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.
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Supported by SFB | TR12 and by the Russian Foundation for Basic Research (grant no. 07-01-00230).
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Vishnyakova, E.G. On complex lie supergroups and split homogeneous supermanifolds. Transformation Groups 16, 265–285 (2011). https://doi.org/10.1007/s00031-010-9114-5
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DOI: https://doi.org/10.1007/s00031-010-9114-5