On complex lie supergroups and split homogeneous supermanifolds

Abstract

It is well known that the category of real Lie supergroups is equivalent to the category of the so-called (real) Harish-Chandra pairs, see [DM], [Kost], [Kosz]. That means that a Lie supergroup depends only on the underlying Lie group and its Lie superalgebra with certain compatibility conditions. More precisely, the structure sheaf of a Lie supergroup and the supergroup morphisms can be explicitly described in terms of the corresponding Lie superalgebra. In this paper we give a proof of this result in the complex-analytic case. Furthermore, if (G, $$ \mathcal{O} $$ G ) is a complex Lie supergroup and H ⊂ G is a closed Lie subgroup, i.e., it is a Lie subsupergroup of (G, $$ \mathcal{O} $$ G ) and its odd dimension is zero, we show that the corresponding homogeneous supermanifold (G/H, $$ \mathcal{O} $$ G/H ) is split. In particular, any complex Lie supergroup is a split supermanifold. It is well known that a complex homogeneous supermanifold may be nonsplit (see, e.g., [OS1]). We find here necessary and sufficient conditions for a complex homogeneous supermanifold to be split.