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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
164 C. Voigtunder this isomorphism. The right H -action is identified withand the right yH -action corresponds to.x ˝ y/ t D x ˝ y.t ( O ı 1 /.x ˝ y/ g D x .2/ .S 2 .y .2/ /*g(S 1 .y .4/ //.S 2 .x .1/ /ı/ ˝ Oı.y .1/ /y .3/ ( O ı 1where ı is the modular function of H . Using this description of A.H / we obtain thefollowing result.Proposition 5.4. The bounded linear map T W A.H / ! A.H / defined byis an isomorphism of A.H /-bimodules.T.x˝ y/ D x .2/ ˝ S 1 .x .1/ /yProof. It is evident that T is a bornological isomorphism with inverse given byT 1 .x ˝ y/ D x .2/ ˝ x .1/ y. We computeT.t .x ˝ y// D T.t .3/ xS.t .1/ / ˝ t .2/ y/D t .5/ x .2/ S.t .1/ / ˝ S 1 .t .4/ x .1/ S.t .2/ //t .3/ yD t .3/ x .2/ S.t .1/ / ˝ t .2/ S 1 .x .1/ /yD t T.x˝ y/and it is clear that T is left yH -linear. Consequently T is a left A.H /-linear map.Similarly, we haveT ..x ˝ y/ t/ D T.x˝ y.t ( O ı 1 // D x .2/ ˝ S 1 .x .1/ /y.t ( O ı 1 / D T.x˝ y/ tand hence T is right H -linear. In order to prove that T is right yH -linear we computeT 1 ..x ˝ y/ g/D T 1 .x .2/ .S 2 .y .2/ /*g(S 1 .y .4/ //.S 2 .x .1/ /ı/ ˝ Oı.y .1/ /y .3/ ( ı O 1 /D x .3/ .S 2 .y .2/ /*g(S 1 .y .4/ //.S 2 .x .1/ /ı/ ˝ Oı.y .1/ /x .2/ .y .3/ ( ı O 1 /D x .3/ .S 2 .y .2/ /*g(S 1 .y .4/ //.S 2 . ı*x O .1/ /ı/ ˝ Oı.y .1/ /.x .2/ y .3/ /( ı O 1which according to Proposition 4.1 is equal toD x .3/ .S 2 .y .2/ /*g(S 1 .y .4/ //.ıS 2 .x .1/ ( O ı// ˝ Oı.y .1/ /.x .2/ y .3/ /( O ı 1D x .6/ .S 2 .x .2/ /S 2 .y .2/ /*g(S 1 .y .4/ /S 1 .x .4/ //.S 2 .x .5/ /ı/˝ Oı.x .1/ y .1/ /.x .3/ y .3/ /( ı O 1D .x .2/ ˝ x .1/ y/ gD T 1 .x ˝ y/ g:We conclude that T is a right A.H /-linear map as well.
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EMS Series of Congress ReportsEMS S
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Editors:Guillermo CortiñasDepartam
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ContentsPreface....................
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IntroductionSince its inception 50
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Introductionxiwith D. Blecher, he c
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Program list of speakers and topics
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Categorical aspects of bivariant K-
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72 H. Emerson and R. MeyerIn this s
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74 H. Emerson and R. MeyerK-theoret
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76 H. Emerson and R. MeyerExample 4
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78 H. Emerson and R. Meyerand exact
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80 H. Emerson and R. Meyercondition
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84 H. Emerson and R. MeyerLet h and
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86 H. Emerson and R. MeyerCorollary
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88 H. Emerson and R. Meyer5 Dual-Di
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On K 1 of a Waldhausen categoryFern
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Page 143 and 144: 128 M. KaroubiThe same method may b
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Page 166 and 167: Equivariant cyclic homology for qua
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Page 191 and 192: 176 C. Voigtalgebra A. The space Cn
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Page 219 and 220: 204 J. CuntzConsider the action of
Page 221 and 222: 206 J. Cuntzprime numbers p and q,
Page 223 and 224: 208 J. Cuntz6 Representations as cr
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214 J. CuntzProposition 7.4. The K-
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On a class of Hilbert C*-manifoldsW
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On a class of Hilbert C*-manifolds
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228 U. Bunke, T. Schick, M. Spitzwe
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Parshin’s conjecture revisitedTho
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428 C. WeibelGiven (0.4) and axiom
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430 C. WeibelAs pointed out in the
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432 C. WeibelConsider the cohomolog
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434 C. WeibelWe need to see that th
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List of contributorsArthur Bartels,
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List of participantsNatalia Abad Sa
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