Groups of diffeomorphisms and Lie theory.
We introduce the concept of homotopy equivalence for Hopf Galois extensions and make a systematic study of it. As an application we determine all -Galois extensions up to homotopy equivalence in the case when is a Drinfeld-Jimbo quantum group.
We employ the notion of the universal differential calculus to propose yet another approach to the Hopf-type cohomology of Hopf algebras.
We first generalize the classical implicit function theorem of Hildebrandt and Graves to the case where we have a Keller -map f defined on an open subset of E×F and with values in F, for E an arbitrary Hausdorff locally convex space and F a Banach space. As an application, we prove that under a certain transversality condition the preimage of a submanifold is a submanifold for a map from a Fréchet manifold to a Banach manifold.
Differential forms on the Fréchet manifold of smooth functions on a compact -dimensional manifold can be obtained in a natural way from pairs of differential forms on and by the hat pairing. Special cases are the transgression map (hat pairing with a constant function) and the bar map (hat pairing with a volume form). We develop a hat calculus similar to the tilda calculus for non-linear Grassmannians [6].
In this paper, we construct a hyperkähler structure on the complexification of any Hermitian symmetric affine coadjoint orbit of a semi-simple -group of compact type, which is compatible with the complex symplectic form of Kirillov-Kostant-Souriau and restricts to the Kähler structure of . By a relevant identification of the complex orbit with the cotangent space of induced by Mostow’s decomposition theorem, this leads to the existence of a hyperkähler structure on compatible with...