Hopf–Cyclic Homology and Relative Cyclic Homology of Hopf–Galois Extensions

Let H be a Hopf algebra and let M s ( H ) be the category of all left H ‐modules and right H ‐comodules satisfying appropriate compatibility relations. An object in M s ( H ) will be called a stable anti‐Yetter–Drinfeld module (over H ) or a SAYD module , for short. To each M ∈ M s ( H ) we associate, in a functorial way, a cyclic object Z * ( 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 ∈ M s ( K ) we define a right H ‐comodule structure onI n d K H M : = H ⊗ K M so thatI n d K H M becomes an object in M s ( H ). Under some assumptions on K and M we compute the cyclic homology ofZ ∗ ( H , I n d 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 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. 2000 Mathematics Subject Classification 16E40 (primary), 16W30 (secondary).