergeometric structures be isometric.

In this work we show that for n ≥ 1 n\geq 1 , every finite ( 2 n + 3 ) (2n+3) -dimensional contact nilpotent Lie algebra g \mathfrak {g} can be obtained as a double extension of a contact nilpotent Lie algebra h \mathfrak {h} of codimension 2. As a consequence, for n ≥ 1 n\geq 1 , every ( 2 n + 3 ) (2n+3) -dimensional contact nilpotent Lie algebra g \mathfrak {g} can be obtained from the 3-dimensional Heisenberg Lie algebra h 3 \mathfrak {h}_3 , by applying a finite number of successive series of double extensions. As a byproduct, we obtain an alternative proof of the fact that a ( 2 n + 1 ) (2n+1) -nilpotent Lie algebra g \mathfrak {g} is a contact Lie algebra if and only if it is a central extension of a nilpotent symplectic Lie algebra.

All solvable, indecomposable, finite-dimensional, complex Lie superalgebras [Formula: see text] whose first derived ideal lies in its nilradical, and whose nilradical is a Heisenberg Lie superalgebra [Formula: see text] associated to a [Formula: see text]-homogeneous supersymplectic complex vector superspace [Formula: see text], are here classified up to isomorphism. It is shown that they are all of the form [Formula: see text], where [Formula: see text] is even and consists of non-[Formula: see text]-nilpotent elements. All these Lie superalgebras depend on an element [Formula: see text] in the dual space [Formula: see text] and on a pair of linear maps defined on [Formula: see text], and taking values in the Lie algebras naturally associated to the even and odd subspaces of [Formula: see text]; namely, if the supersymplectic form is even, the pair of linear maps defined on [Formula: see text] take values in [Formula: see text], and [Formula: see text], respectively, whereas if the supersymplectic form is odd these linear maps take values on [Formula: see text]. When the supersymplectic form is even, a bilinear, skew-symmetric form defined on [Formula: see text] is further needed. Conditions on these building data are given and the isomorphism classes of the resulting Lie superalgebras are described in terms of appropriate group actions.