A survey on Hopf-cyclic cohomology and Connes-Moscovici characteristic map

A survey on Hopf-cyclic cohomology and Connes-Moscovici characteristic map
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  This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. Contemporary Mathematics A survey on Hopf-cyclic cohomology and Connes-Moscovicicharacteristic map Atabey Kaygun To Henri with admiration and gratitude. A bstract . In 1998 Alain Connes and Henri Moscovici invented a cohomology theory forHopf algebras and a characteristic map associated with the cohomology theory in orderto solve a specific technical problem in transverse index theory. In the following decade,the cohomology theory they invented developed on its own under the name Hopf-cycliccohomology . But the history of Hopf-cyclic cohomology and the characteristic map theyinvented remained intricately linked. In this survey article, we give an account of thedevelopment of the characteristic map and Hopf-cyclic cohomology. Introduction One can claim that following its inception in the seminal article [ 10 ] by Alain Connesand Henri Moscovici, Hopf cyclic cohomology and the characteristic map associated withit have since become a standard cohomological tool in noncommutative geometry. Theparticular cohomology theory they defined has proved itself to be a robust analogue of the Gelfand-Fuchs cohomology complete with an appropriate analogue of the classicalcharacteristic map.In this survey article, we are going to trace the srcins of the theory and investigate thedevelopments in the subject in an attempt to give a brief but focused account of the pastand the current research. Here is the plan of this article: In the first section, we will detailthe main aspects of [ 10 ] focusing mainly on the construction of Hopf-cyclic cohomologyand the characteristic map as defined by Connes and Moscovici in op. cit. In the secondsectionwe detailthe busyaftermath of  op. cit. untilthe introduction of SAYDmodulesintothe theory by Hajac, Khalkhali, Rangipour and Sommerh¨auser in their ground-breakingwork [ 23 ]. The third section is a study of [ 23 ] and [ 24 ], and their generalizations. Inthe fourth section we take a quick detour to the dual Hopf-cyclic cohomology as definedby Khalkhali and Rangipour in [ 34 ] which is needed for the cup product interpretationof the Connes-Moscovici characteristic map. In the same section, we also list the main 2010 Mathematics Subject Classification. Primary 16E40; Secondary 16T05, 19D53, 55B34. c  0000 (copyright holder) 1   Contemporary MathematicsVolume 546 , 2011c  2011 American Mathematical Society 171  This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 2 ATABEY KAYGUN calculations made in Hopf-cyclic cohomology (and its dual) for a number of relevant andimportant Hopf algebras. In the fifth section, starting from [ 37 ] we investigate all of themajor cup product interpretations of Connes-Moscovici characteristic map. In the samesection, we also explain how these di ff  erent interpretations were seemingly incompatibleand how this problem was finally resolved by the author in [ 30 ]. 1. The beginnings If we were to summarize succinctly why Hopf-cyclic cohomology and the character-istic map associated with it were invented, we can simply say for computing the index of atransversally elliptic operator on a foliation. For a foliated manifold ( V  , F  ) and a transver-sally elliptic operator D on V  we have the index pairing yieldingIndex(  D ) =  ch(  D ) , ch(  E  )  via the Chern-Connes character for any E  ∈ K  ( V  / F  ) [ 4, 6 ]. In [ 9 ] Connes and Moscovicishowed that ch(  D ) reduces to a finite sum of expressions of the form(1.1)    a 0 [  D , a 1 ] ( k  1 ) ··· [  D , a m ] ( k  m ) |  D | − ( m + 2 k  1 + ··· + 2 k  m ) where [  D , a ] ( k  ) denotes the k  -th iterated commutator of  D 2 with [  D , a ] and   is a Diximiertrace or a Wodzicki residue. See also [ 8 ]. It may seem that the formula is computable byvirtue of being local , but in [ 10 ] Connes and Moscovici observe that ... although the general index formula easily reduces to the local formoftheAtiyah-Singerindextheoremwhen D issayaDiracoperatoronamanifold, the actualexplicitcomputation ofallthe terms(1.1) involvedin the cocycle ch(  D ) is a rather formidable task. As an instance of thislet us mention that even in the case of codimension one foliations, theprinted form of the explicit computation of the cocycle takes aroundone hundred pages. Each step in the computation is straightforwardbut the explicit computation for higher values of  n is clearly impossiblewithout a new organizing principle which allows [one] to bypass them.The organizing principle for codimension n foliations, it turns out, is a particular Hopf algebra denoted by H  n . Abstractly, the setup involves a unital algebra A on which H  n actscompatibly as(1.2) h ( a · b ) = h (1) ( a ) · h (2) ( b )and an algebra map δ : H  n → C which satisfies(1.3) h = δ ( h (1) S ( h (3) )) S 2 ( h (2) )and finally a trace map τ : A → C which satisfies(1.4) τ ( h ( a )) = δ ( h ) τ ( a )for any a , b ∈ A and h ∈ H  n . Then one can account for all of the terms (1.1) in the localindex formula by considering terms of the form(1.5) τ ( a 0 · h 1 ( a 1 ) ··· h m ( a m ))   172  This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. A SURVEY ON HOPF-CYCLIC COHOMOLOGY AND CONNES-MOSCOVICI CHARACTERISTIC MAP 3 for h 1 ,..., h m ∈ H  n in this setup. Then we consider the standard cyclic module of analgebra A  and rewrite (1.5) as a map(1.6) τ : H  ⊗ mn −→ Hom C ( A ⊗ m + 1 , C )by using the compatible action (1.2) and the invariance property (1.4) for every m ≥ 0.This yields a cocyclic structure on the graded module  m ≥ 0 H  ⊗ mn such that τ becomes amorphism of cocyclic modules. Explicitly, the cocyclic structure on the graded module H   n : = ⊕ m ≥ 0 H  ⊗ mn is defined by ∂ i ( h 1 ⊗···⊗ h m ) = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ (1 ⊗ h 1 ⊗···⊗ h m ) if  i = 0( h 1 ⊗···⊗ h (1) i ⊗ h (2) i ⊗···⊗ h m ) if 0 < i ≤ m ( h 1 ⊗···⊗ h m ⊗ 1) if  i = m + 1 σ  j ( h 1 ⊗···⊗ h m ) =   ( h i + 1 )( h 1 ⊗···⊗ h i ⊗ h i + 2 ⊗···⊗ h m ) τ m ( h 1 ⊗···⊗ h m ) = S ( h ( m + 1)1 ) h 2 ⊗···⊗ S ( h (3)1 ) h m ⊗ S ( h (2)1 ) δ ( h (1)1 )The e ff  ect of the morphism (1.6) in periodic cyclic cohomology  HP ∗ ( τ ): HP ∗ ( H   n ) → HP ∗ ( A )is the Connes-Moscovici characteristic map.We must warn the reader that the short description we gave above does little justice tothe intricacies of the subject since the technical details of the subject as developed in [ 10 ]are rather complicated, but still illuminating.In [ 10 ] Connes and Moscovici also investigate in detail the Hopf algebra structure of the class of Hopf algebras H  n and prove that the periodic cyclic cohomology of  H  1 isessentially the Gelfand-Fuchs cohomology of the Lie algebra of formal vector fields on R . They show that the primary characteristic classes of codimension-1 foliations, namelythe transverse fundamental class and the Godbillion-Vey class, do lie in the image of theircharacteristic map. More importantly, they show that H  1 has only two non-trivial periodicclasses based on their identification with the Gelfand-Fuchs cohomology, and they are sentto the transverse fundamental class and the Godbillion-Vey class under the characteristicmap. This fact indicates, at least cohomologically, that H  1 is a universal algebraic objectclassifying codimension-1 foliations.At this point, let us give an explicit description of  H  1 in terms of generators andrelations before we describe the characteristic classes Connes and Moscovici expressed interms of Hopf-cyclic cocycles. As a noncommutative algebra H  1 is countably generatedby symbols X  , Y  and δ m for m ≥ 1. The relations are(1.7) [ Y  ,  X  ] = X  , [ Y  ,δ m ] = m δ m , [  X  ,δ m ] = δ m + 1 , [ δ m ,δ   ] = 0for every , m ≥ 1. The coproduct on the generators are given by Δ ( Y  ) = (1 ⊗ Y  ) + ( Y  ⊗ 1) , Δ ( δ 1 ) = (1 ⊗ δ 1 ) + ( δ 1 ⊗ 1) , Δ (  X  ) = (1 ⊗  X  ) + (  X  ⊗ 1) + ( δ 1 ⊗ Y  )One can write explicit formulas for Δ ( δ m ) using the fact that Δ is multiplicative and thecommutator relation [  X  ,δ m ] = X  δ m − δ m  X  = δ m + 1 for every m ≥ 1. In order to complete   173  This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 4 ATABEY KAYGUN the setup we need a character δ : H  1 → C , which is defined on the generators by(1.8) δ ( Y  ) = 1 , δ (  X  ) = 0 , δ ( δ m ) = 0 for every m ≥ 1Now, we are ready to determine the primary characteristic classes: the transverse funda-mental class is represented by the cyclic 2-cocycle(1.9) (  X  ⊗ Y  ) − ( Y  ⊗  X  ) − ( δ 1 Y  ⊗ Y  )and the Godbillion-Vey class by the 1-cocycle δ 1 .The Hopf algebra H  1 and its higher dimensional analogues H  n are deformations of theLie algebra of the group of a ffi ne transformations of  R n for n ≥ 1. We note that this classof Hopf algebras is di ff  erent than previously defined deformations of Lie algebras  /  groupssuch as quantum groups. The full generalization of deformations of this type for arbitraryprimitive Lie pseudo-groups is given by Moscovici and Rangipour in [ 42 ], with applica-tions in foliations with certain transverse symmetries in [ 43 ]. 2. Early works Besides the srcinal framework Connes has developed for cyclic cohomology [ 5 ]which uses cyclic invariant Hochschild cocycles, there are other computational paradigmssuch as the Cuntz-Quillen framework of  X  -complexes [ 18 ]. An early development in thesubject came from Crainic in the preprint [ 16 ] which was later published as [ 17 ] wherehe reframed Connes and Moscovici’s cohomology theory within the Cuntz-Quillen frame-work for arbitrary Hopf algebras provided that the invariant character δ satisfies (1.3). Healso found an analogue of Bott’s characteristic map [ 2 ] k  : H  ∗ ( WO q ) → H  ∗ ( V  / F  ) of a fo-liated manifold ( V  , F  ) of codimension q by developing a noncommutative analogue W  ( H  )of the Weil algebra WO q for an arbitrary Hopf algebra H  . This approach was later gen-eralized by Sharygin [ 46 ], and Nikonov and Sharygin [ 44 ] utilizing the full generality of coe ffi cients in stable anti-Yetter-Drinfeld modules [ 24 ].The similarly highly geometric approach also forms the basis of Gorokhovsky’s re-markable preprint [ 19 ] which was later published as [ 20 ]. In this paper Gorokhovsky ex-tends to scope of the Hopf-cyclic theory to di ff  erential graded Hopf algebras, but his maingoal was to answer the following simple question. Can one obtain the secondary charac-teristic classes of foliations from non-periodic Hopf-cyclic cohomology similar to the wayConnes and Moscovici obtained the Godbillion-Vey class and the transverse fundamen-tal class from periodic Hopf-cyclic cohomology? In that paper, Gorokhovsky provides anexplicit a ffi rmative answer for a certain class of foliated manifolds. Later Kaminker andXiang in [ 27 ] considered these secondary classes for Riemannian-foliated flat bundles byusing an extension of the cohomology theory for Hopf algebroids similar to [ 12 ] and [ 36 ].On the other hand, the first attempt at computing non-periodic cohomology classesof the Connes-Moscovici Hopf algebra H  1 the writer is aware of is Antal’s unpublishedthesis [ 1 ]. There Antal shows that the first non-periodic Hopf-cyclic cohomology of  H  1 isgenerated by the Godbillion-Vey class δ 1 and the class δ  1 called the Schwarzian which isdefined as(2.1) δ  1 = δ 2 − 12 δ 21   174  This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. A SURVEY ON HOPF-CYCLIC COHOMOLOGY AND CONNES-MOSCOVICI CHARACTERISTIC MAP 5 The arduous task of computing all of the non-periodic classes of the Connes-MoscoviciHopf algebra H  1 was completed by Moscovici and Rangipour in [ 42 ] based on methodsthey developed in [ 41 ]. 3. Stable anti-Yetter-Drinfeld modules In [ 11 ] Connes and Moscovici defined our next important object: modular pairs ininvolution . A modular pair in involution in a Hopf algebra H  is a pair ( σ,δ ), where σ ∈ H  is a group-like element and δ : H → C is a character such that(3.1) δ ( σ ) = 1 and δ ( h (1) S ( h (3) )) S 2 ( h (2) ) = σ h σ − 1 They show that the cocyclic module they defined for the class of Hopf algebras H  n canbe defined for an arbitrary Hopf algebra H  if the Hopf algebra H  has a modular pair ininvolution. The technical condition (3.1) is needed to make sure that the cyclic identity τ mm = id is satisfied for any m ≥ 1 in the cocyclic module H   they similarly defined.But one of the most exciting developments in the theory of Hopf-cyclic cohomologycame from Hajac, Khalkhali, Rangipour and Sommerh¨auser [ 24, 23 ]. They made the im-portant discovery that one has to consider H  as a H  -module coalgebra over itself whenwe consider Hopf-cyclic cohomology of  H  . The second fundamental discovery they madeis that a modular pair in involution ( σ,δ ) in a Hopf algebra H  is really a 1-dimensional H  -module  /  comodule σ C δ = C with comodule structure λ : σ C δ → H ⊗ σ C δ defined by λ (1) = σ ⊗ 1 and with module structure ρ : H ⊗ σ C δ → σ C δ defined by h · 1 = δ ( h )1for any h ∈ H  . Then the Hopf-cocyclic module of  H  is obtained from the ordinary co-cyclic module of  H  viewed as a coalgebra which is twisted by the H  -module  /  comodule σ C δ . In [ 23 ] the authors determine what specific conditions are required for a higher di-mensional H  -module  /  comodule M  to yield cocyclic modules for arbitrary Hopf algebras.These conditions are m ( − 1) m (0) = m and(3.2)( hm ) ( − 1) ⊗ ( hm ) (0) = h (1) m ( − 1) S − 1 ( h (3) ) = h (2) m (0) for any m ∈ M  and h ∈ H  . Such modules are called stable anti-Yetter-Drinfeld (SAYD)modules [ 24 ].Following [ 23 ], in [ 28 ] Kaygun showedthat if one trivializes the diagonal action of theunderlying Hopf algebra H  on the cocyclic module of  H  viewed as a module coalgebraover itself twisted by a SAYD module M  then one obtains the cocyclic module of  H  asdefined in [ 23 ]. If the coe ffi cient module is not SAYD, the quotient need not be a cocyclicmodule. But one can always trivialize the diagonal action and force the quotient to be acocyclic module at the same time. The immediate consequence of this approach was toextension of the theory to bialgebras and arbitrary coe ffi cient modules [ 28 ]. 4. The dual theory and cyclic duality The discovery of Hopf-cyclic cohomology by Connes and Moscovici came also withthe identification of the cohomology with Gelfand-Fuchs cohomology of formal vectorfields over R for H  1 [ 10 ]. There are two other important non-trivial computations of Hopf-cyclic cohomology: for the quantum group U  q ( sl 2 ) by Kusterman, Rognes and Tuset   175
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