Hopf–cyclic Homology and Relative Cyclic Homology of Hopf–galois Extensions

The determination of cyclic (co)homology of a given algebra is a quite important and difficult problem. Let us briefly recall some of the results obtained that are somehow related to our paper. The cyclic homology of group algebras over fields of characteristic 0 was computed by Burghelea, [3]. For a complete algebraic proof of Burghelea’s result the reader is referred to [19], while a relative variant of this computation can be found in [25]. Crossed products (with trivial cocycle) are generalizations of group algebras. Let B be an algebra on which a group G acts by algebra automorphisms. On A, the free left B module having a basis {eg | g ∈ G}, we define an algebra structure by (xeg ) · (yeh) := x(g.y)egh , where x, y ∈ B, and g.y denotes the action of g on y. Feigin and Tsygan [8] and, independently, Nistor [22] showed that HC∗(A) decomposes canonically as a direct sum ⊕ σ∈T (G) HC∗(A)σ , where T (G) denotes the set of conjugacy classes of G. They also constructed a spectral sequence converging to HC∗(A)1, the component corresponding to σ = {1}. For related work on this case see also [9]. The cyclic homology of enveloping algebras of Lie algebras is also known. As a matter of fact, C. Kassel in [14] described the cyclic homology of all almost symmetric algebras (A is said to be almost symmetric if it is non-negatively filtered such that grA is the symmetric algebra of gr1A). The computation of cyclic homology of U(g) was also performed in [8]. Cyclic (co)homology of Hopf algebras was introduced by Connes and Moscovici in order to compute the index of transversally elliptic operators of foliations. To every Hopf algebra H and every modular pair in involution (σ, δ) they associated a cocyclic module H (σ,δ). Recall that (σ, δ) is a modular pair in involution if σ is a group-like element in H, δ : H → k is a morphism of algebras, and the twisted antipode Sδ is involutive; see [5, 6]. The cyclic cohomology ofH # (σ,δ) is called the cyclic cohomology of H and it is denoted HC(σ,δ)(H). One of the features of HC(σ,δ)(H) is that, for a given algebra A on which H acts and a given H-invariant trace τ : A → k, there is a canonical morphism γ∗ τ : HC ∗ (σ,δ)(H) −→ HC∗(A).