process

Equivariant cyclic homology for quantum groups 153A module M over H is called essential if the module action induces an isomorphismH y˝H M Š M . Moreover an algebra homomorphism f W H ! M.K/ is essentialif f turns K into an essential left and right module over H . Assume that W H !M.H y˝ H/ is an essential homomorphism. The map is called a comultiplicationif it is coassociative, that is, if . y˝ id/ D .id y˝ / holds. Moreover the Galoismaps l ; r ; l ; r W H y˝ H ! M.H y˝ H/for are defined by l .x ˝ y/ D .x/.y ˝ 1/; l .x ˝ y/ D .x ˝ 1/.y/; r .x ˝ y/ D .x/.1 ˝ y/; r .x ˝ y/ D .1 ˝ x/.y/:Let W H ! M.H y˝ H/ be a comultiplication such that all Galois maps associatedto define bounded linear maps from H y˝ H into itself. If ! is a bounded linearfunctional on H we define for every x 2 H a multiplier .id y˝ !/.x/ 2 M.H/ by.id y˝ !/.x/ y D .id y˝ !/ l .x ˝ y/;y .id y˝ !/.x/ D .id y˝ !/ l .y ˝ x/:In a similar way one defines .! y˝ id/.x/ 2 M.H/. A bounded linear functional W H ! C is called left invariant if.id y˝ /.x/ D .x/1for all x 2 H . Analogously one defines right invariant functionals.Let us now recall the definition of a bornological quantum group.Definition 2.1. A bornological quantum group is an essential bornological algebra Hsatisfying the approximation property with a comultiplication W H ! M.H y˝ H/such that all Galois maps associated to are isomorphisms together with a faithful leftinvariant functional W H ! C.The definition of a bornological quantum group is equivalent to the definition of analgebraic quantum group in the sense of van Daele [24] in the case that the underlyingbornological vector space carries the fine bornology. The functional is unique up toa scalar and referred to as the left Haar functional of H .Theorem 2.2. Let H be a bornological quantum group. Then there exists an essentialalgebra homomorphism W H ! C and a linear isomorphism S W H ! H which isboth an algebra antihomomorphism and a coalgebra antihomomorphism such that. y˝ id/ D id D .id y˝ /and.S y˝ id/ r D y˝ id; .id y˝ S/ l D id y˝ :Moreover the maps and S are uniquely determined.