Mathematics – Complex Variables
Scientific paper
2010-12-20
Mathematics
Complex Variables
This is an extended version of the previous preprint "Almost complex structures on Lie supergroups" with applications to radia
Scientific paper
Integrable invariant almost complex structures are introduced in the setting of Lie-Hopf superalgebras. This leads to a convenient description of the subsheaf of holomorphic superfunctions in the sheaf of smooth superfunctions yielding an identification of complex Lie supergroups and complex Lie-Hopf superalgebras. Applications to the construction of a universal complexification of a real Lie supergroup and, in the case of type I supergroups, to the existence of a local product decomposition defined by conjugation at a generic point of a maximal torus are given. Deducing a Helgason type formula on the radial part of the Laplacian in the super setting we are able to give a very comprehensive approach to Berezin's formula on radial parts of Laplace-Casimir operators on special type I Lie supergroups.
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