On indecomposable solvable Lie superalgebras having a Heisenberg nilradical

All solvable, indecomposable, finite-dimensional, complex Lie superalgebras whose first derived ideal lies in its nilradical, and whose nilradical is a Heisenberg Lie superalgebra associated to a ℤ2-homogeneous supersymplectic complex vector superspace V, are here classified up to isomorphism. It is shown that they are all of the form g=h ⊕ a, where a is even and consists of non-ad g-nilpotent elements. All these Lie superalgebras depend on an element γ in the dual space a∗ and on a pair of linear maps defined on a, and taking values in the Lie algebras naturally associated to the even and odd subspaces of V; namely, if the supersymplectic form is even, the pair of linear maps defined on a take values in GP(V0), and (V1), respectively, whereas if the supersymplectic form is odd these linear maps take values on (V0) ≈ (V1). When the supersymplectic form is even, a bilinear, skew-symmetric form defined on 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. © 2016 World Scientific Publishing Company.

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