process

Equivariant cyclic homology for quantum groups 161and since yF r is an isomorphism this impliesF r .S 2 .x// F l .ı 1 .x ( O ı 1 /ı/: (4.5)Assembling these relations we obtain the following result.Proposition 4.1. Let H be a bornological quantum group and let ı and O ı be the modularelements of H and yH , respectively. ThenS 4 .x/ D ı 1 . O ı*x( O ı 1 /ıfor all x 2 H .Proof. Using equation (4.2) and equation (4.5) we computeF l .S 4 .x// F r . O ı*S 2 .x// D F r .S 2 . O ı*x// F l .ı 1 . O ı*x( O ı 1 /ı/which impliesS 4 .x/ ı 1 . ı*x( O ı O 1 /ıfor all x 2 H since F l is an isomorphism. The claim follows from the observation thatboth sides of the previous equation define algebra automorphisms of H .5 Anti-Yetter–Drinfeld modulesIn this section we introduce the notion of an anti-Yetter–Drinfeld module over a bornologicalquantum group. Moreover we discuss the concept of a paracomplex.We begin with the definition of an anti-Yetter–Drinfeld module. In the context ofHopf algebras this notion was introduced in [12].Definition 5.1. Let H be a bornological quantum group. An H -anti-Yetter–Drinfeldmodule is an essential left H -module M which is also an essential left yH -module suchthatt .f m/ D .S 2 .t .1/ /*f (S 1 .t .3/ // .t .2/ m/:for all t 2 H; f 2 yH and m 2 M . A homomorphism W M ! N between anti-Yetter–Drinfeld modules is a bounded linear map which is both H -linear and yH -linear.We will not always mention explicitly the underlying bornological quantum groupwhen dealing with anti-Yetter–Drinfeld modules. Moreover we shall use the abbreviationsAYD-module and AYD-map for anti-Yetter–Drinfeld modules and their homomorphisms.According to Theorem 3.2 a left yH -module structure corresponds to a right H -comodule structure. Hence an AYD-module can be described equivalently as a bornologicalvector space M equipped with an essential H -module structure and an H -comodulestructure satisfying a certain compatibility condition. Formally, this compatibility conditioncan be written down asfor all t 2 H and m 2 M ..t m/ .0/ ˝ .t m/ .1/ D t .2/ m .0/ ˝ t .3/ m .1/ S.t .1/ /