process

Equivariant cyclic homology for quantum groups 155H -equivariant if it commutes with the module actions. We denote the category of essentialH -modules and equivariant linear maps by H -Mod. Using the comultiplicationof H one obtains a natural H -module structure on the tensor product of two H -modulesand H -Mod becomes a monoidal category in this way.We will frequently use the regular actions associated to a bornological quantumgroup H .Fort 2 H and f 2 yH one definest*f D f .1/ f .2/ .t/; f ( t D f .1/ .t/f .2/and this yields essential left and right H -module structures on yH , respectively.Dually to the concept of an essential module one has the notion of an essentialcomodule. Let H be a bornological quantum group and let V be a bornological vectorspace. A coaction of H on V is a right H -linear bornological isomorphism W V y˝H ! V y˝ H such that the relationholds..id ˝ r / 12 .id ˝ 1r / D 12 13Definition 3.1. Let H be a bornological quantum group. An essential H -comodule isa bornological vector space V together with a coaction W V y˝ H ! V y˝ H .A bounded linear map f W V ! W between essential comodules is called H -colinear if it is compatible with the coactions in the obvious sense. We write Comod- Hfor the category of essential comodules over H with H -colinear maps as morphisms.The category Comod- H is a monoidal category as well.If the quantum group H is unital, a coaction is the same thing as a bounded linearmap W V ! V y˝ H such that . y˝ id/ D .id y˝ / and .id y˝ / D id.Modules and comodules over bornological quantum groups are related in the sameway as modules and comodules over finite dimensional Hopf algebras.Theorem 3.2. Let H be a bornological quantum group. Every essential left H -moduleis an essential right yH -comodule in a natural way and vice versa. This yields inverseisomorphisms between the category of essential H -modules and the category of essentialyH -comodules. These isomorphisms are compatible with tensor products.Since it is more convenient to work with essential modules instead of comoduleswe will usually prefer to consider modules in the sequel.An essential H -module is called projective if it has the lifting property with respectto surjections of essential H -modules with bounded linear splitting. It is shown in [27]that a bornological quantum group H is projective as a left module over itself. Thiscan be generalized as follows.Lemma 3.3. Let H be a bornological quantum group and let V be any essentialH -module. Then the essential H -modules H y˝ V and V y˝ H are projective.