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Contemporary Mathematics
A survey on Hopfcyclic cohomology and ConnesMoscovicicharacteristic 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
Hopfcycliccohomology
. But the history of Hopfcyclic cohomology and the characteristic map theyinvented remained intricately linked. In this survey article, we give an account of thedevelopment of the characteristic map and Hopfcyclic 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 GelfandFuchs 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 Hopfcyclic 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 groundbreakingwork [
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 Hopfcyclic cohomology as deﬁnedby Khalkhali and Rangipour in [
34
] which is needed for the cup product interpretationof the ConnesMoscovici 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
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2 ATABEY KAYGUN
calculations made in Hopfcyclic 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 ConnesMoscovici 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 Hopfcyclic cohomology and the characteristic 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 transversally elliptic operator
D
on
V
we have the index pairing yieldingIndex(
D
)
=
ch(
D
)
,
ch(
E
)
via the ChernConnes 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 formoftheAtiyahSingerindextheoremwhen
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
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A SURVEY ON HOPFCYCLIC COHOMOLOGY AND CONNESMOSCOVICI 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 ConnesMoscovici 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 GelfandFuchs cohomology of the Lie algebra of formal vector ﬁelds on
R
. They show that the primary characteristic classes of codimension1 foliations, namelythe transverse fundamental class and the GodbillionVey class, do lie in the image of theircharacteristic map. More importantly, they show that
H
1
has only two nontrivial periodicclasses based on their identiﬁcation with the GelfandFuchs cohomology, and they are sentto the transverse fundamental class and the GodbillionVey class under the characteristicmap. This fact indicates, at least cohomologically, that
H
1
is a universal algebraic objectclassifying codimension1 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 Hopfcyclic 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
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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 fundamental class is represented by the cyclic 2cocycle(1.9) (
X
⊗
Y
)
−
(
Y
⊗
X
)
−
(
δ
1
Y
⊗
Y
)and the GodbillionVey class by the 1cocycle
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 pseudogroups is given by Moscovici and Rangipour in [
42
], with applications 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 CuntzQuillen 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 CuntzQuillen framework 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 foliated 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 generalized by Sharygin [
46
], and Nikonov and Sharygin [
44
] utilizing the full generality of coe
ﬃ
cients in stable antiYetterDrinfeld modules [
24
].The similarly highly geometric approach also forms the basis of Gorokhovsky’s remarkable preprint [
19
] which was later published as [
20
]. In this paper Gorokhovsky extends to scope of the Hopfcyclic theory to di
ﬀ
erential graded Hopf algebras, but his maingoal was to answer the following simple question. Can one obtain the secondary characteristic classes of foliations from nonperiodic Hopfcyclic cohomology similar to the wayConnes and Moscovici obtained the GodbillionVey class and the transverse fundamental class from periodic Hopfcyclic 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 Riemannianfoliated 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 nonperiodic cohomology classesof the ConnesMoscovici Hopf algebra
H
1
the writer is aware of is Antal’s unpublishedthesis [
1
]. There Antal shows that the ﬁrst nonperiodic Hopfcyclic cohomology of
H
1
isgenerated by the GodbillionVey class
δ
1
and the class
δ
1
called
the Schwarzian
which isdeﬁned as(2.1)
δ
1
=
δ
2
−
12
δ
21
174
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A SURVEY ON HOPFCYCLIC COHOMOLOGY AND CONNESMOSCOVICI CHARACTERISTIC MAP 5
The arduous task of computing all of the nonperiodic classes of the ConnesMoscoviciHopf algebra
H
1
was completed by Moscovici and Rangipour in [
42
] based on methodsthey developed in [
41
].
3. Stable antiYetterDrinfeld 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 grouplike 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 Hopfcyclic cohomologycame from Hajac, Khalkhali, Rangipour and Sommerh¨auser [
24, 23
]. They made the important discovery that one has to consider
H
as a
H
module coalgebra
over itself whenwe consider Hopfcyclic 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 1dimensional
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 Hopfcocyclic module of
H
is obtained from the ordinary cocyclic 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 dimensional
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 antiYetterDrinfeld (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 Hopfcyclic cohomology by Connes and Moscovici came also withthe identiﬁcation of the cohomology with GelfandFuchs cohomology of formal vectorﬁelds over
R
for
H
1
[
10
]. There are two other important nontrivial computations of Hopfcyclic cohomology: for the quantum group
U
q
(
sl
2
) by Kusterman, Rognes and Tuset
175