On Complex Lie Supergroups and Homogeneous Split 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. 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 $H\subset G$ 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 non-split. We find here necessary and sufficient conditions for a complex homogeneous supermanifold to be split.

Version 3 - exposition expanded, references added, 24 pages

References in corpus (1)

• On the Structure of Complex Homogeneous Supermanifolds

Cited by in corpus (9)

• On holomorphic functions on a compact complex homogeneous supermanifold
• Super Distributions, Analytic and Algebraic Super Harish-Chandra pairs
• Non-split supermanifolds associated with the cotangent bundle
• Real forms of complex Lie superalgebras and supergroups
• SUSY structures, representations and Peter-Weyl theorem for $S^{1|1}$
• Almost complex structures on real Lie supergroups
• Highest weight Harish-Chandra supermodules and their geometric realizations
• SUSY $N$-supergroups and their real forms
• Non-integrable supersymmetries and their classification for $\mathfrak{gl}(1,1)$ and $\mathfrak{sl}(1,1)$