On complex lie supergroups and split homogeneous supermanifolds
On complex lie supergroups and split homogeneous supermanifolds
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关于复谎言超群和分裂齐次超流形
DOI：
10.1007/s00031-010-9114-5
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发表时间：
2009
影响因子：
0.7
通讯作者：
E. Vishnyakova
中科院分区：
文献类型：
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作者：
E. Vishnyakova
众所周知，实李超群的范畴相当于所谓的（实）Harish-Chandra 对的范畴，参见 [DM]、[Kost]、[Kosz]。这意味着李超群仅依赖于基础李群及其具有某些兼容性条件的李超代数。更准确地说，李超群的结构束和超群态射可以用相应的李超代数来明确描述。在本文中，我们在复分析情况下证明了这一结果。此外，如果 (G,G) 是复李超群且 H ⊂ G 是闭李子群，即它是 (G,G) 的李子超群且其奇维数为零，则我们证明相应的齐次超流形 (G/H,G/H) 是分裂的。特别是，任何复数李超群都是分裂超流形。众所周知，复齐次超流形可能是非分裂的（参见，例如，[OS1]）。我们在这里找到了复杂齐次超流形分裂的充分必要条件。
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,G) is a complex Lie supergroup and H ⊂ G is a closed Lie subgroup, i.e., it is a Lie subsupergroup of (G,G) and its odd dimension is zero, we show that the corresponding homogeneous supermanifold (G/H,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.