Cyclic Homology and Quantum Orbits

TL;DR

This paper establishes a cyclic homology isomorphism for Galois extensions, introduces a spectral sequence for computing homology in this context, and explores dualities linking cyclic homology to various advanced mathematical theories.

Contribution

It presents a novel isomorphism between cyclic objects in Galois extensions and develops a spectral sequence for their homology, extending classical Hochschild homology computations.

Findings

Established a cyclic homology isomorphism for Galois extensions.

Constructed a spectral sequence generalizing Hochschild homology.

Linked cyclic duality with dual results in invariant cyclic homology.

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.