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.

Odd-quadratic Lie superalgebras with a weak filiform module as an odd part

“…Proof. We have the result by the paper [15], since by Proposition 3.1 g0 is a nilpotent Lie algebra, and π(x) := (ad g (x))| g1 is nilpotent for all x ∈ g0.…”

Quadratic symplectic Lie superalgebras with a filiform module as an odd part

“…Let us remark that the structure constants C k ij determine the structure of a nilpotent Lie algebra on g0 and E k ij make the representation ρ g nilpotent. Consequently, g0 is nilpotent and also the corresponding adjoint representation of g0 over g1, which leads to the fact that the Lie superalgebra g = g0 ⊕ g1 is nilpotent [26].…”

Cohomologically rigid solvable Lie superalgebras with model filiform and model nilpotent nilradical

“…Studying solvable Lie superalgebras, on the other hand, represents more difficulties than studying solvable Lie algebras, see [32]. For instance, Lie's theorem does not hold true in general and neither its corollaries.…”