Cyclic Homology and Quantum Orbits

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.


Introduction
It is well known due to Takeuchi [28] (vastly extended by van Oystaeyen-Zhang [29] and Schauenburg [27]) that given a Hopf algebra H there is a Galois 1-1 correspondence between its left coideal subalgebras, for which H is a faithfully flat algebra extension, and its quotient right module coalgebras, for which H is a faithfully coflat coalgebra coextension.
This can be viewed as a relation between some extensions of comodule algebras and some coextensions of module coalgebras. Both related subjects have their specific homological invariants (Hochschild, cyclic, periodic cyclic and negative cyclic homology and homology with coefficients) computed from appropriate cyclic objects.
On the other hand, for module coalgebras, it is a cyclic object with SAYD coefficients which is cyclic dual to the cocyclic object also introduced in [18], and for coalgebra extensions it is the cyclic dual of the Pontryagin dual analogue of the Kadison's cyclic object.
Therefore it is very natural to ask whether these different types of cyclic objects are related in the context of the aforementioned Takeuchi-Galois correspondence.
This question goes far beyond Galois theory (herein the theory of so called homogeneous quotient coalgebra-Galois extensions [9]) and reaches topology. Adapted to the case of smooth functions on compact Lie groups and their homogeneous spaces, relative periodic cyclic homology computes the vector bundle of de Rham cohomology of the stabilizer over the homogeneous space. This bundle is equipped with the Gauss-Manin connection (see [15] for noncommutative fibrations with commutative base) determining a local system of coefficients whose cohomology appears in the second page of the Leray spectral sequence interpolating between cohomology of the compact Lie group and cohomology of the homogeneous space. Then the above Takeuchi-Galois 1-1 correspondence boils down to that one between stabilizers and orbits, and the algebra extension describes the orbital map.
On the other hand, the cyclic object with SAYD coefficients of a module coalgebra is cyclic dual to the cocyclic object generalizing the one used by Connes and Moscovici in the proof of a generalized transversal local index theorem for foliated spaces, where the index computation relies on a symmetry governed by a Hopf algebra [12,18].
The aim of the present paper is to prove in the aforementioned context of Takeuchi-Galois correspondence that, after choosing appropriate SAYD coefficients, the corresponding two types of homological invariants become isomorphic in a natural way.
It is worth noticing that even in the case of such a basic homogeneous H-Galois extension as k → H computing the left hand side, being then Hochschild homology of the Hopf algebra itself, is nontrivial and of fundamental interest [14,16,17,5,10,2]. Using our result we give an independent proof of the classical computation of Feng-Tsygan [14], reproved by another argument by Bichon [2]. Therefore our natural isomorphism can be viewed as a generalization of their results to the case of arbitrary homogeneous coalgebra-Galois extensions.
The Podleś quantum deformation of the Hopf fibration by circles of a 3-sphere over a 2-sphere, carrying at the same time two Pontryagin dual structures, provides an example of a homogeneous coalgebra-Galois extension of algebras, and a co-homogeneous algebra-Galois coextension of coalgebras [7], illustrating the Pontryagin duality principle discussed here.

Preliminaries
In this section we recall the material we will use in the sequel. In the first subsection we recall coalgebra-Galois extensions. In the second subsection we recall the Pontryagin like duality between discrete and linearly compact vector spaces and after recalling the Takeuchi-Galois correspondence between left comodule subalgebras and right module quotient coalgebras of a Hopf algebra we show Pontryagin selfduality of this correspondence. In the third subsection we recall the relative cyclic homology of algebra extensions. Finally, the Hopf-cyclic homology of coalgebras with coefficients is recalled in the fourth subsection.

Coalgebra-Galois extensions
In this subsection we recall the definition and basic properties of coalgebra-Galois extensions from [9].
Let C be a coalgebra, and A an algebra. Let A also be a right C-comodule via ρ : A → A ⊗ C, which we denote by ρ(a) = a 0 ⊗ a 1 . Then the coaction invariants of A is defined to be the algebra of endomorphisms of A regarded as a right Amodule right C-comodule It follows, from the second equality, that A co C is a subalgebra of A.
determines a right H/I-coaction on A. Finally, a Galois H/I-extension A co H/I → A is called a quotient coalgebra-Galois extension. An important property of quotient coalgebra-Galois extensions shared with Hopf-Galois extensions is that the algebra of invariants of the coaction can be expressed diagrammatically in vector spaces as where one arrow in the parallel pair in the equalizer is ρ defined as in (2.3), and another one is the composition The quantum instanton bundle introduced in [4] is a quotient coalgebra-Galois extension, [3], and it uses the full generality of the quotient coalgebra-Galois extensions: A = H and I = 0.
For I = 0, the quotient coalgebra-Galois extensions recover the Hopf-Galois extensions, and in case of A = H they are called homogeneous coalgebra-Galois extensions. There is a celebrated example of this construction due to Podleś [6,25] which is a quantum spherical fibration SU q (2) → S 2 q,µ,ν , see [7], a quantum deformation of the classical Hopf fibration of a 3-sphere over a 2-sphere into circles.

Formal Pontryagin duality and Galois-Takeuchi correspondence
In this subsection we will first summarize the basic facts on the dualization functor on vector spaces from [1]. We will then recall a one-to-one correspondence between the coideal subalgebras and quotient coalgebras, known as Galois-Takeuchi correspondence [28].
From the point of view of linear topological vector spaces with continuous linear mappings as morphisms, the dualization functor defines an equivalence between the opposite category of discrete vector spaces and the category of linearly compact vector spaces [1,Prop. 24.8]. Moreover, it transforms naturally the algebraic tensor product of discrete vector spaces into a completed one of linearly compact spaces [1,Cor. 24.25]. In other words, dualization is a strong monoidal functor. Therefore one can regard on equal footing all structures defined by diagrams in vector spaces together with their dual counterparts obtained by reversing all arrows in all necessary diagrams. This regards linear subspaces and quotient spaces, (co)algebras and Hopf algebras, as well as their bi(co)modules or one sided (co)modules, one-sided and two-sided (co)ideals, and their (co)tensor products. Hence, having a diagrammatical proof of a theorem in a symmetric monoidal category of linear (topological) spaces, one has automatically a dual theorem after an appropriate dualization of the structures. It is well known that the notion of Hopf algebra is self-dual. In particular, it transforms the Hopf group-algebra of a discrete group into a linearly compact topological Hopf algebra of functions on that group, customary regarded as a group algebra of a dual compact quantum group. Therefore we will call this duality Pontryagin to distinguish it from cyclic duality which will appear later on. We will show that the notion of SAYD module is Pontryagin self-dual up to interchanging left and right. According to [7,8] the notion of coalgebra-Galois extension of algebras is Pontryagin dual to the notion of algebra-Galois coextension of coalgebras. Both dualities play a role in the present paper.
where the parallel pair of left arrows in the coequalizer and the parallel pair of right arrows in the equalizer read as composites respectively. It is hence evident that this correspondence is Pontryagin self-dual up to an interchange of left and right in dual structures.

Relative cyclic homology of algebra extensions
In this subsection we recall the homological preliminaries that will be needed in the sequel. More precisely, we shall first include a very brief summary of the homological bar complex and its relative version, as well as their relations with Hochschild homology [22,24]. We will then provide with a short discussion on the relative cyclic homology of algebras from [20,26].
Let A be an associative algebra, X a right A-module, and Y a left A-module. Then, for n ≥ 1, is called the two sided bar complex. If i : B → A is a morphism of associative unital algebras, then A becomes a B-bimodule via the map i : B → A, and we can consider the relative bar complex (2.10) with the differential (2.9).
Let A be an algebra, and M an A-bimodule. Then the Hochschild homology of A with coefficients in M, denoted by H * (A, M), is defined to be the homology of the complex . . ⊗ a n + (−1) n+1 a n · m ⊗ a 1 ⊗ . . . ⊗ a n−1 . We proceed with a quick detour on the relative cyclic homology of algebra extensions. To this end we shall first need to recall the notion of cyclic tensor product from [26]. The cyclic tensor product of B-bimodules M 1 , . . . , M n is defined by As it is remarked in [20], The relative cyclic homology of an algebra extension B → A is computed by the cyclic object equipped, for all n ≥ 0, with the morphisms t(a 0 ⊗ B · · · ⊗ B a n ) = a n ⊗ B a 0 ⊗ B · · · ⊗ B a n−1 .
Hochschild homology of the complex (2.14) is called the Hochschild homology of the extension, and is denoted by HH * (A | B). Similarly, cyclic (resp. periodic cyclic, negative cyclic) homology of the cyclic object (2.14) is called the relative cyclic (resp. periodic cyclic, negative cyclic) homology of the extension B → A, and it is denoted by HC * (A | B) (resp. HP * (A | B), HN * (A | B)).

Hopf-cyclic homology of H-module coalgebras
In this subsection we recall the relative Hopf-cyclic homology with coefficients, for coalgebras, using the cyclic duality principle [11,23].
Let us first recall the definition of a left-right stable anti-Yetter-Drinfeld module (SAYD module) over a Hopf algebra H from [19]. On the other hand, a right-left SAYD module is defined by Now we see that reversing the arrows and interchanging the pairs ∆ and µ, λ and ρ, and finally left and right, we obtain the right-left SAYD module compatibility.
Let us now recall the Hopf-cyclic cohomology of a Hopf-module coalgebra with coefficients. Let H be a Hopf algebra, C a right H-module coalgebra, i.e., a right H-module such that for all n ≥ 0, is a cocyclic module with the operators and Cyclic cohomology of this cocyclic module is called the Hopf-cyclic cohomology of the H-module coalgebra C with coefficients in the left-right SAYD module M over H, and is denoted by HC * (C, M) H .
Applying the cyclic duality procedure [11,23] we obtain on (2.26) the cyclic module structure given by the faces

Relative cyclic homology as Hopf-cyclic homology with coefficients
In this section we achieve our main result identifying the cyclic homology of a homogeneous C-Galois extension B → H with the Hopf-cyclic homology, with coefficients, of the right H-module coalgebra C. Then, by the Pontryagin duality, we automatically have the identification of the cyclic homology of a B-Galois coextension H → C with the cyclic homology of the Hopf-cyclic homology, with (the Pontryagin dual) coefficients, of the H-comodule algebra B. We shall conclude the section developing spectral sequences to shed further light on the relative homology groups.

Isomorphisms of cyclic objects
In this subsection we will construct an explicit isomorphism from the relative homology complex of a homogeneous C : Let H be a Hopf algebra and I ⊆ H a coideal right ideal of H. Then H/I becomes a coaugmented quotient coalgebra in a canonical way, where h := h + I, and the canonical coaugmentation of H/I is given by the grouplike 1.

Let B → H be a homogeneous H/I-Galois extension given by the canonical right H/I-coaction
On the other hand, it is clear that ε(B + H) = 0, hence B + H is a coideal in H. (2), (3.5) is well defined and provides the inverse to the canonical map

Now,
hence the extension B → H is a H/I-Galois extension. The canonical map and its inverse can be inductively extended to H-bimodule isomorphisms (3.7) and respectively, for any H-bimodule M.
Next, recall for the translation map for any h ∈ H. The latter implies also that under the isomorphism We also recall that [H] B , equipped with the (left) H-action (3.14) and the (right) H-coaction given by the comultiplication, is a left-right SAYD module over H, see [20,Ex. 4.3]. All that is used in the following lemma. implemented by with the inverse (3.17) Proof. First of all, we have to prove that the maps are well defined. Let us begin with (3.16). We observe that (3.18) As for (3.17), on the other hand, we have Next we observe for any p ∈ H that (3.20) Accordingly, ψ n is well-defined for any n ≥ 0.
Finally, we prove that ϕ n and ψ n are inverses to each other for any n ≥ 0. On one hand we have, while on the other hand (3.22) We are now ready to state our main result. (3.24) (3.25) Finally for the last face operator we have (3.26) We next investigate the interaction between the degeneracy operators. To this end, for 0 ≤ j ≤ n − 1 we observe (3.27) Let us finally check the cyclic operators. To this end we have (3.28) We will conclude this subsection with a Pontryagin dual of this result. Proof. The proof of Theorem 3.2, using only the structural maps and relations equivalent to the commutativity of appropriate diagrams, is in fact diagrammatical. Hence, in view of the Pontryagin duality, interchanging the left and right, reversing the arrows and applying the cyclic duality we obtain a diagrammatical proof of the claim.

Spectral sequences
We note that the left hand side of Theorem 3.2, resp. Theorem 3.3, compute the homology of the extension A relative to B, resp. coextension D corelative to C, while the right hand side computes the cyclic dual homology of the Pontryagin dual objects. In order to be able to investigate the latter homologies, in this subsection we shall develop computational tools.
We will focus on the Hochschild homology groups of the relative homology of the extension. where Tor is computed for the left H-module structure on M coming from its SAYD module structure. In particular, we have a five-term exact sequence Proof. Let us consider the cyclic dual C • to the standard cocyclic object of the coalgebra C, consisting of the tensor powers of C, C p = C p+1 , with the boundary map ∂ : C p → C p−1 ∂(c 0 ⊗ · · · ⊗ c p ) = p i=0 (−1) i c 0 ⊗ · · · ⊗ ε(c i )c i+1 ⊗ · · · ⊗ c p where the cyclic order of length p + 1 is assumed. Note that ∂ is a morphism of right H-modules. The operator h : C p → C p+1 h(c 0 ⊗ · · · ⊗ c p ) := 1 ⊗ c 0 ⊗ · · · ⊗ c p is a homotopy contracting this complex to k concentrated at zero degree, see for instance [23]. Let M • be a flat resolution of the left H-module M, let us consider the total complex C • ⊗ H M • . We have two spectral sequences abutting to the total homology. The first page of the first one reads  d → E 2 0,1 → H 1 → E 2 1,0 → 0 to finish the proof. The second arrow is the boundary map of the second page of the first spectral sequence, the next one is induced by the augmentation C • → k. Now we show that the above theorem generalizes the classical result of Feng-Tsygan [14] (reproved by another argument by Bichon [2]), from the case of the homogeneous H-Galois extension k → H to arbitrary homogeneous quotient coalgebra-Galois extensions B → H. The following can be regarded as a third independent proof of this classical result.