On indecomposable solvable Lie superalgebras having a Heisenberg nilradical

Abstract

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.