process

Equivariant cyclic homology for quantum groupsChristian Voigt1 IntroductionEquivariant cyclic homology can be viewed as a noncommutative generalization ofequivariant de Rham cohomology. For actions of finite groups or compact Lie groups,different aspects of the theory have been studied by various authors [4], [5], [6], [17],[18]. In order to treat noncompact groups as well, a general framework for equivariantcyclic homology following the Cuntz–Quillen formalism [8], [9], [10] has been introducedin [25]. For instance, in the setting of discrete groups or totally disconnectedgroups this yields a new approach to classical constructions in algebraic topology [26].However, in contrast to the previous work mentioned above, a crucial feature of theconstruction in [25] is the fact that the basic ingredient in the theory is not a complexin the usual sense of homological algebra. In particular, the theory does not fitinto the traditional scheme of defining cyclic homology using cyclic modules or mixedcomplexes.In this note we define equivariant periodic cyclic homology for quantum groups.This generalizes the constructions in the group case developed in [25]. Again we workin the setting of bornological vector spaces. Correspondingly, the appropriate notionof a quantum group in this context is the concept of a bornological quantum groupintroduced in [27]. This class of quantum groups includes all locally compact groupsand their duals as well as all algebraic quantum groups in the sense of van Daele [24].As in the theory of van Daele, an important ingredient in the definition of a bornologicalquantum group is the Haar measure. It is crucial for the duality theory and also explicitlyused at several points in the construction of the homology theory presented in this paper.However, with some modifications our definition of equivariant cyclic homology couldalso be adapted to a completely algebraic setting using Hopf algebras with invertibleantipodes instead.From a conceptual point of view, equivariant cyclic homology should be viewedas a homological analogon to equivariant KK-theory [15], [16]. The latter has beenextended by Baaj and Skandalis to coactions of Hopf-C -algebras [3]. However, inour situation it is more convenient to work with actions instead of coactions.An important ingredient in equivariant cyclic homology is the concept of a covariantmodule [25]. In the present paper we will follow the terminology introduced in [12]and call these objects anti-Yetter–Drinfeld modules instead. In order to construct thenatural symmetry operator on these modules in the general quantum group setting weprove a formula relating the fourth power of the antipode with the modular functions ofa bornological quantum group and its dual. In the context of finite dimensional Hopfalgebras this formula is a classical result due to Radford [23].