process

152 C. VoigtAlthough anti-Yetter–Drinfeld modules occur naturally in the constructions oneshould point out that our theory does not fit into the framework of Hopf-cyclic cohomology[13]. Still, there are relations to previous constructions for Hopf algebrasby Akbarpour and Khalkhali [1], [2] as well as Neshveyev and Tuset [22]. Remarkin particular that cosemisimple Hopf algebras or finite dimensional Hopf algebras canbe viewed as bornological quantum groups. However, basic examples show that thehomology groups defined in [1], [2], [22] only reflect a small part of the informationcontained in the theory described below.Let us now describe how the paper is organized. In Section 2 we recall the definitionof a bornological quantum group. We explain some basic features of the theoryincluding the definition of the dual quantum group and the Pontrjagin duality theorem.This is continued in Section 3 where we discuss essential modules and comodulesover bornological quantum groups as well as actions on algebras and their associatedcrossed products. We prove an analogue of the Takesaki–Takai duality theorem in thissetting. Section 4 contains the discussion of Radford’s formula relating the antipodewith the modular functions of a quantum group and its dual. In Section 5 we studyanti-Yetter–Drinfeld modules over bornological quantum groups and introduce the notionof a paracomplex. Section 6 contains a discussion of equivariant differential formsin the quantum group setting. After these preparations we define equivariant periodiccyclic homology in Section 7. In Section 8 we show that our theory is homotopy invariant,stable and satisfies excision in both variables. Finally, Section 9 contains a briefcomparison of our theory with the previous approaches mentioned above.Throughout the paper we work over the complex numbers. For simplicity we haveavoided the use of pro-categories in connection with the Cuntz–Quillen formalism to alarge extent.2 Bornological quantum groupsThe notion of a bornological quantum group was introduced in [27]. We will workwith this concept in our approach to equivariant cyclic homology. For information onbornological vector spaces and more details we refer to [14], [20], [27]. All bornologicalvector spaces are assumed to be convex and complete.A bornological algebra H is called essential if the multiplication map induces anisomorphism H y˝H H Š H . The multiplier algebra M.H/ of a bornological algebraH consists of all two-sided multipliers of H , the latter being defined by the usual algebraicconditions. There exists a canonical bounded homomorphism W H ! M.H/.A bounded linear functional W H ! C on a bornological algebra is called faithfulif .xy/ D 0 for all y 2 H implies x D 0 and .xy/ D 0 for all x implies y D 0.If there exists such a functional the map W H ! M.H/ is injective. In this case onemay view H as a subset of the multiplier algebra M.H/.In the sequel H will be an essential bornological algebra with a faithful bounded linearfunctional. For technical reasons we assume moreover that the underlying bornologicalvector space of H satisfies the approximation property.