process

Equivariant cyclic homology for quantum groups 163Proposition 5.3. Let H be a bornological quantum group. Then the category ofAYD-modules over H is isomorphic to the category of essential left A.H /-modules.Proof. Let M Š A.H / y˝A.H / M be an essential A.H /-module. Then we obtaina left H -module structure and a left yH -module structure on M using the canonicalhomomorphisms H W H ! M.A.H // and y H W yH ! M.A.H //. Since the action ofH on A.H / is essential we have natural isomorphismsH y˝H M Š H y˝H A.H / y˝A.H / M Š A.H / y˝A.H / M Š Mand hence M is an essential H -module. Similarly we haveyH y˝ yH M Š yH y˝ yH A.H / y˝A.H /y˝ M Š A.H / y˝A.H / M Š Msince A.H / is an essential yH -module. These module actions yield the structure of anAYD-module on M .Conversely, assume that M is an H -AYD-module. Then we obtain an A.H /-modulestructure on M by setting.f ˝ t/ m D f .t m/for f 2 yH and t 2 H . Since M is an essential H -module we have a naturalisomorphism H y˝H M Š M . As in the proof of Proposition 5.2 we obtain aninduced essential yH -module structure on H y˝H M and canonical isomorphismsA.H / y˝A.H / M Š yH y˝ yH .H y˝H M/ Š M . It follows that M is an essentialA.H /-module.The previous constructions are compatible with morphisms and it is easy to checkthat they are inverse to each other. This yields the assertion.There is a canonical operator T on every AYD-module which plays a crucial rolein equivariant cyclic homology. In order to define this operator it is convenient topass from yH to H in the first tensor factor of A.H /. More precisely, consider thebornological isomorphism W A.H / ! H y˝ H given by.f ˝ y/ D yF l .f / ˝ y( O ı 1where O ı 2 M. yH/is the modular function of yH . The inverse map is given by 1 .x ˝ y/ D SG l .x/ ˝ y( O ı:It is straightforward to check that the left H -action on A.H / corresponds toand the left yH -action becomest .x ˝ y/ D t .3/ xS.t .1/ / ˝ t .2/ yf .x ˝ y/ D .f * x/ ˝ y