process

Equivariant cyclic homology for quantum groups 159Proposition 3.7. Let H be a bornological quantum group and let A be an H -algebra.Then the map A W A Ì H Ì yH ! A y˝ K H is an equivariant algebra isomorphism.For algebraic quantum groups a discussion of Takesaki–Takai duality is containedin [11]. More information on similar duality results in the context of Hopf algebras canbe found in [21].If H D D.G/ is the smooth convolution algebra of a locally compact group G thenan H -algebra is the same thing as a G-algebra. As a special case of Proposition 3.7 oneobtains that for every G-algebra A the double crossed product AÌH Ì yH is isomorphicto the G-algebra A y˝ K G used in [25].4 Radford’s formulaIn this section we prove a formula for the fourth power of the antipode in terms of themodular elements of a bornological quantum group and its dual. This formula wasobtained by Radford in the setting of finite dimensional Hopf algebras [23].Let H be a bornological quantum group. If is a left Haar functional on H thereexists a unique multiplier ı 2 M.H/ such that. y˝ id/.x/ D .x/ıfor all x 2 H . The multiplier ı is called the modular element of H and measuresthe failure of from being right invariant. It is shown in [27] that ı is invertible withinverse S.ı/ D S 1 .ı/ D ı 1 and that one has .ı/ D ı ˝ ı as well as .ı/ D 1.In terms of the dual quantum group the modular element ı defines a character, that is,an essential homomorphism from yH to C. Similarly, there exists a unique modularelement O ı 2 M. yH/for the dual quantum group which satisfies. O y˝ id/ O.f / D O.f / O ıfor all f 2 yH .The Haar functionals of a bornological quantum group are uniquely determined upto a scalar multiple. In many situations it is convenient to fix a normalization at somepoint. However, in the discussion below it is not necessary to keep track of the scalingof the Haar functionals. If ! and are linear functionals we shall write ! if thereexists a nonzero scalar such that ! D . We use the same notation for elements in abornological quantum group or linear maps that differ by some nonzero scalar multiple.Moreover we shall identify H with its double dual using Pontrjagin duality.To begin with observe that the bounded linear functional ı*on H defined byis faithful and satisfies.ı * /.x/ D .xı/..ı * / y˝ id/.x/ D . y˝ id/..xı/.1 ˝ ı 1 // D .xı/ıı 1 D .ı * /.x/: