Lie Supergroups Obtained from 3-Dimensional Lie Superalgebras Associated to the Adjoint Representation and Having a 2-Dimensional Derived Ideal

A possible notion of superspace associated to a given 3-dimensional Lie algebra g0 might be a Lie superalgebra g g0 ⊕ g1. If one is to introduce a minimal set of assumptions, it seems quite natural to consider those superalgebras for which g1 g0 and for which the g0 action on g1 is the adjoint representation. A complete classification of the real and complex 3, 3 -dimensional Lie superalgebras having precisely this restriction was recently obtained in 1 . The classification was given in terms of the dimension of the derived ideal g0 g0, g0 . The most trivial case was that corresponding to dim g0 3 whereas the most difficult case was that of dim g ′ 0 0. Our aim in this paper is to explicitly produce the real and complex Lie supergroups associated to the Lie superalgebras classified in 1 having a 2-dimensional derived ideal. This is just one step forward in our understanding of the real and complex 3, 3 -dimensional Lie supergroups. In order to maintain our exposition as self-contained as possible, we will summarize the basic statements from 1 . Once we fix the adjoint representation, it is well known see 1, 2 that there are as many Lie superalgebras as bilinear symmetric maps Γ : g0×g0 → g0 satisfying