Journal of Lie Theory

It is a classical result that any complex analytic Lie supergroup G is split (see J.-L. Koszul, Graded manifolds and graded Lie algebras, Proceeding of the International Meeting on Geometry and Physics (Bologna), Pitagora, 71--84 (1982)), that is, its structure sheaf is isomorphic to the structure sheaf of a certain vector bundle. However, there do exist non-split complex analytic homogeneous supermanifolds.
We study the question how to find out whether a complex analytic homogeneous supermanifold is split or non-split. Our main result is a description of left invariant gradings on a complex analytic homogeneous supermanifold G/H in the terms of H-invariants. As a corollary to our investigations we get some simple sufficient conditions for a complex analytic homogeneous supermanifold to be split in terms of Lie algebras.