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

Abstract

Let $H$ be a Hopf algebra and let $\mathcal{M}_s (H)$ be the category of all left $H$-modules and right $H$-comodules satisfying appropriate compatibility relations. An object in $\mathcal{M}_s (H)$ will be called a stable anti-Yetter–Drinfeld module (over $H$) or a SAYD module, for short. To each $M \in \mathcal{M}_s (H)$ we associate, in a functorial way, a cyclic object $\mathrm{Z}_\ast (H, M)$. We show that our construction can be used to compute the cyclic homology of the underlying algebra structure of $H$ and the relative cyclic homology of $H$-Galois extensions.Let $K$ be a Hopf subalgebra of $H$. For an arbitrary $M \in \mathcal{M}_s (K)$ we define a right $H$-comodule structure on $\mathrm{Ind}_K^H M := H \otimes_K M$ so that $\mathrm{Ind}_K^H M$ becomes an object in $\mathcal{M}_s (H)$. Under some assumptions on $K$ and $M$ we compute the cyclic homology of $\mathrm{Z}_\ast (H, \mathrm{Ind}_K^H M)$. As a direct application of this result, we describe the relative cyclic homology of strongly graded algebras. In particular, we calculate the cyclic homology of group algebras and quantum tori.Finally, when $H$ is the enveloping algebra of a Lie algebra $\mathfrak{g}$, we construct a spectral sequence that converges to the cyclic homology of $H$ with coefficients in a given SAYD module $M$. We also show that the cyclic homology of almost symmetric algebras is isomorphic to the cyclic homology of $H$ with coefficients in a certain SAYD module.

Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category

“…Namely, for α = β = id H , we obtain the usual Yetter-Drinfeld modules; for α = S 2 , β = id H , we obtain the so-called anti-Yetter-Drinfeld modules, introduced in [7], [8], [10] as coefficients for the cyclic cohomology of Hopf algebras defined by Connes and Moscovici in [5], [6]; finally, an (id H , β)-Yetter-Drinfeld module is a generalization of the object H β defined in [4], which has the property that, if H is finite dimensional, then the map β → End(H β ) gives a group anti-homomorphism from Aut Hopf (H) to the Brauer group of H. It is natural to expect that (α, β)-Yetter-Drinfeld modules have some properties resembling the ones of the three kinds of objects we mentioned. We will see some of these properties in this paper (others will be given in a subsequent one), namely the ones directed to our main aim here, which is the following: if we denote by H YD H (α, β) the category of (α, β) -Yetter-Drinfeld modules and we define YD(H) as the disjoint union of all these categories, then we can organize YD(H) as a braided T-category (or braided crossed group-category, in the original terminology of Turaev, see [16]) over the group G = Aut Hopf (H) × Aut Hopf (H) with multiplication (α, β) * (γ, δ) = (αγ, δγ −1 βγ).…”

A Note on Cyclic Duality and Hopf Algebras

“…In this paper we show that C H * (H, M ) is in fact isomorphic, via a non-trivial map, to the cyclic dual of C * H (H, M ) (Theorem 3.1). The cyclic dual of C * H (H, M ) appears naturally in the study of the relative cyclic homology of Hopf-Galois extensions [7]. Thus our result shows that this theory is a special case of the invariant cyclic homology defined in [6] for general coefficients and in [8] for a restricted class of coefficients.…”

“…Note that what we call an stable anti-Yetter-Drinfeld module in the present paper (and in [6]), is called a modular crossed module in [7]. It follows from Theorem 3.1 above that Theorem 4.13 in [7] is a consequence of Theorem 3.1 in [6] (by choosing A = H).…”

“…The cyclic module K H * (H, M ) is used by Jara and Stefan in their study of relative cyclic homology of Hopf-Galois extensions [7]. Note that what we call an stable anti-Yetter-Drinfeld module in the present paper (and in [6]), is called a modular crossed module in [7].…”

“…Recently we have seen a proliferation of cyclic and cocyclic modules attached to Hopf algebras [5,11,9,8,6,7], extending the pioneering work of Connes and Moscovici [3,4]. Recall that in [6] a cocyclic module C * H (H, M ) and a cyclic module C H * (H, M ) is defined for any Hopf algebra H and an stable anti-Yetter-Drinfeld H-module M (see Section 3 for definitions).…”