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Equivariant cyclic homology for quantum groups 161and since yF r is an isomorphism this impliesF r .S 2 .x// F l .ı 1 .x ( O ı 1 /ı/: (4.5)Assembling these relations we obtain the following result.Proposition 4.1. Let H be a bornological quantum group and let ı and O ı be the modularelements of H and yH , respectively. ThenS 4 .x/ D ı 1 . O ı*x( O ı 1 /ıfor all x 2 H .Proof. Using equation (4.2) and equation (4.5) we computeF l .S 4 .x// F r . O ı*S 2 .x// D F r .S 2 . O ı*x// F l .ı 1 . O ı*x( O ı 1 /ı/which impliesS 4 .x/ ı 1 . ı*x( O ı O 1 /ıfor all x 2 H since F l is an isomorphism. The claim follows from the observation thatboth sides of the previous equation define algebra automorphisms of H .5 Anti-Yetter–Drinfeld modulesIn this section we introduce the notion of an anti-Yetter–Drinfeld module over a bornologicalquantum group. Moreover we discuss the concept of a paracomplex.We begin with the definition of an anti-Yetter–Drinfeld module. In the context ofHopf algebras this notion was introduced in [12].Definition 5.1. Let H be a bornological quantum group. An H -anti-Yetter–Drinfeldmodule is an essential left H -module M which is also an essential left yH -module suchthatt .f m/ D .S 2 .t .1/ /*f (S 1 .t .3/ // .t .2/ m/:for all t 2 H; f 2 yH and m 2 M . A homomorphism W M ! N between anti-Yetter–Drinfeld modules is a bounded linear map which is both H -linear and yH -linear.We will not always mention explicitly the underlying bornological quantum groupwhen dealing with anti-Yetter–Drinfeld modules. Moreover we shall use the abbreviationsAYD-module and AYD-map for anti-Yetter–Drinfeld modules and their homomorphisms.According to Theorem 3.2 a left yH -module structure corresponds to a right H -comodule structure. Hence an AYD-module can be described equivalently as a bornologicalvector space M equipped with an essential H -module structure and an H -comodulestructure satisfying a certain compatibility condition. Formally, this compatibility conditioncan be written down asfor all t 2 H and m 2 M ..t m/ .0/ ˝ .t m/ .1/ D t .2/ m .0/ ˝ t .3/ m .1/ S.t .1/ /
162 C. VoigtWe want to show that AYD-modules can be interpreted as essential modules overa certain bornological algebra. Following the notation in [12] this algebra will bedenoted by A.H /. As a bornological vector space we have A.H / D yH y˝ H and themultiplication is defined by the formula.f ˝ x/ .g ˝ y/ D f.S 2 .x .1/ /*g(S 1 .x .3/ // ˝ x .2/ y:There exists an algebra homomorphism H W H ! M.A.H // given by H .x/ .g ˝ y/ D S 2 .x .1/ /*g(S 1 .x .3/ / ˝ x .2/ yand.g ˝ y/ H .x/ D g ˝ yx:It is easily seen that H is injective. Similarly, there is an injective algebra homomorphismH y W yH ! M.A.H // given byas well asH y .f / .g ˝ y/ D fg ˝ y.g ˝ y/ y H .f / D g.S2 .y .1/ /*f (S 1 .y .3/ // ˝ y .2/and we have the following result.Proposition 5.2. For every bornological quantum group H the bornological algebraA.H / is essential.Proof. The homomorphism H y induces on A.H / the structure of an essential left yH -module. Similarly, the space A.H / becomes an essential left H -module using thehomomorphism H . In fact, if we write yH for the space yH equipped with the trivialH -action the map c W A.H / ! yH y˝ H given byc.f ˝ x/ D S.x .1/ /*f (x .3/ ˝ x .2/is an H -linear isomorphism. Actually, the actions of yH and H on A.H / are definedin such a way that A.H / becomes an AYD-module. Using the canonical isomorphismH y˝H A.H / Š A.H / one obtains an essential yH -module structure on H y˝H A.H /given explicitly by the formulaf .x ˝ g ˝ y/ D x .2/ ˝ .S.x .1/ /*f (x .3/ /g ˝ y:Correspondingly, we obtain a natural isomorphism yH y˝ yH .H y˝H A.H // Š A.H /.It is straightforward to verify that the identity map yH y˝ .H y˝ A.H // Š yH y˝ H y˝yH y˝ H ! A.H / y˝ A.H / induces an isomorphism yH y˝ yH .H y˝H A.H // !A.H / y˝A.H / A.H /. This yields the claim.We shall now characterize AYD-modules as essential modules over A.H /.
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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 221 and 222: 206 J. Cuntzprime numbers p and q,
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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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List of contributorsArthur Bartels,
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List of participantsNatalia Abad Sa
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