process

Equivariant cyclic homology for quantum groups 173andtr A .x ˝ .v 0 ˝ a 0 ˝ w 0 /d.v 1 ˝ a 1 ˝ w 1 //D b..S 1 .x .1/ / w 1 ;v 0 /b.w 0 ;v 1 /x .2/ ˝ a 0 da 1 :In these formulas we implicitly use the twisted trace tr x W l.b/ ! C for x 2 H definedby tr x .v ˝ w/ D b..S 1 .x/ w; v/. The twisted trace satisfies the relationtr x .T 0 T 1 / D tr x.2/ ..S 1 .x .1/ / T 1 /T 0 /for all T 0 ;T 1 2 l.b/. Using this relation one verifies that tr A defines a chain map. It isclear that tr A is yH -linear and it is straightforward to check that tr A is H -linear. Let usdefine t A D tr A ı X H . A / and show that Œt A is an inverse for Œ A . Using the fact that uis H -invariant one computes Œ A Œt A D 1.WehavetoproveŒt A Œ A D 1. Considerthe following equivariant homomorphisms l.bI A/ ! l.bI l.bI A// given byi 1 .v ˝ a ˝ w/ D u ˝ v ˝ a ˝ w ˝ u;i 2 .v ˝ a ˝ w/ D v ˝ u ˝ a ˝ u ˝ w:As above we see Œi 1 Œt l.bIA/ D 1 and we determine Œi 2 Œt l.bIA/ D Œt A Œ A . Let h tbe the linear map from l.bI A/ into l.bI l.bI A// given byh t .v ˝ a ˝ w/ D cos.t=2/ 2 u ˝ v ˝ a ˝ w ˝ u C sin.t=2/ 2 v ˝ u ˝ a ˝ u ˝ wi cos.t=2/ sin.t=2/u ˝ v ˝ a ˝ u ˝ wC i sin.t=2/ cos.t=2/v ˝ u ˝ a ˝ w ˝ u:The family h t depends smoothly on t and we have h 0 D i 1 and h 1 D i 2 . Since u isinvariant the map h t is in fact equivariant and one checks that h t is a homomorphism.Hence we have indeed defined a smooth homotopy between i 1 and i 2 . This yieldsŒi 1 D Œi 2 and hence Œt A Œ A D 1.We derive the following general stability theorem.Proposition 8.5 (Stability). Let H be a bornological quantum group and let A bean H -algebra. Moreover let V be an essential H -module and let b W V V ! Cbe a nonzero equivariant bilinear pairing. Then there exists an invertible element inHP0 G .A; l.bI A//. Hence there are natural isomorphismsHP H .l.bI A/; B/ Š HP Hfor all H -algebras A and B..A; B/ HPH .A; B/ Š HP H .A; l.bI B//Proof. Let us write ˇ W H H ! C for the canonical equivariant bilinear pairingintroduced in Section 3. Moreover we denote by b the pairing b W V V ! Cwhere V is the space V equipped with the trivial H -action. We have an equivariantisomorphism W l.b I l.ˇI A// Š l.ˇI l.bI A// given by.v ˝ .x ˝ a ˝ y/ ˝ w/ D x .1/ ˝ x .2/ v ˝ a ˝ y .1/ w ˝ y .2/