## Cyclic homology and quantum orbits.(English)Zbl 1365.19002

Authors’ abstract: A natural isomorphism between the cyclic object computing the relative cyclic homology of a homogeneous quotient-coalgebra-Galois extension, and the cyclic object computing the cyclic homology of a Galois coalgebra with SAYD coefficients is presented. The isomorphism can be viewed as the cyclic-homological counterpart of the Takeuchi-Galois correspondence between the left coideal subalgebras and the quotient right module coalgebras of a Hopf algebra. A spectral sequence generalizing the classical computation of Hochschild homology of a Hopf algebra to the case of arbitrary homogeneous quotient-coalgebra-Galois extensions is constructed. A Pontryagin type self-duality of the Takeuchi-Galois correspondence is combined with the cyclic duality of Connes in order to obtain dual results on the invariant cyclic homology, with SAYD coefficients, of algebras of invariants in homogeneous quotient-coalgebra-Galois extensions. The relation of this dual result with the Chern character, Frobenius reciprocity, and inertia phenomena in the local Langlands program, the Chen-Ruan-Brylinski-Nistor orbifold cohomology and the Clifford theory is discussed.

### MSC:

 19D55 $$K$$-theory and homology; cyclic homology and cohomology 57T15 Homology and cohomology of homogeneous spaces of Lie groups 06A15 Galois correspondences, closure operators (in relation to ordered sets) 46A20 Duality theory for topological vector spaces
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