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THE ABDUS SALAM INTERNATIONAL CENTREFORTHEORETICALPHYSICS

NONCOMMUT ATIVE CHERN-CONNES CHARACTERSOF SOME NONCOMPACT QUANTUM ALGEBRAS

Do NgocDiep1

Instituteof Mathematics,NationalCentre for ScienceandTechnology of Vietnam,P.O.Box631,Bo Ho 10.000,Hanoi,Vietnam

andTheAbdusSalamInternationalCentre for Theoretical Physics,Trieste, Italy

and

AderemiO. Kuku2

TheAbdusSalamInternationalCentre for Theoretical Physics,Trieste, Italy.

Abstract

We prove in this paperthat theperiodiccyclic homologyof thequantizedalgebrasof functionson

coadjointorbitsof connectedandsimply connectedLie group,areisomorphicto theperiodiccyclic ho-

mologyof thequantizedalgebrasof functionsoncoadjointorbitsof compactmaximalsubgroups,with-

out localization.Somenoncompactquantumgroupsandalgebraswereconstructedandtheir irreducible

representationswereclassifiedin recentworks of Do Ngoc Diep andNguyenViet Hai [DH1]-[DH2]

by usingdeformationquantization.In this paperwe computetheir K-groups,periodiccyclic homology

groupsandtheirCherncharacters.

MIRAMARE – TRIESTE

September2001

1E-mail: [email protected]: [email protected]

Intr oduction

Let�

be a connectedLie group, � a fixed maximalcompactsubgroupof�

and � a locally convex

algebraover the complex numbers. Oneof the major resultsin of V. Nistor in [N1] is Theorem1.1

of [N1], sayingthatup to localizationat somemaximalideal � of thealgebra�� of bi-invariant

functions � , � �� , � �"! �$# �, theperiodiccyclic homologyof thecrossedproduct �&% �

andthatfor �'%(� , areisomorphic,i.e.)�*,+.- �&% ��0/213 )�*,+.-�465 �7%(� �0/98 ;: �In thecasewhere � 3=<

andtheactionof�

on<

is trivial, thecrossedproductbecomesconvolution

andthe restrictionsof elementsof ��> �� , asfunctionson�

, to the coadjointorbits give algebrasof

quantizedfunctionson thecoadjointorbits. We prove the isomorphism(*) in thatcasewithout any lo-

calization.Themainreasonis thatoncoadjointorbits,thebi-invariantfunctionscorrespondto constants

andtheir localizationarethesameconstants.This will bedonein thefirst section.It is interestingthat

with this isomorphism(without any localization)for the quantizedalgebrasof functionson coadjoint

orbits,we canreducethecomputationof thenoncommutative Chern-Connescharactersto somemore

easilycomputedonesfor maximalcompactsubgroups.Ourmainobservation is thattheconjugacy in�

correspondsto theadjointactionon ? andto thecoadjointactionon ? - . It is especiallyimportantin the

concretecasesin thelasttwo sections,involving theLie groups@BA �C � and @BA < � , (seeD 3 and D 4).

Indeed,the homogeneousclassicalmechanicalsystemswith fixed Lie groupsof symmetrywere

classifiedascoadjointorbitsof theLie groupsof symmetryor their centralextensionby C , in thevector

spacedual to theLie algebras,see[K1]. For somespecialcases,whereall thenontrivial orbits areof

dimensionequalto thedimensionof thegroup(classEGF ), all suchLie groupshave beenclassifiedand

all theorbitsexplicitly computed.They reduceto thecasesof thegroupsof all affine transformationsof

therealor complex lines,see[D1].

Thegroupof affinetransformationsof therealline hastwo 2-dimensionalcoadjointorbits: theupper

andlowerhalf-planes,see[D1]. Usingdeformationquantization,aquantumanalogueof thehalf-planes

wasconstructedin [DH1]. The groupof affine transformationsof the complex line hasoneorbit of

complex dimension2: namely, the punctured(withdraw a complex line throughthe origin) complex

plane. Its quantumalgebrawasconstructedin [DH2]. We computethe K-groups,the periodiccyclic

homologyof thesequantumalgebrasandthecorrespondingChern-Connescharacters.

In order to obtain theseresults,we usethe methodsfrom [DKT1]-[DKT2] and the methodsand

resultsfrom [N1]: Wefirst constructsomediffeomorphimsrealizingcanonicalcoordinateson coadjoint

orbits andthenreducethe quantumalgebrasto the onesrelatedto thecorrespondingquantumgroups.

Wethenreducethesequantumalgebrasof quantumfunctiononcoadjointorbitsto thequantumalgebras

relatedto themaximalcompactsubgroups,thataremoreeasilycomputed.This computationis realized

in thelasttwo sections.

Noteson Notation: As usualwedenoteby capitalletterssomeLie groups,namely�

, � , etc.Their

2

correspondingLie algebraswill be denotedby thecorrespondingGothic letters,namely ? , H , etc. The

dualspaceto aLie algebraor avectorspacewill bedenotedby thesameletterwith : , e.g. ? - or I - is the

dualspaceof theLie algebra? or thevectorspaceI .)J*K+.-

will denotetheperiodiccyclic homology,

following [N1]. If � is a locally convex<

-algebra on which a Lie group�

actssmoothly, then �L% �denotesthecrossedproductof � with

�. M ON � and P ON � will meantheimaginaryandrealpartsof the

complex numberN , while as Q RTS and UVR�W wedenotetheintegralandfractionalpartsof R . C and<

means

thefield of realor complex numbers,respectively. For any<

-algebra A, denote�,X 3 UV� 3 �KY 4 � Wthewell-known Connes-Tsygancomplex.

1 Localization and coadjoint orbits

Let�

be a connectedand simply connectedLie group and � ��Z�T �� the convolution algebraof bi-

invariantfunctionson�

, see[N1]. We prove in this sectionthat localizationof theconvolution algebra� �� atamaximalidealcorrespondsto the(quantized)convolutionalgebraof functionswith compact

supporton thecorrespondingorbit. Wefirst modify someresultsobtainedin thework [N1] of V. Nistor.

1.1 Preparation

Recallthata quasi-cyclic objectin anAbeliancategory [ is a gradedobject �\ � T]"^ , \ #`_�ab [ �togetherwith morphismsc �ed \ gf \ �� , for h 3ji ! 8k8k8 !�l and m 4 � d \ nf \ satisfyingthe

following two axioms:

porq � c � cZs 3 cZs 4 � c � ! for hJtvu � q � c � m 4 � 3xw m c � �� for q�y h y lc for h 3zisee[N1] for moredetails.V. Nistorpointedout theexamplesof quasi-cyclic objectslike:

(i) thecyclic objects,

(ii) { X , where

{ d 3 �'% � 3 � �> � !|� � 3 U�} # � � � !|� ��~ } is of compactsupportW�!�aLie group, � alocally convex algebraonwhich

�actssmoothly, with thetwistedconvolution

product } :�� d 3�� } �� K �� c � !wherec � is afixedleft invariantHaarmeasureon

�,� d � f @�� asmoothactionof

�on �

in thesensethat themap ��f ��is continuousandunital andthemap ��f �� R � , for all R # �

is smooth.In thatcase,wehave { X 3 �&% �� X 3 � �> � 4 � !|��Y 4 � � with theoperations

c s } � � ^ !�� ! 8k8k8 !�� d 33

�� c sJ� �qJ� 8k8k8 �zq9� �� q�� 8k8k8 �zq � } � ^ ! 8k8k8 !�� s �� ! � ! � �� s ! 8k8k8 !�� c � !u 3 q ! 8k8k8 !�l(� q , c } � � ^ !�� ! 8k8k8 !�� d 3� � c � � �zq 8k8k8 �zq � } �� ^ !�� ! 8k8k8 !�� ! � � c �and �� 4 � } � � ^ ! 8k8k8 !�� d 3 � 4 � } � � ! 8k8k8 !�� !�� ^ ��As remarked in [N1], the operatorscZs�! � 4 � on the right handside are the onesof the cyclic

Connes-Tsygancomplex � X (see[C].

In particular, if � 3z<andtheactionof

�on

<is trivial, we have thealgebraof convolution.

(iii) Let � ,�

beasin (ii) above. Then UV� ;� ! � � � W is aquasi-cyclic object, where �7� �is anopen

set,� � is someotherLie groupandfor any grouphomomorphism� d � f � � , � � ! � � � d 3��> �L�G� 4 �� !|� Y 4 � � with similaroperationsc�s�!�m 4 � (see[N1] for moredetails).

1.2 An � -relativecohomologycomplex

Letusnow introducesomenew examplesof quasi-cyclic objects,relatedwith somequotientmaps.In (ii)

above,� � is agroup, � ahomomorphism.However in whatfollows,

� � is replacedby thehomogeneous

space ¢¡ � and � is just thequotientmap.

Considerthecanonicalquotientmap � d � f £¡ � , for somesubgroup . Consideranopenset�7� �. Define � ;� !| ¢¡ �� d 3 � �> ;� � ¤¡ �� 4 � !|� Y 4 � �

anddefinealso

c�sZ} � �� !| ¥� ^ ! 8k8k8 !�¦�§s�! 8k8k8 !| (� � d 3 � � cZs } �� !| (� ^ ! 8k8k8 !| ¥� �� cT¨ �ks � !u 3 q ! 8k8k8 !�l , cT¨ � is thequotientmeasureon thequotientspace ©¡ � ,

c ^ } � �� !9¦� ^ !| (� � ! 8k8k8 !| ¥� � d 3 �� c ^ } �� !| ¥� ^ !| ¥� � ! 8k8k8 !| ¥� �� c�¨ � ^ �and m 4 � } � �� !| ¥� ^ ! 8k8k8 !| ¥� � d 3 �qJ� � ��ª � 8k8k8 ! �«q � � 4 � } �� !| ¥� � ! 8k8k8 !| ¥� !| (� ^ ��Proposition 1.1 UV� ;� !| ¤¡ �� W�!|cZsT!|c ^ !�m 4 � � is a quasi-cyclicobject.

Proof. By similarargumentsto thosein thework of V. Nistor, see[N1], it is easyto seethatwehavealso

aquasi-cyclic object. ¬Thisquasi-cyclic objectis relatedto thequantumalgebrasof functionsof orbits,asweshallseelater.

4

Let usnow defineanactionof�

on thequasi-cyclic object UV� ;� !| ¡ �� W�!|cZsT!|c ^ !�m 4 � � . For a

fixedaction� d � f @�� , define® d � f � � � ;� !| ¤¡ �� by

® ª } � �� !| ¥� ^ !| ¥� � ! 8k8k8 !| (� � d 3 � Y 4 �ª } �� !| (� ^ � ! 8k8k8 !| (� � �� !for all � ! � � in

�, � ^ ! 8k8k8 !�� in

�, } in � ;� !| ¤¡ �� .

In particular, if 3 UZ¯bW and � 3 �, we have £¡ �x13 �

anda map ° d � � ! �� f �±% �� X ,definedby °6} �� ^ ! 8k8k8 ! � � d 3 ��³² ® ª } � � !�� ^ ! 8k8k8 !�� !where� ^Kd 3 � ^ , � � d 3 � ^ � � , ��´ d 3 � ^ � � � ´ , .... � ed 3 � ^ � � � ´ 8k8k8 � ,

² d 3 ��µ � ��¶ � 8k8k8·�¸� µ ��¹ � �� .Wenow definethemap º whichgivesriseto anisomorphismin Hochschildhomology, (see[N1]).

Consideragainan openset � which is @¼» � -invariant in�

, and define º d � ;� !| ½¡ �� f� ;� !¾UZ¯bW � by º�} � �� d 3 �"¿ÁÀJÂ��Ã µ·Ä ¹ } �� !| ¥� ^ ! 8k8k8 !| (� � c�¨ � ^ � 8k8k8 c�¨ � �Notethatwe hereusec � � to denotetherelative quasi-invariantmeasureon thequotientspace ©¡ � .

Lemma 1.2 º is a morphismof quasi-cyclicobjectsand Å º � is an isomorphismof thecorrespond-

ing Hochschild homology groups.

Proof. In [N1] the similar assertionwasproven for the absolutecase. In the relative case,we have a

similarargument,whichwe omit here. ¬ .

Supposewe have somecontinuoushomomorphismof acompactgroup � into�

. Wehave then

Proposition 1.3*,+r- � �G!| £¡ ��Æ��Ç13 *,+.- � � �L%(� ��Æ�� À �

Proof. We have from [N1] the isomorphism*,+ - � �¥! ��ÆÈ�g13 *K+ - � � �É%Ê� ��ÆK��

. From the

definition of complexes defining*K+.- � �¥! �� Æ � and

*K+B- � � �Ë%Ì� � Æ ��, we have isomorphic -

invarianthomologygroup*,+.- � � �G! �� Æ � À ��13 *,+r- � � �±%$� � Æ � À �

. But*K+.- � � �¥! �� Æ � À �K13*,+.- :� � �G!| Ë¡ �� Æ � ¬

Let�

be a connectedLie group, Í a smooth�

-module, Î 3 »�ÏÑÐ ��Ò � �. We now definethe�

-equivarianthomology:Considerthecomplex of relative Lie algebrahomologyi f pÓ 5 ? Ò H � � À Í � � Æ <ÕÔ� f 8k8k8 Ô� f pÓ ^ ? Ò H � � À Í � � Æ < � f Í � � < � f i ! OÖ �where × d pÓ s ? Ò H � � À Í � � Æ < � f pÓ s ��V ? Ò H � � À Í � � Æ <

is definedas

× ,Ø\ � Ó 8k8k8 ÓÙØ\ s �`Ú � d 3 sÛ �ÝÜ � Ø\ � Ó 8k8k8 Ó ¦\ � Ó 8k8k8 Ø\ s �`\ � �Ú � �� Û ��Þ s � q � �

46ß Q \ � ! \ ß S ÓÙØ\ � Ó 8k8k8 Ó ¦\ � Ó 8k8k8 Ó ¦\ ß Ó 8k8k8 ÓÙØ\ s �`Ú !andfor \ � # ?"! Ø\ � is theclassof \ � in ? Ò H , and ¦\ � indicatesthat Ø\ � is omitted.In thecomplex (I) Í is

regardedasasmooth -module.It is nothardto prove

5

Proposition 1.4 Thiscomplex is acyclic.

For acyclicity of a similar complex with<

insteadof , see[N1](Prop. 3.6). In our relative case

this is truealso,ascanbeverified.

1.3 A relativecomplexand the correspondingMain Lemma

In the main body of the paperof V. Nistor is the constructionof complexescomputingthe mentioned

isomorphism, )�*,+.- �&% ��0/213 )�*,+.-�465 �7%(� �0/98Wenow introducea relative complex satisfyingall theconditionsof theMain Lemmaof Nistor.

1.3.1 The Main Lemma

Following [N1], let usconsiderthefollowing data:à Two exactsequencesof quasi-cyclic objectsof completelocally convex spaces

Oáeq � inâ ��6�jã â ��6�Éä ¿ ^ Ã Ôâ ��å� 8k8k8 Ôâ ��å�æä ¿ 5 �� Ã Ôâ ��6�Ôâ ��å� ä ¿ 5 Ã â ��6� i OáÈç � i �6�� f ä ¿ ^ Ã è�6�� f 8k8k8 è�å�� f ä ¿ 5 �� Ã è�6�� f ä ¿ 5 Ã �6�� f�6�� f é �6�� f i

à A � � -actionof C on ä ¿ s Ã for any u suchthat ê � 3 m 4 � 4 � andthederivative ë 3£ì�í|îì|ïJð ï Ü6^ of ê at� 3ziis equalto ë 3 ×Vñ�òóñ¸× , with theconventionthat × ä ¿ ^ Ã � 3 ñ ä ¿ 5 Ã � 3zi

à Theendomorphismsq �ôm 4 � 4 � areinjective on ä¿ s Ã for all u 3zi ! 8k8k8 !|Î andfor all l2õ i

.

Let usrecallthataprecyclic objectis aquasi-cyclic object � �\ � �]"^ !|c s !�m 4 � � suchthat m 4 � 4 � 3 q .Givenaprecyclic objectoneconstructstheConnes-Tsygancomplex asin thecyclic case,see[N1]. The

homologyof the 2-periodictotal complex ö�÷��ø �\ �associatedto the bi-complex ø �\ �

is definedas

theperiodiccyclic homology)�*,+ �\ �

of thecomplex \ , (see[N1], Definition 2.2,andthedefinition

thereafter).

Wenow associateto thedatasatisfyingtheabove definition,somenew objectsùä ¿ ^ Ã d 3 ä ¿ ^ Ã Ò ëúä ¿ ^ Ã ! ùä ¿ s Ã d 3 ä ¿ s Ã Ò ëúä ¿ s Ã ò`ñ�ä ¿ s �� Ã � !û��u 3 q ! 8k8k8 !|Î 8andstatethefollowingsLemmadueto V. Nistor (see[N1]

Lemma 1.5(The Main Lemma) (i) For any u 3 q ! 8k8k8 !|Î , theobjectùä ¿ s Ã is a precyclicobject.

(ii) Thecomplex inâ ��6� ã â ��6� ùä ¿ ^ Ã Ôâ ��6� ùä ¿ � Ã Ôâ ��6� 8k8k88k8k8 Ôâ ��6� ùä ¿ 5 �� Ã Ôâ ��6� é 13 ä ¿ 5 Ã Ò ñ�ä ¿ 5 �� Ã â ��6� iis acyclic.

6

(iii))�*,+.- ùä ¿ s Ã � 3zi

for any u 3zi ! 8k8k8 !|Î .(iv)

)�*,+.- ã ��13 )�*,+.-�465 é ��8It wasshown in [N1] thatall theconditionsof theMain Lemmaaresatisfiedfor theabsoluteNistor’s

complex. Wenow verify conditionsof this lemmafor the -relative complex.

1.3.2 Relative form of ä ¿ s Ã and ×For any Lie group

�, let

<ýübe the one-dimensionalrepresentationof

�by multiplication with the

modularfunction þ ,��ÿ 3 U �2# � ð � � 3 �"� W , ? ÿ 3�� Ï�� ÿ , � amaximalcompactsubgroupof

�and�7� �

, � ÿ 3 �� ÿ . Let þ�� denotethemodularfunctionfor��ÿ

,

Í � d 3 w < ü� � � ;� �O! � ÿ � ! if � � # � ÿ !< ü� � � ;� � ! ��ÿT��Ò �q � � � � ü� � � ;� � ! ��ÿ�� ! if � � Ò# � ÿ d 3 w � ÿ � ! if � � # � ÿ !Themaximalcompactsubgroupin

� ÿ � Ò �� ! if � � Ò# � ÿand � d 3�� Ï��

.

Definethe -relative ? ÿ -cohomologycomplex U�� sbW of Í�� by � s d 3 pÓ s ? ÿ � Í�� À and × ^�d � s f� s �� with

× ^ �\ ^ Ó 8k8k8 Ó�\ s �`Ú � 3 sÛs Ü � \ � Ó 8k8k8 Ó ¦\ � Ó 8k8k8 Ó�\ s �`\ � �Ú �� Û�� Þ ß s � q � ß Q \ � ! \ ß S Ó�\ � Ó 8k8k8 Ó ¦\ � Ó 8k8k8 Ó�\ ß Ó 8k8k8 Ó \ s �`Ú

Now wedefineä ¿ s Ã d 3 � s Ò ��s , where��s is generatedby � Ó � s �� and �� q � � s for all �Å# � ÿ�� .It is nothardto seethat

× ^ �\¢Ó�� 3 \` �� \¢Ó × ^ �� !û� \ # ? ÿ !û� � # � s �� !where\ó �� meansthecontractionof \ and � with valuesin � s �� . Define × to bethequotientmapof× ^ on ä ¿ s Ã d 3 � s Ò ��s .

Definealso c s d � ß f � ß !0u 3 q ! 8k8k8 !�� by

c�s �\ � Ó 8k8k8 Ó \ ß �� `Ú � d 3 \ � Ó 8k8k8 Óú\ ß ��È� cZs Ú !� \ � Ó 8k8k8 Ó�\ ß #ÅÓ ß ? ÿ !û� �¥# <Çü !û� Ú³#�� .

1.3.3 Definition of ñ in the relative case

Let � s9ñ beasin 1.3.1and1.3.2.For every element� # � � ;� � !�? ÿT� , define � �� \ � d 3 \ and

ñ ^Kd � ß f � ß¾4 � ~ ñ ^ �� d 3 � Ó � !Define ñ d ä ¿ ß Ã f ä ¿ ßk4 � Ã to bethequotientmapsof ñ ^ mode � �s .

7

1.3.4 Relative form of ê and the equation ë 3 ñ¸×.òÌ×ZñLet

��ÿ, beasbefore, � 3 �� Kÿ . Firstdefineanaction ê � of C on � ;� � !| ©¡ ��ÿT� as ê �ï } � �� \ !| � � ^ ! 8k8k8 !| � � � d 3 ®"!$#&% ¿ ï�' Ã �� (� �\ � !| � � ^ ! 8k8k8 !| � � � ! ;:�: �

for any } # � ;� � !| � ¡ ��ÿT� , �9#(C . It is naturalto extendthis actionto � s by� 8 �\ � Ó 8k8k8 Ó�\ ß �)�È�`Ú � d 3 \ � Ó 8k8k8 Ó�\ ß �� ê �ï �Ú � !for all \ � ! 8k8k8 ! \ ß # ? ÿ , �v# <Çü

and Ú # � ;� � !| £¡ ��ÿT� . Let ë ^ bethederivative of theactionof ê �ïabove at0. By asimilar computationasin [N1], onecanseethat

ë ^ 3 × ^ ñ ^ òóñ ^ × ^ 8It is alsonot hardto seethateach� �s is invariantunderthis actionandthereforewe getandaction ê ofC in ä ¿ s Ã .

Finally for theabove data,in away analogousto thatin [N1], we define

ã d 3 < ü� � À � ;� � !| � ¡ ��ÿ�� * <n3 � �7% �Kÿ�� X+ � � Àand é±d 3xw � ;� �,� � ÿ !| -��¡ � ÿ � � � Æ * � < if � � # � ÿ� ;� � � ^ !| � ¡ � ÿ � � �/. <

if � � Ò# � ÿ !where

^ is theinverseimageof

in� ÿ �

of themaximalcompactsubgroupof� ÿ � Ò �� .

By a similar argumentto that in [N1], we can concludethat for the -relative complex all the

conditionsof themainlemmaof [N1] arealsosatisfied.

1.4 Passageto coadjoint orbits

After thestatementof his mainresult(Theorem1.1,p. 4 in [N1]), V. Nistor statedasfollows. “This fits

with Mackey’s methodof orbits,exceptthatnow for reasonswe do not yet understand,we obtainorbits

on � Ï�� ratherthanin � Ï0� �� - . An interestingfeatureof theresultis worthwhilestressing:thereis

no � -obstructionin cyclic cohomology.” This meansthathedidn’t work with thecoadjointorbits. We

now explain thatit is naturalto passto coadjointorbitsandthatthelocalizationdisappearson coadjoint

orbits.

Let�

be a connectedandsimply connectedLie group, ? 3�� Ï0� �� , ? - the dual of ? . Let � be a

fixedpoint in ? - , ��1 thestabilizerof � . Let usconsiderthenaturalprojection�32 �/1 ¡ � 3�4)5 f ? -

definedas � ^ �f ��1 � ^ 3 ù� ^ # ?76 � ^ # ? - . Supposethat � ^ 3 ��(� ù� ^ � , i.e.ù� ^ 398;: � ^ , then

from thewell-known VanCampbell-Hausdorff-Dynkin formulafor80: ��(� �\ � �� =< �� , wededucethat80: �� ^ � �� 3 @¼» ÿ ù� ^ . Wehave thereforethefollowing

Lemma 1.6 Under themap�>2 ��1 ¡ � 3?4@5 f ? - , theelement� � ^ � �� in

�goesto theelement@�» �� ^ , andtheconjugacyorbit of � ^ goesto thecoadjointorbit4 1�¶ 3 UV@�» �� ^ ð �Å# � W 8

8

1.5 Localization on coadjoint orbits

Weapplytheconstructionof thesubsection1.2to thecaseof coadjointorbit4g3 ��1 ¡ � .

Lemma 1.7 Let�

be a connectedand simplyconnectedLie group. There is a natural isomorphism��> ? ��13 ��> ;� � , where � is anopensetin�

.

Proof. Theexponentialmap �� d ? f �is a local diffeomorphismandthe image � 3 ��(�.? of ? is

openin�

. ¬ .

Lemma 1.8 There is a natural isomorphismbetweenconvolutionalgebras

� �> ? ��13 � �> ? - �Proof. It is naturalto identify ? with ? - . ¬Lemma 1.9 There is a natural isomorphismbetweenconvolutionalgebras

� �> ? � � 13 � �> ? - � �Proof. Undertheisomorphism? 13 ? - , theadjointaction @¼» � � becomesthecoadjointaction@�» -� ��¸�� .¬Lemma 1.10 There is a one-to-onecorrespondencebetweenthelocalizationof thealgebra �� at

maximalidealsandthecentral characters of theLie group�

.

Proof. On thecoadjointorbits,bi-invariantfunctions } # �� areconstant.Therefore,localization

at any maximal ideal at pointsof the coadjointorbits arethe sameasthe constants.The valueof the

constantfunctiononcoadjointorbitsarethesameasthecentralcharacterof therepresentationassociate

to theseorbits,(seefor example[K1]). ¬Thefollowing lemmas1.11and1.12areimplicit in [K1]:

Lemma 1.11 For almostall connectedand simplyconnectedLie groups,namely“almost algebraic”

or solvableLie groups,there is a one-to-onecorrespondencebetweenthe central characters and the

irreducibleunitary representations

Proof. For largeclassesof connectedandsimply connectedLie groups,thestatementhasbeenverified.

It wasverifiedalso,(see[K1]) for almostall connectedandsimplyconnectedLie groups. ¬Lemma 1.12 For almostall connectedandsimplyconnectedLie groups,e.g. almostalgebraic or solv-

ableLie groups,there is a one-to-onecorrespondencebetweentheirreducibleunitary representationsof�andthecoadjointorbits or their coverings

9

Proof. For largeclassesof connectedandsimply connectedLie groupsthis statementhasbeenverified,

see[K1] for moredetails. ¬Wenow have themaintheorem.Let usfirst fix somenotations.For any Lie group

�andsubgroups� ,

4n3A4 1acoadjointorbit of

�, passingthroughafixedpoint � in ? - , weshallwrite

4 ð Æ 3A4 1�B Æ for

thecoadjointK-orbit passingthrough � ð C . We alsowrite � 1�B D for thestabilizerof coadjoint � -action

at � ð C .Theorem 1.13(Main Theorem) Let

�be a connectedand simplyconnectedLie group,

4 3E4 1a

coadjoint orbit of�

passingthrough a fixed point � in ? - , � a maximalcompactsubgroup of�

,Î 3 »"ÏÝÐ ��Ò � �and

4 Æ thecoadjointorbit passingthrough � ð C . Let ��> 4 � (resp. ��> 4 ð Æ �� bethe

quantizedalgebra of functionson4

(resp.,4 ð Æ ). Thenwehavean isomorphism)�*,+ - � �> 4 � ��13 )J*K+ -�465 � �> 4 Æ ��8

Proof. First observe thata coadjointorbit4 13F ? - canbe identifiedwith thehomogeneousspace ¢¡ � , where 3 ��1

is thestabilizerof anarbitrarypoint � in4 1

. Wehave therefore

� �> 4 1Ç��13 � �> ��1 ¡ ��13 � �> �� À 8 GLet usindicatewith sub-index � therestrictionsonto � . Wehavethesamenotationsfor afixedmaximal

compactsubgroup� , � �> 4 1,D9�Ç13 � �> � 1,D ¡9� ��13 � �> � � À D. FromProposition1.3,wehave*K+ - :� � �¥!| Ë¡ �� Æ �Ç13 *K+ - :� � �&%¥� � Æ � À 8

Also we have from Proposition1.4thatthecomplexi f pÓ 5 ? Ò H � � À Í � � Æ < Ô� f 8k8k8 Ô� f pÓ ^ ? Ò H � � À Í � � Æ < � f Í � � < � f iis acyclic. UsingLemma1.9,we canidentify � �> 4 1�� with somealgebraof functionson a conjugacy

orbit of�

in ? with adjointaction. A function in ��Z�� meansa functionwhich is constanton con-

jugacy classesof�

. So from Lemmas1.7-1.9,it is thesameasa function,which is constanton each

coadjointorbit. FromLemmas1.10-1.12,a maximalidealin thespaceof centralfunctionscorresponds

exactly to a coadjointorbit. The localizationat the ideal � thereforemeanstakingrestrictionsof func-

tionsontheassociatedorbit. Now wecanapplythetheorem1.1of Nistor to concludethatthehomology

groupswith localizationareisomorphicto thegroupswithout localization. ¬2 NoncommutativeChern-Connescharacters

Let usnow briefly recalltheconstructionof noncommutative Chern-Connescharacters.Wethenpresent

an interestingresult(Theorem2.1) for the noncommutative Chern-Connescharactersof the quantized

algebrasof functionson coadjointorbits.

10

2.1 K-gr oups.Connes-Kasparov-Rosenberg Theorem

In the following, � -groupsshallmeanthe H Ò �ç � -gradedalgebraic� -groupsof algebrasover thefield

of complex numbers. For connectedsolvable Lie groups,the so called Connes-Kasparov conjecture

assertingsimilar isomorphismshasbeenproved (see,e.g. [J]) andit is known for large classesof Lie

groupsasConnes-Kasparov-Rosenberg Theorem,i.e.

� - � �> ��13 � -�465 � � � �� !whereÎ 3 »�ÏÑÐ ��Ò � �

.

2.2 NoncommutativeChern-Connescharacters

For the generalnotion of Chern-Connescharacters,readersarereferredto the work of J. Cuntz[Cu].

Fromtheworksof J.Cuntz[Cu] andV. Nistor [N1]-[N2] we candeducethatthereis a naturalnoncom-

mutative Chern-ConnescharacterI � with valuesin thelocalizationsof)J*K+.- � � , asI � d � - � �> �� f )�*,+r- � �> ��0/.8

2.3 The commutativediagrams

Thereis anaturalcommutative diagram:

� - ��> �� > �KJ�å�"� f )J*K+r- ��> ��0/LLM LLM� -�465 ��> � �� > � D�å�"� f )�*,+.-�465 ��> � ��0/wherethefirst verticalrow is theConnes-Kasparov-Rosenberg isomorphism,andthesecondverticalone

is theisomorphismof V. Nistor. Thiscanbereducedto themaximaltoruscase

� -�465 ��> � �� > � D�6�� f )J*K+r-�465 ��> � ��0/LLM LLM� -�465 � �> =N ��PO > �RQ�6�� f )J*K+.-�465 � �> =N ��PO/where S 3 S =N � is theWeyl groupcorrespondingto themaximaltorus N andthesub-indicesof I �indicatethe correspondingtarget groups. In this secondcommutative diagramthe first vertical row is

thewell-known resultof theK-theoryof compactLie groupsandthesecondoneis thereductionof V.

Nistor in [N1]. Thehorizontalrow on thebottomis anisomorphism,asis well-known in topology. We

have thereforethefollowing interestingconsequence

Theorem 2.1 Let�

be a connectedand simplyconnectedLie group,4±3�4 1

a coadjointorbit of�

passingthrougha fixedpoint � , � a maximalcompactsubgroup of�

,4 ð Æ thecoadjointorbit of �

passingthrough � ð C , T themaximaltorusof�

in � and S 3VU =N ��Ò N theWeyl groupcorresponding

to N . For anyco-adjointorbit4

, let ��> 4 � be thequantizedalgebra of functionson4

with compact

11

support.Then,there is a commutativediagramfor thenoncommutativeChern-Connescharacters of the

quantizedalgebra of functionson coadjointorbits

� - ��> 4 �� > �XW�6�� f )J*K+ - ��> 4 ��Y LLM LLM YZYZY� -�465 � �> 4 Æ �� > �XW D�6�� f )�*,+.-�465 � �> 4 Æ ��Y�Y LLM LLM Y�[� -�465 ��> =N �� O > �RQ�6�� f )J*K+ -�465 ��> =N �� O

andmodulotorsion,thenoncommutativeChern-Connescharacters are isomorphisms.

Proof. Let usfirst considerthecommutative diagram

� - ��> �� > �KJ�å�"� f )J*K+r- ��> ��0/LLM LLM� -�465 ��> � �� > � D�å�"� f )�*,+.-�465 ��> � ��0/and � -�465 ��> � �� > � D�6�� f )J*K+ -�465 ��> � �� /LLM LLM� -�465 ��> =N �� O > �RQ�6�� f )J*K+.-�465 ��> =N �� O/Localizing � �> �� at theideals� , correspondingto theorbit

4yields � �> 4 � , aswasexplainedin the

proofof Theorem1.13.Thenby doingsimilarcomputationsto whatwasdonein [DKT1]-[DKT2], since

therepresentationsof � aredefinedby their restrictionsto maximaltori, we have the isomorphism(I),

(II) and(IV). Theisomorphism(III) is themaintheorem1.13. ¬Notethatin thisdiagramwe don’t needto take localizationbecause,asexplainedin theproof of the

maintheorem1.13,localizationat anideal � in � ��Z�� give usthequantizedalgebrasof functionson

coadjointorbits.

Note that)�*,+.- 8 � here is formally different from

*�)J- 8 � in [DKT1]-[DKT2] in the sensethat

in the definition of)�*,+.- 8 � productsare usedin placeof direct sumsin the definition of the total

complexes.However, thisdoesnotaffect thedefinitionof thenoncommutativeChern-Connescharacters,

becausefor theentirecyclic homology*]\ - 8 � , oneusesconvergentseriesof finite degreecycles. For

thealgebraicversion*�)J- 8 � , we usedtheCuntz-Quillenä -complexes,which reducedalsoto products

in thebicomplexes.

In thenext two sectionswe shallapplyTheorem1.13to deduceisomorphismsof cohomologiesof

quantizedalgebrasof functionson coadjointorbits. In orderto do this,we recalltheresultsfrom [DH1]

aboutthealgebrasandthencomputethe K-theory, periodiccyclic homologyandthe noncommutative

Cherncharacters.

12

3 Quantum half-planes

Applying themaintheorem1.13,wecomputein thissectionthenoncommutative Chern-Connescharac-

tersfor thequantumalgebrasof functionson thehalf-planes.

3.1 Deformation quantization

Let usrecallsomeresultsfrom [DH1]: RecallthattheLie algebra? 3_^ A �C � of affine transformations

of the real straightline is describedasfollows, seefor example[D1]: TheLie group @BA �C � of affine

transformations: �G# C �f R � ò a ! for someparametersR ! aK# C !|Ra`3zi 8is known to bea two-dimensionalLie groupwhich is isomorphicto thegroupof matrices

@BA �C �¼13 U b R ai q-c ð R"! aK# C ! Ra`3zi W 8Weconsiderits connectedcomponent@ A ^ �C � of theidentityelementgivenby� 3 @ A ^ �C � 3 U b R ai q c ð R"! aK#(C !�Red i W 8Its Lie algebra ? 3�^ A �C � 13 U b � ®i i c ð � !�® # C Wadmitsabasisof two generators\ ! < with theonly nonzeroLie bracket Q \ ! < S 3 < , i.e.

? 3�^ A �C �B13 U � \ ò`® < ð Q \ ! < S 3 < ! � !�® # C W 8Theco-adjointactionof

�on ? - is given(seee.g.[K1]) byf � � � � !g��h 3 f � !|@�» � �� h�!û� �=# ? - !�� # �

and � # ? 8Denotetheco-adjointorbit of

�in ? , passingthrough� by4 1 3 � �� d 3 UV� � � � ð � # � W 8

Becausethegroup� 3 @BA ^ �C � is exponential(see[D1]), thenfor �=# ? - 3�^ A �C � - , we have4 1 3 UV� ��(� ;� � � ð �L# ^ A �C � W 8

andhencethat f � �� !g�]h 3 f � !�� ^ » + � ��h 8It is easythereforeto seethat

� ��(� � � � 3 f � !�� ^ » + � \ h \ - ò f � !�� ^ » + � < h < - 813

For ageneralelement� 3 � \ òó® <x# ? , we have�� ^ » + � 3 �ÛbÜ6^ ql�i b i i® � � c 3 b q i� ¯ �kj c !where� 3 � òó® ò jl �q � ¯ l � . Thismeansthat

� �� 3 m� ò`¨¸� � \ - ò ¨�¯ �kj � < - 8Fromthis formulaonededuces[D1] thefollowing descriptionof all co-adjointorbitsof

�in ? - :à If ¨ 3ji

, eachpoint �� 3 � !�n 3 i �on the abscissaordinatecorrespondsto a 0-dimensional

co-adjointorbit 4po�3 U ��\ - W�! �G# C 8à For ¨�`3zi

, therearetwo 2-dimensionalco-adjointorbits: theupperhalf-planeU m� !�¨ � ð � !�¨ #C !�¨)d i W correspondsto theco-adjointorbit4 4 d 3 U � 3 m� ò`¨¸� � \ - ò ¨�¯ �kj � < - ð ¨qd i W�! (1)

andthelowerhalf-planeU m� !�¨ � ð � !�¨ # C !|¨gt i W correspondsto theco-adjointorbit4 � d 3 U � 3 m� ò`¨¸� � \ - ò ¨�¯ �kj � < - ð ¨2t i W 8 (2)

Weshallwork henceforthon thefixedco-adjointorbit4 4

. Thecaseof theco-adjointorbit4 � couldbe

similarly treated.Firstwestudythegeometryof thisorbit andintroducesomecanonicalcoordinatesin it.

It is well-known from theorbit method[K1] thattheLie algebra? 3�^ A �C � is realizedby thecomplete

right-invariantHamiltonianvectorfieldson co-adjointorbits4 1213 ��1 ¡ � with flat (co-adjoint)action

of theLie group� 3 @BA ^ �C � . Ontheorbit

4 4wechooseafix point � 3 < - . It is well-known from the

orbit methodthatwecanchooseanarbitrarypoint � on4 1

. It iseasyto seethatthestabilizerof this(and

thereforeof any) point is trivial, i.e.�/1 3 UZ¯bW . Weidentify therefore

�with

��rs ¡ � . Thereis anatural

diffeomorphismt�»vu � ��(� 8 � from thestandardsymplecticspaceC ´ with symplectic2-form ck° Ó c�Î in

canonicalDarboux °�!|Î � -coordinates,ontotheupperhalf-planew 4 13 C % C 4 with coordinates °�! ¯ 5 � ,which is, from theabove coordinatedescription,alsodiffeomorphicto theco-adjointorbit

4 4. We can

usetherefore °�!|Î � asthestandardcanonicalDarbouxcoordinatesin4 rxs

. Therearealsonon-canonical

Darbouxcoordinates �� !�n � 3 °�! ¯ 5 � on4 rxs

. Weshow now thatin thesecoordinates �� !�n � , theKirillo v

form lookslike � rs �� !�n � 3 �y c �JÓ czn , but in thecanonicalDarbouxcoordinates °�!|Î � , theKirillo v form

is just thestandardsymplecticform ck° Ó c�Î . This meansthat therearesymplectomorphismsbetween

thestandardsymplecticspaceC ´ !|ck° Ó c�Î � , theupperhalf-plane w 4 ! �y c � Ó czn � andtheco-adjointorbit 4 rs ! � rxsk� . Eachelement� # ? canbe consideredasa linear functionalù� on co-adjointorbits, as

subsetsof ? - , whereù� =� � d 3 f � !g��h . It is well-known that this linear function is just theHamiltonian

functionassociatedwith theHamiltonianvectorfield ÚX{ , which represents� # ? following theformula

�Ú { � � �� d 3 cc � � �� ð ï Ü6^ !û�6� # � � 4 4 ��814

TheKirillo v form � 1 is definedby theformula� 1 �Ú { ! Ú�| � 3 f � !VQ}�B!�mBS�h�!û�~�B!�m # ? 3�^ A �C �¾8 (3)

This form definesthe symplecticstructureand the Poissonbrackets on the co-adjointorbit4 4

. For

the derivative alongthe direction Ú { andthe Poissonbracket we have relation Ú { � � 3 U ù��! ��W�!û�6� #� � 4 4 � . It is well-known in differentialgeometrythat thecorrespondence�x�f Ú { !g� # ? definesa

representationof our Lie algebraby vectorfields on co-adjointorbits. If theactionof�

on4 4

is flat

[D1], we have the secondLie algebrahomomorphismfrom strictly Hamiltonianright-invariant vector

fieldsinto theLie algebraof smoothfunctionsontheorbitwith respectto theassociatedPoissonbrackets.

Denoteby � theindicatedsymplectomorphismfrom C ´ onto4 4

°�!|Î � # C ´ �f �� °�!|Î � d 3 °�! ¯ 5 � # 4 4It wasprovenin [DH1] that:à Hamiltonianfunction � { 3 ù� in canonicalcoordinates °�!|Î � of theorbit

4 4is of theformù� � �� °�!|Î � 3 � °«ò`®�¯ 5 ! if � 3 b � ®i i c 8

à In thecanonicalcoordinates °�!|Î � of theorbit4 4

, theKirillo v form � r s is just thestandardform� 3 ck° Ó cTÎ .Let usdenoteby � the2-tensorassociatedwith thecanonicalKirillo v standardform � 3 ck° Ó cTÎ

in the canonicalDarbouxcoordinates.Recall the deformedG-productof two smoothfunctions ��!�� #�� 4�� G�� 3 � 8 �,ò Û � ] � q� b �ç h c��

��!�� ! G�G��where

� � ��!�� 3 � � ¹�� s ¹ � �0� � s � 8k8k8 � �� s �&� � ¹ 8k8k8 � � � � � s ¹ 8k8k8 � s � �"!

with theordinarymulti-index notationsof thepartialderivations. Note that in [DH1], G�G�� wasshown

for thenormalizedPlanckconstant� 3 q . Thesituationis thesamefor anarbitrarynonzerovalueof � .

It wasshown thatto every element\É# ? correspondsa functionù\ on ? - andthereforeon theK-orbits4 1

.

It wasshown in [DH1](Proposition3.1)thath� ù\ G�� h� ùmn� h� ùm G�� h� ù\ 3 h��Q \ !�m S0!û�~�B!�m # ^ A �C �·8Onethereforehasa representation \ � f h� ù\ G��

15

of the Lie algebra ��> 4 � � by the leftG��

-multiplication. On the half-planewith the fixed Darboux Î !O° � -coordinates,onefixestheFouriertransformationin ° -coordinate

Í�� êå!|Î � d 3 qç�� u �� .hÁ°"ê � � °�!|Î � c§°andobtainthatfor theelement

ù� 3 � °�ò�®�¯ 5 , theoperator� { actingonthedensesubspace� ´ �C ´ ! ì � ì 5´P� � �of smoothfunctionsby left

G��-multiplicationby h ù� G�� , i.e. � { � � d 3 �� ù� G�� . It waspreciselycomputed

(seeProposition3.4 in [DH1]) that

¦� { � � d 3 Í�� { � Í �� 3 � qç � 5 � � � � � 8It wasalsoprovenin Theorem4.2of [DH1] that:

Therepresentation�� ¦� { � of thegroup� 3 @BA ^ �C � is exactly theirreducibleunitaryrepresenta-

tion m~� Ä of� 3 @ A ^ �C � associated,following theorbit methodconstruction,to theorbit

4 4, which is

theupperhalf-planew 13 C % C - , i. e.

��(� ¦� { � � � n � 3 m~� Ä � � � � n � 3 ¯x�� y � Rzn � !û�6� # � ´ �C - ! cznn � !where� 3 ��(�� 3 b R ai q�c 83.2 K-gr oupsand periodic cyclic homologyand Chern-Connescharacters

Let usapplynow thegeneralnotionof Chern-Connescharactersto theexampleof thequantumalgebras

of functionsonthecoadjointorbitsof thegroupsof affine transformationof thereal(in thissection)and

complex (in thenext section)lines. Recallthat thenoncommutative Chern-Connescharactersaresome

homomorphismsI � from the K-groups � - 8 � to the correspondingperiodic cyclic homologygroups)J*K+ - 8 � .Lemma 3.1 In the groups @ A �C � , the maximalcompactsubgroups is � 13 H.´ 3 H Ò �ç H � . In its

connectedcomponentof identity @BA ^ �C � , themaximalcompactsubgroupis trivial, i.e. � 13 UZ¯bW .Proof. Theproof is clear. ¬Proposition 3.2 Let

4 4bethecoadjointorbit which is theupperhalf-plane. Then

� - � �> 4 4 ��13 )�*,+.- � � �> 4 4 ��13 UZ¯bW 8andtherefore thenoncommutativeChern-Connescharacters are isomorphisms.

Proof. Becausethemaximalcompactsubgroupsof�

aretrivial, wecanconcludethattheK-groupsand

the)�*,+r-

-groupsarealsotrivial. ¬

16

4 Quantum punctured complexplane

In this sectionwe demonstrateanotherapplicationof themaintheoremfor thegroupof affine transfor-

mationsof thecomplex line. Thedeformationof thecoadjointorbitsof thisgroupis in somesensemore

complicatedthantheonein therealcase,seee.g.[DH2].

4.1 Deformation quantization

Thegroup @ A < � is definedas

@BA < � 3 U b R ai q c ð R"! a�# < !|R�`3�i W 8It is isomorphicto thesemi-directproductof thecomplex line

<andthepuncturedcomplex line

< - 3< ¡ i � . Thegroupis connectedbut not simplyconnected.andtheexponentmap�� d < f < - ~ N �f ¯X�givesriseto theuniversalcovering�@ A < � 13 < % < 13 U ON !�� ð N !�� # < Wof @BA < � with multiplication

ON !�� ON � !�� d 3 ON ò N � !�� òÌ¯ � � � ��8As a realLie group,it is 4-dimensionalandwe denoteits Lie algebraby

^ A < � 33� Ï��¸@ A < � Thedual

space? - of ? 3�� Ï��¸@ A < � canbeidentify with C�� with coordinates � !�®�! � ! × � , see[D1]. Thecoadjoint

orbitsof �@ A < � in ? - passingthougha point � 3 � \ -� òÌ® \ -´ ò ��< -� òg× < -´ is denoteby4 1

, where\ -� ! \ -´ ! < -� ! < -´ form thebasisof ? - dualto thebasis\ � ! \ ´�! < � ! < ´ of ? with thebrackets

Q \ � ! < � S 3 < � !VQ \ � ! < ´·S 3 < ´Z!VQ \ ´�! < � S 3 < ´�!VQ \ ´�! < ´ S 3 � < � 8Thenà Eachpoint � ! i ! i ! × � is a 0-dimensionalcoadjointorbit, denoted

4 ¿ j � ^ � ^ � Ô Ã ,à Theopenset ® ´ ò � ´ `3 iis thesingle4-dimensionalcoadjointorbit

4)� < � < -, thepunctured

complex plane.

Note that the orbit4

is not simply connectedand thereis no diffeomorphismfrom somesymplectic

vectorspaceonto it. In [DH2], somesystemof diffeomorphismswasconstructed.Let us recall them.

Considerw ß d 3 U�� 3 Î � ò`h0ÎV´ # < ð �¡ Ùt Î � tnò¢ Ê! ç � � tgÎ§´Kt ç � � t Î§´Kt �ç �Èò q � � W�!for each� 3zi !g£ q ! 8k8k8 . Let

< ß d 3�< ¡r� , where� is thepositive realline

� 3 U§� ¯ �0¤ # < ð i t �út¥ Ê!�} 3�i W 817

Thereis anaturalmap < � < f 4 13 < � < - ~ ON !�� f ON ! ¯�¦ � !whoserestrictiongivesadiffeomorphism

} ß d < � w ß f < � < - 8On

< � w ß wehave thenaturalsymplecticform� ^�d 3 qç Q c N�Ó cz�ÌòÌc~§N¼Ó c/§� Sp!inducedfrom thestandardsymplecticform on

< ´ with coordinates ON !�� . Thecorrespondingsymplectic

form matrix is

Ó 3©¨ªª« i � q i iq i i ii i i qi i � q i¬®¯ and Ó �� 3©¨ªª« i q i i� q i i ii i i � qi i q i

¬®¯ 8ThecorrespondingPoissonbracketsof functions �å!�� # � � 4 � is

UZ�å!��W 3 Ó � s � �� s 3 � �� ° � � �� Î � � � �� Î � � �� ° � � � �� ° ´ � �� Î§´ ò � �� ÎV´ � �� ° ´ 8For anarbitraryelement� # ^ A < � it wascomputedin [DH2](Proposition2.4) that:à thecorrespondingfunctionon

4isù� � } ß ON !�� 3 qç Q � N ò`®�¯ ¦ ò@§� §N ò §®ý¯±°¦ S 8

à In thelocalcoordinates ON !�� of theorbit4

, theKirillo v form4

coincideswith thestandardform� ^Kd 3 qç Q c NKÓ c,� òóc²§N�Ó c�§� S 8It wasalsoproven in [DH2](Proposition3.1) that for all �È!Z³ # ^ A < � , theMoyal

G��productsatisfies

therelation h� ù� G�� h� ù³7� h� ù³ G�� h� ù� 3 h� Q �È!Z³ÈS 8This meansthat we have somerepresentation� ¿ ß Ã´ d � �f �� ù� G�� of Lie algebra A < � on the space�� 4 � . Denoteby Í � theFouriertransformation

Í � � � �Ú !�� d 3 qç�� u � ��(� �Bh;P �Ú §N �� ON !�� ck° � ck° ´andtheinverseFouriertransformation

Í �� ù� � ON !�� d 3 qç�� u � �� h;P �Ú §N �� ù� �Ú !�� ck° � ck° ´By thesamecomputationasin [DH2] we have for each� 3 b � ®i i c # ^ A < � andfor eachcom-

pactlysupportedsmoothfunction � # � �> < � w ß � ,¦� ´ � 3 Í � � � ¿ ß Ã´ � Í �� 3 Q � qç � ¦ � � °µ � ò@§� qç � °¦ � � µ � ò hç ®ý¯ ¦ � ¹� °µ ò §® ¯ °¦ � ¹��¶ � � S 8

18

It wasshown [DH2](Theorem4.2) that the representation��(� ¦� ¿ ß Ã´ �of the universalcovering r group�@ A < � is is coincidedwith theirreducibleunitaryrepresentationm~· of �@BA < � associatedwith

4by the

obit method,i.e. �� ¦� ¿ ß Ã´ � � �� 3 Q m�· �� S �� !realizingthethespace� ´ �C ��¸ � � andactingas

Q m~· ON !�� S �� 3 �� h� P � � � ò ç��~¹ Q M �� ò N �ç�� S �� »º`N � !where ON !�� # �@ A < � , �G#(C � ¸ � 3z< ¡ i � , � # � ´ �C ��¸ � � ! �¼º$N d 3 P �� ò N � ò ç�� h�U ÿ 4 �´P� W�!�Q R�S is

theintegral partof R and UVR"W is thedecimalpartof R .

4.2 K-gr oupsand periodic cyclic homology

Lemma 4.1 Themaximalcompactsubgroup � 3 U¾½ ¯ �0¤ ii qÀ¿ W of @BA < � is isomorphicto¸ � .

Proof is easyandis omitted. ¬Fromthisonededucesthefollowing results.

Proposition 4.2 Let4

b thecoadjointorbit of @BA < � , which is thepuncturedcomplex plane. Then,

� ^ � �> 4 ��13 H and � � � �> 4 ��13 U i W 8)J*K+ ^ � �> 4 ��13 H and)�*,+ � � �> 4 ��13 U i W 8

Proof. Becauseof Theorem1.13andLemma4.1, � - ��> 4 �� arethesameas � - �� ¸ � ��13 � - ¸ � � ,and

)�*,+ - ��> 4 �� areisomorphicto)�*,+ - ��> ¸ � �� . Theassertionsbecomeclear. ¬

4.3 Chern-Connescharacters

Proposition 4.3 Let4

be thecoadjointorbit of @ A < � , which is the punctured complex plane. Then,

theChern-Connescharacter I � d � - � �> 4 �� f )J*K+.- � �> 4 ��is an isomorphism.

Proof. FromTheorem2.1,wehave thecommutative diagram:

� - ��> 4 �� > �XW�6�� f )J*K+ - ��> 4 ��Y LLM LLM YZYZY� -�465 � �> 4 Æ �� > �XW D�6�� f )�*,+.-�465 � �> 4 Æ ��Y�Y LLM LLM Y�[� -�465 � �> =N ��PO > �RQ�6�� f )J*K+.-�465 � �> =N ��PO/

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It wasshown [DKT1]-[DKT2] thattheChern-Connescharactersarereducedto theclassicalChernchar-

actersof commutative tori. For thetori, I �kÁ is anisomorphismmodulotorsions,andthereforeI � � D andI � � arealsoisomorphismsmodulotorsions.In our caseof thequantumpuncturedcomplex plane,the

groupsareeither0 or H . Hence,Chern-ConnescharacterI � � d � - � �> 4 �� f )�*,+.- � �> 4 ��is anisomorphism. ¬Acknowledgments

This work wascompletedduring the time the first authorvisited AbdusSalamICTP. The first author

expresseshissincerethanksfor thehospitalityandfor theexcellentconditionsprovided.

References

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[DH1] DO NGOC DIEP AND NGUYEN V IET HAI, Quantumhalf-planesvia deformationquantiza-

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[DKT1] DO NGOC DIEP, ADEREMI O. KUKU AND NGUYEN QUOC THO, NoncommutativeChern

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[DKT2] DO NGOC DIEP, ADEREMI O. KUKU AND NGUYEN QUOC THO, NoncommutativeChern

characters for compactquantumgroups, Journalof Algebra,226(2000),311–331.

[J] JUDITH P. JESUDASON, TransformationgroupC*-algebras: A selectivesurvey, in Contem-

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[N1] V. NISTOR, Cyclic cohomology of crossedproducts by algebraic groups, Invent. Math.

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