Operations On Hopf Cyclic Cohomology

Operations on Hopf Cyclic Cohomology

(Spine title: Operations on Hopf cyclic cohomology)

(Thesis format: Monograph)

by

Mohammad Hassanzadeh

Graduate Programin

Mathematics

A thesis submitted in partial fulfillmentof the requirements for the degree of

Doctor of Philosophy

School of Graduate and Postdoctoral StudiesThe University of Western Ontario

London, Ontario, Canada

c⃝ Mohammad Hassanzadeh 2010

i

Certificate of Examination

THE UNIVERSITY OF WESTERN ONTARIO

SCHOOL OF GRADUATE AND POSTDOCTORAL STUDIES

Chief Adviser: Examining Board:

Professor Masoud Khalkhali Professor Gerry McKeon

Advisory Committee: Professor Atabey Kaygun

Professor Andre Boivin

Professor Ajneet Dhillon

The thesis by

Mohammad Hassanzadeh

entitled:

Operations on Hopf cyclic (co)homologies ...

is accepted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

Date:Chair of Examining BoardFirstname Lastname

ii

Abstract

In the first three chapters of this thesis we recall the preliminaries of Hopf algebras,

cyclic cohomology and cup products. In Chapters four and five we introduce the

tools that we will use for our main results. Specially in Chapter five we introduce

Kunneth formula for periodic cyclic (co)homology. In Chapters six, seven and eight

we introduce our coproducts and cup products for Hopf cyclic theory and mention

several applications. In Chapter nine we define a coproduct for cocyclic modules. In

the last chapter we mention the headlines of all the new results in this thesis.

Keywords: Noncommutative Geometry, Hopf Algebra, cup product and coproduct,

Hopf cyclic cohomology, Operations in Hopf cyclic theory.

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Acknowledgements

First, I would like to express my deep and sincere gratitude to my supervisor, professorMasoud Khalkhali. He provided encouragement, sound advice, good teaching, goodcompany, and lots of good ideas.

It is a pleasure to thank the many people who made this thesis possible. Ithank Professor Andre Boivin, Graduate chair of the mathematics department, whowas always supportive during my PhD program. Also I wish to appreciate Carl andAgnes Santoni for their great supports in my last year of PhD when I won theirgraduate scholarship in mathematics as a part of my Ontario Graduate Scholarshipfor Science and Technology(OGSST). I would like to thank my external examinerprofessor Atabey Kaygun and my examiners professors Andre Boivin, Ajneet Dhillonand Gerry Mckeon for their valuable suggestions in my thesis. At the end, I wouldlike to thank my wife Nahid and my father, mother, sister and brother who supportedme during these years.

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To my family Specially to my beautifulwife, Nahid, and my mother, Molki, my

father, Hassan, my sister, Fati, mybrother, Mehdi and my beautifulnephews, Kimia and Kamiar.

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Table of Contents

Certificate of Examination . . . . . . . . . . . . . . . . . . . . . . . . . ii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 Preliminaries of Hopf algebra . . . . . . . . . . . . . . . . . . . . . . 11.1 Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Preliminaries of cyclic (co)homology . . . . . . . . . . . . . . . . . . 92.1 (Co)cyclic modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Hochschild (co)homology of (co)cyclic modules . . . . . . . . . . . . . 122.3 Cyclic (co)homology of (co)cyclic modules . . . . . . . . . . . . . . . 13

2.3.1 First description of cyclic (co)homology: Connes complex . . . 132.3.2 Second description of cyclic (co)homology: (b, b′)-bicomplex . 152.3.3 Third description of cyclic (co)homology: (b, B)-bicomplex . . 16

2.4 Periodic cyclic (co)homology of (co)cyclic module . . . . . . . . . . . 182.4.1 First description of periodic cyclic cohomology . . . . . . . . . 192.4.2 Second description of periodic cyclic cohomology . . . . . . . . 202.4.3 Third description of periodic cyclic cohomology . . . . . . . . 202.4.4 First and second description of periodic cyclic homology . . . 21

2.5 Normalized complex . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 Hopf cyclic cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 242.7 Dual Hopf cyclic cohomology . . . . . . . . . . . . . . . . . . . . . . 27

2.7.1 Non localized cyclic module for dual Hopf cyclic homology . . 312.8 Hopf cyclic cohomology with coefficients . . . . . . . . . . . . . . . . 33

3 Preliminaries of cup product and coproduct . . . . . . . . . . . . . 373.1 Group (co)homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Cup products and coproducts for group (co)homology . . . . . . . . . 413.3 Lie algebra (co)homology . . . . . . . . . . . . . . . . . . . . . . . . 433.4 Cup products and coproducts for Lie algebra (co)homology . . . . . 473.5 Cup products for cyclic cohomology of algebras . . . . . . . . . . . . 53

vi

4 Eilenberg-Zilber isomorphisms . . . . . . . . . . . . . . . . . . . . . 554.1 Diagonal and tensor product complex . . . . . . . . . . . . . . . . . . 554.2 Shuffle and Alexander-Whitney maps . . . . . . . . . . . . . . . . . 564.3 The Eilenberg-Zilber isomorphisms for Hochschild (co)homology of

(co)cyclic modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4 The Eilenberg-Zilber isomorphisms for cyclic (co)homology of (co)cyclic

modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.5 The Eilenberg-Zilber isomorphisms for periodic cyclic (co)homology of

(co)cyclic modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 Kunneth formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.1 Kunneth formulas for Hochschild (co)homology of (co)cyclic modules 685.2 Kunneth formulas for cyclic (co)homology of (co)cyclic modules . . . 695.3 Kunneth formulas for periodic cyclic (co)homology of (co)cyclic modules 71

6 Coproducts for Hopf cyclic cohomology with coefficients . . . . . . 776.1 Eilenberg-Zilber isomorphisms and Kunneth formulas for Hopf cyclic

cohomology with coefficients . . . . . . . . . . . . . . . . . . . . . . . 776.2 Coproducts for Hochschild, cyclic and periodic cyclic cohomology of

Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.3 Applications of the coproducts in Hopf cyclic cohomology . . . . . . 87

7 Coproducts for the dual Hopf cyclic homology . . . . . . . . . . . . 927.1 Applications of the coproducts for the dual Hopf cyclic homology . . 96

8 Cup products for the dual Hopf cyclic homology . . . . . . . . . . 98

9 Coproducts for cocyclic modules . . . . . . . . . . . . . . . . . . . . 1029.1 Coalgebra structure for cocyclic modules . . . . . . . . . . . . . . . . 1029.2 Coproducts for cyclic cohomology of algebras endowed with trace . . 106

10 The headlines of the new results in this thesis . . . . . . . . . . . 108

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

vii

Chapter 1

Preliminaries of Hopf algebra

1.1 Hopf algebras

In this thesis vector spaces are assumed to be over a field denoted by k. In this chapter

we recall the preliminaries of Hopf algebras and provide some examples [23, 33]. Let

us start by recalling the definition of coalgebra which can be obtained by dualizing

the notion of an unital associative algebra. A coalgebra is a k-vector space C with

k-linear maps ∆ : C −→ C ⊗ C, called comultiplication, and ε : C −→ k, called

counit, such that

(id⊗∆)∆ = (∆⊗ id)∆, (Coassociativity), (1.1.1)

(id⊗ ε)∆ = (ε⊗ id)∆ = idC , (Counit). (1.1.2)

In this thesis we use the Sweedler’s notation for coproduct, with summation under-

stood, i.e.,

∆nh = h(1) ⊗ h(2) ⊗ h(3) ⊗ . . .⊗ h(n+1), n ≥ 1. (1.1.3)

With this notation, the relations (1.1.1) and (1.1.2) can be written as follows:

c(1) ⊗ c(2)(1) ⊗ c(2)(2) = c(1)(1) ⊗ c(1)(2) ⊗ c(2)

ε(c(1))c(2) = ε(c(2))c(1) = c.

1

2

A bialgebra is, simultaneously, an algebra and a coalgebra, with some compatibility

conditions. More precisely:

Definition 1.1.1. A bialgebra B is an unital associative algebra (B,m, η) with m a

multiplication map, η a unit, and it is also a counital coassociative coalgebra (B,∆, ε),

with ∆ a comultiplication, and ε a counit, such that the algebra and coalgebra struc-

tures are compatible in the sense that the comultiplication and counit maps are both

algebra morphisms.

In the above definition, the map ∆ being an algebra morphism means:

∆(1) = 1⊗ 1,

(hk)(1) ⊗ (hk)(2) = h(1)k(1) ⊗ h(2)k(2), ∀h, k ∈ B,

where the last equality in terms of commutative diagrams is:

B ⊗B ∆m //

∆⊗∆

B ⊗B

B ⊗B ⊗B ⊗Bid⊗τ⊗id

// B ⊗B ⊗B ⊗B

m⊗mOO

Here the linear map τ : B ⊗B → B ⊗B is given by τ(h⊗ k) = k ⊗ h.

The map ε being an algebra morphism means:

ε(hk) = ε(h)ε(k), ε(1) = 1

where the last identity can be illustrated by the following commutative diagram:

B ⊗B m //

ε⊗ε

B

εuulllllllllllllll

k ⊗ k = k

3

Definition 1.1.2. A Hopf algebra H is a bialgebra equipped with a linear map S :

H → H, called antipode, satisfying the following relation

S(h(1))h(2) = h(1)S(h(2)) = ε(h)1, ∀h ∈ H.

Definition 1.1.3. A Hopf algebra H is called commutative if it is commutative as

an algebra, i.e.,

hk = kh ∀h, k ∈ H.

Definition 1.1.4. A Hopf algebra H is called cocommutative if it is a cocommutative

as a coalgebra, i.e.,

h(1) ⊗ h(2) = h(2) ⊗ h(1) ∀h ∈ H,

Remark 1.1.1. Let H be a cocommutative Hopf algebra, then referring to equation

(1.1.3) we have

h(1)⊗...⊗h(i)⊗...⊗h(j)⊗...⊗h(n) = h(1)⊗...⊗h(j)⊗...⊗h(i)⊗...⊗h(n), 1 ≤ i ≤ j ≤ n.

Proposition 1.1.1. For any commutative or cocommutative Hopf algebra we have

S2 = id. The converse is not true.

Proposition 1.1.2. The antipode map, S, of a Hopf algebra H is an antialgebra and

anticoalgebra map. More precisely, for all h and k in H:

S(kh) = S(h)S(k), S(1) = 1,

S(h)(1) ⊗ S(h)(2) = S(h(2))⊗ S(h(1)),

ε(S(h)) = ε(h).

Proposition 1.1.3. Let H and K be two Hopf algebras. Then H ⊗ K is a Hopf

4

algebra with the following antipode, comultiplication, multiplication and counit:

SH⊗K(h⊗ k) = SH(h)⊗ SK(k)

∆H⊗K(h⊗ k) = (h(1) ⊗ k(1))⊗ (h(2) ⊗ k(2))

mH⊗K((h1 ⊗ k1)⊗ (h2 ⊗ k2)) = h1h2 ⊗ k1k2

εH⊗K(h⊗ k) = ε(h)ε(k).

Example 1.1.1. For any discrete group G, let H = kG be the group algebra of G

over the field k. A Hopf algebra structure on H can be defined by:

∆(g) = g ⊗ g, S(g) = g−1, and ε(g) = 1,

for all g ∈ G. This Hopf algebra is cocommutative. It is commutative if and only if

G is commutative.

Example 1.1.2. Let g be a Lie algebra over a field k, and let H = U(g) be the

universal enveloping algebra of g. Recall that U(g) as an algebra, isT (g)I , where T (g)

is the tensor algebra of g and I is the two sided ideal generated by all elements of the

form

x⊗ y − y ⊗ x− [x, y], x, y ∈ g.

More precisely the algebra structure of U(g) is given by setting:

[(g1 ⊗ ...⊗ gn)][(g′1 ⊗ ...⊗ g′m)] = [g1 ⊗ ...⊗ gn ⊗ g′1 ⊗ ...⊗ g

′m].

One can define a Hopf algebra structure on U(g) by:

∆(g) = g ⊗ 1 + 1⊗ g, ε(g) = 0, and S(g) = −g,

for all g ∈ g.

5

One can see that

S(g1 ⊗ ...⊗ gn) = (−1)ngn ⊗ ...⊗ g1,

∆(g1 ⊗ ...⊗ gn) =∑σ

sign(σ)(gσ(1) ⊗ ...⊗ gσ(p))⊗ (gσ(p+1) ⊗ ...⊗ gσ(p+q)),

where, σ runs over all (p, q)-shuffles with p + q = n. With the above structure

(U(g),∆, ε, S) is a cocommutative Hopf algebra. Also if g is an abelian Lie algebra,

then U(g) is commutative and moreover U(g) = S(g), the symmetric algebra of g.

Now let us give an example of a Hopf algebra which is neither commutative nor

cocommutative.

Example 1.1.3. Let k be a field of characteristic zero and q ∈ k, q = 0 and q

not a root of unity. The quantized universal enveloping algebra of functions on the

quantum group SLq(2, k), which is denoted by A(SLq(2, k)), is a Hopf algebra which

is generated, as an algebra, by symbols x, u, v, y subject to the following relations:

ux = qxu, vx = qxv, yu = quy, uv = vu, xy − q−1uv = yx− quv = 1.

One can also define a coproduct, a counit and an antipode for A(SLq(2, k)) by:

∆(x) = x⊗σx+u⊗, ∆(y) = v⊗u+y⊗y, ∆(u) = x⊗u+u⊗y, ∆(v) = v⊗x+y⊗v,

S(x) = y, S(y) = x, S(u) = −qu, S(v) = −q−1v

ϵ(x) = ϵ(y) = 1, ϵ(u) = ϵ(v) = 0.

Notice that S2 = id.

Definition 1.1.5. An element h in a Hopf algebra H is called primitive if

∆(h) = 1⊗ h+ h⊗ 1.

6

Also h ∈ H is called a group-like element if

∆(h) = h⊗ h.

Example 1.1.4. Let H and K be two Hopf algebras. Let σ1 ∈ H and σ2 ∈ K be two

group-like elements. Then σ1 ⊗ σ2 is a group-like element in H ⊗K.

Examples (1.1.1) and (1.1.2) were both examples of cocommutative Hopf al-

gebras. This suggests a close relation between cocommutativite Hopf algebras and

groups and Lie algebras. For any Hopf algebra H, the set of all primitive elements

of H, P (H), is a Lie algebra. Also the set of all group like elements of H, G(H), is

a group. The next two structure propositions will completely characterize all cocom-

mutative Hopf algebras over an algebraically closed field of characteristic zero.

Proposition 1.1.4. ( Kostant and, independently, Cartier) Any cocommutative Hopf

algebra H over an algebraically closed field k of characteristic zero is isomorphic, as a

Hopf algebra, to a crossed product algebra H = U(P (H))oG(H) where G(H) acts on

P (H) by inner automorphism given by (g, h) 7−→ ghg−1 for g ∈ G(H) and h ∈ P (H).

Definition 1.1.6. A Hopf algebra H is called connected if for any h ∈ H, there is a

natural number n such that ∆n(h) = 0, where H is the kernel of the counit map ε,

and

∆ : H −→ H ⊗H,

∆(h) = ∆(h)− 1⊗ h− h⊗ 1.

Proposition 1.1.5. (Cartier- Milnor-Moore) A cocommutative Hopf algebra over a

field of characteristic zero is isomorphic, as a Hopf algebra, to the enveloping algebra

of a Lie algebra if and only if it is connected.

7

Definition 1.1.7. Let H be a Hopf algebra and A be an algebra on which H coacts

from the right by β : A −→ A⊗H. i.e., the following diagrams commute:

Aβ //

β

A⊗HIA⊗∆

A⊗H

β⊗IH// A⊗H ⊗H

Aβ //

∼= ##GGGGGGGGG A⊗H

IA⊗ϵ

A⊗ k

The algebra A is called a right H-comodule algebra if the map β is an algebra map

where the algebra structure of A⊗H is the tensor product of the algebras A and H.

We use the following Sweedler notation for the coaction of H on A:

β(a) = a(0) ⊗ a(1)

One can, similarly, define a left comodule algebra.

Definition 1.1.8. Let H and A be as in the above definition. Let δ : H −→ k be a

character for H, and σ ∈ H be a group-like element. A linear map, Tr : A → k is

called δ-trace if

Tr(ab) = Tr(ba(0))δ(a(1)) ∀a, b ∈ A.

It is called σ-invariant if for all a, b ∈ A,

Tr(a(0)b) (a(1)) = Tr(ab(0))Sσ(b(1)),

or equivalently,

Tr(a(0))a(1) = Tr(a)σ.

Example 1.1.5. Let G be a discrete group and A = H = kG be its group algebra. It

is clear that H is a H-comodule algebra by β = ∆. Let σ be a central element in G.

8

It is obvious that the following trace is a δ-trace which is σ-invariant,

Tr(x) =

1 if x = σ

0 otherwise.

Chapter 2

Preliminaries of cyclic (co)homology

In this chapter we recall some background materials on the cyclic (co)homology of

a (co)cyclic modules. We, also, give some important examples of (co)cyclic modules

related to algebras, coalgebras and Hopf algebras [9, 5, 7, 30, 31].

2.1 (Co)cyclic modules

In this section we recall the definitions of cyclic and cocyclic modules. These defi-

nitions are, in fact, special cases of a more general concept of (co)cyclic objects in

any abelian category [9, 31]. We do not mention those general concepts as we will

not make use of them in this thesis. Recall that a cosimplicial module is given by

datum (Cn, δi, si), where, Cn, n ≥ 0 is a k-module. The maps δi : Cn −→ Cn+1

are called cofaces, and si : Cn −→ Cn−1 called codegeneracies. These are k-module

maps satisfying the following cosimplicial relations:

δnj δn−1i = δni δ

n−1j−1 if i < j,

snj sn+1i = sni s

n+1j+1 if i ≤ j,

snj δn+1i =

δni s

n−1j−1 if i < j

id if i = j or i = j + 1

δni−1sn−1j if i > j + 1.

(2.1.1)

Definition 2.1.1. [30, 31] A cocyclic module is a cosimplicial module equipped with

extra morphisms, τn : Cn −→ Cn, called cocyclic maps such that the following extra

9

10

relations hold.

τnδni = δni−1τn−1 1 ≤ i ≤ n

τnδn0 = δnn

τnsni = sni−1τn+1 1 ≤ i ≤ n

τnsn0 = snnτ

2n+1 (2.1.2)

τn+1n = id.

In a dual manner, one can define a cyclic module as a simplicial module with

extra cyclic maps. More precisely:

Definition 2.1.2. [31] A cyclic module is given by data, C = (Cn, δi, si, τn), where

Cn, n ≥ 0 is a k-module, and δni : Cn −→ Cn−1, σni : Cn → Cn+1, 0 ≤ i ≤ n,

and τn : Cn → Cn, are called faces, degeneracies and cyclic maps, respectively, are

k-module maps, satisfying the following relations:

δn−1i δnj = δn−1j−1 δni if i < j,

sn+1i snj = sn+1

j+1 sni if i ≤ j,

δn+1i snj =

sn−1j−1 δ

ni if i < j

id if i = j or i = j + 1

sn−1j δni−1 if i > j + 1

(2.1.3)

11

and

δni τn = τn−1δn−1i−1 1 ≤ i ≤ n,

δn0 τn = δnn,

sni τn = τn+1sn+1i−1 1 ≤ i ≤ n,

sn0τn = τ2n+1sn+1n ,

τn+1n = id. (2.1.4)

Example 2.1.1. To any unital algebra A, one can associate a cyclic module A given

by

Cn(A) := A⊗(n+1), n ≥ 0,

with face, degeneracy and cyclic operators defined by:

δi(a0 ⊗ ...⊗ an) =

(a0 ⊗ ...⊗ aiai+1 ⊗ ...⊗ an) 0 ≤ i < n

(ana0 ⊗ ...⊗ an−1) i = n

σi(a0 ⊗ ...⊗ an) = (a0 ⊗ ...⊗ ai ⊗ 1⊗ ai+1 ⊗ ...⊗ an), 0 ≤ i ≤ n,

τn(a0 ⊗ ...⊗ an) = (an ⊗ a0 ⊗ ...⊗ an−1).

Example 2.1.2. To any coalgebra (C,∆, ε), we associate a cocyclic module C defined

by:

Cn(C) := C⊗(n+1), n ≥ 0,

with the following coface, codegeneracy and cyclic maps:

δi(c0 ⊗ ...⊗ cn−1) =

(c0 ⊗ ...⊗ c

(1)i ⊗ c

(2)i ⊗ ...⊗ cn−1) 0 ≤ i < n

(c(2)0 ⊗ c1 ⊗ ...⊗ cn−1 ⊗ c

(1)0 ) i = n

12

σi(c0 ⊗ ...⊗ cn+1) = (c0 ⊗ ...⊗ ci ⊗ ε(ci+1)⊗ ...⊗ cn+1), 0 ≤ i ≤ n,

τn(c0 ⊗ ...⊗ cn) = (c1 ⊗ ...⊗ cn ⊗ c0).

Proposition 2.1.1. Let (Cn, δni , sni , τn) and (C ′n, δ′ni , s

′ni , τ

′n) be two (co)cyclic mod-

ules. We construct a new (co)cyclic module ((C × C ′)n, dni , sni , tn), called diagonal

(co)cyclic module C × C ′, by:

(C × C ′)n := Cn ⊗ C ′n,

dni := δni ⊗ δ′ni ,

sni := sni ⊗ s′ni ,

tn := τn ⊗ τ ′n.

In the following sections we shall see more examples of cyclic and cocyclic

modules.

2.2 Hochschild (co)homology of (co)cyclic

modules

In this section, we recall the definition of Hochschild (co)homology of a (co)cyclic

module.

Definition 2.2.1. Let C = (Cn, δi, σi), n ≥ 0 be a cosimplicial module. The

Hochschild cohomology of C denoted by HH∗(C) is the cohomology of the cochain

complex C∗HH :

C0 b−−−→ C1 b−−−→ C2 b−−−→ C3 . . . ,

13

where b : Cn−1 −→ Cn is defined by

b =n∑

i=0

(−1)iδni .

Similarly we can define the Hochschild homology of a simplicial module. One

should notice that to define the Hochschild (co)boundary b only faces are used.

Example 2.2.1. In Example (2.1.1) we have defined a cyclic module for unital alge-

bras. Let A = C be the ground field, then it can be easily checked that:

HH0(k) = k, HHn(k) = 0 for n ≥ 1.

2.3 Cyclic (co)homology of (co)cyclic modules

In this section we describe three equivalent definitions for cyclic (co)homology of

(co)cyclic modules. First description is based on the so called Connes complex. sec-

ond and third descriptions are in terms of (b, b′)-bicomplex and (b, B)-bicomplex,

respectively.

2.3.1 First description of cyclic (co)homology: Connes

complex

Definition 2.3.1. [5, 7, 15] Let C = (Cn, δi, σi, τn), n ≥ 0 be a cocyclic module.

The cyclic cohomology of C, denoted by HC∗(C), is the cohomology of the following

cochain complex, known as the Connes complex C∗λ:

C0λ

b−−−→ C1λ

b−−−→ C2λ

b−−−→ C3λ . . . ,

where

λn = (−1)nτn,

14

and

Cnλ := Ker (1− λn).

The coboundary map b =∑n

i=0(−1)iδni is, in fact, the Hochschild coboundary re-

stricted to Cnλ .

Definition 2.3.2. Let C = (Cn, δi, σi, τn), n ≥ 0 be a cyclic module. The cyclic

homology of C denoted by HC∗(C) is the homology of the chain complex

Cλ0

b←−−− Cλ1

b←−−− Cλ2

b←−−− Cλ3 · · · ,

where, as before, λn = (−1)nτn and

Cλn :=

Cn

Im (1− λn),

The boundary map b : Cn → Cn−1 is defined by

b =n∑

i=0

(−1)iδi.

Example 2.3.1. Let A be an unital algebra and let C∗(A) be the cyclic module defined

in Example (2.1.1). The cyclic homology of this cyclic module is called the cyclic

homology of the algebra A.

Example 2.3.2. Let C be a counital coalgebra and C∗(C) be the cocyclic module

defined in Example (2.1.2). The cyclic cohomology of this cocyclic module is called

the cyclic cohomology of the coalgebra C.

15

2.3.2 Second description of cyclic (co)homology:

(b, b′)-bicomplex

To obtain the cyclic cohomology of the cocyclic module C = (Cn, δni , σni , τn) (2.3.1),

one can alternatively construct the (b, b′)-bicomplex, denoted by C∗∗(C), as follows:

......

...

C2 1−λ−−−→ C2 N−−−→ C2 1−λ−−−→ · · ·xb

x−b′ xb

C1 1−λ−−−→ C1 N−−−→ C1 1−λ−−−→ · · ·xb

x−b′ xb

C0 1−λ−−−→ C0 N−−−→ C0 1−λ−−−→ · · ·

where

b′ =n−1∑i=0

(−1)iδi

λn = (−1)nτn,

N = 1 + λ+ λ2 + ...+ λn.

It is easy to check the following relations hold:

b2 = b′2 = 0, (1− λ)b = b′(1− λ), and (1− λ)N = N(1− λ) = 0.

One can verify that the cyclic cohomology of C = (Cn, δni , σni , τn) is isomorphic to

the cohomology of the total complex TotC∗∗(C), i.e.,

HCn(C) ∼= Hn(TotC∗∗(C)).

Notice that the even columns in the above (b, b′)-bicomplex are Hochschild cochains

with Hochschild coboundaries. In a dual manner, the cyclic homology of the cyclic

module C = (Cn, δni , σni , τn) is isomorphic to the homology of the total complex of

16

the following (b, b′)-bicomplex denoted by C∗∗(C):

......

...

C21−λ←−−− C2

N←−−− C21−λ←−−− · · ·yb

y−b′ yb

C11−λ←−−− C1

N←−−− C11−λ←−−− · · ·yb

y−b′ yb

C01−λ←−−− C0

N←−−− C01−λ←−−− · · ·

Here we give a short sketch for the proof of the above statement: The natural surjec-

tion

ρ : TotC∗∗(C) −→ Cλ∗

which, in the level of bicomplexes, is the quotient map Cn −→ Cλn on the first column

and zero on other columns, defines a quasi-isomorphism.

2.3.3 Third description of cyclic (co)homology:

(b, B)-bicomplex

For the third description of the cyclic cohomology of a cocyclic module C = (Cn, δni , σni , τn),

we introduce the so called (b, B)-bicomplex, denoted by B∗∗(C), as follows

......

...

C2 B−−−→ C1 B−−−→ C0

b

x b

xC1 B−−−→ C0

b

xC0

17

where the map B is defined by:

B = Ns(1− λ).

In this formula the operator s, called the extra degeneracy, is given by:

s = σnτn+1 : Cn+1 → Cn.

The relations: b2 = B2 = 0 and bB +Bb = 0 are satisfied. It can be shown that:

Hn(TotC∗∗(C)) ∼= Hn(TotB∗∗(C)).

Example 2.3.3. (Cyclic cohomology of algebras). Let Cn(A) = Homk(A⊗(n+1), k)

and define cofaces, codegeneracies and cocyclic maps as follows:

δiφ(a0 ⊗ ...⊗ an) =

φ(a0 ⊗ ...⊗ aiai+1 ⊗ ...⊗ an) 0 ≤ i < n

φ(ana0 ⊗ a1 ⊗ ...⊗ an−1) i = n

σiφ(a0 ⊗ ...⊗ an) = φ(a0 ⊗ ...⊗ ai ⊗ 1⊗ ai+1 ⊗ ...⊗ an), 0 ≤ i ≤ n

τnφ(a0 ⊗ ...⊗ an) = φ(an ⊗ a0 ⊗ ...⊗ an−1).

The last degeneracy s is given by:

sφ(a0 ⊗ ...⊗ an) = φ(1⊗ a0 ⊗ ...⊗ an).

The cyclic homology of a cyclic module C = (Cn, δni , σni , τn), can be computed

18

from the following (b, B)-bicomplex:

......

...

C2B←−−− C1

B←−−− C0

b

y b

yC1

B←−−− C0

b

yC0

where

B = (1− λ)sN,

and

s = τn+1σn : Cn → Cn+1.

More precisely, one can prove that:

Hn(TotC∗∗(C)) ∼= Hn(TotB∗∗(C)).

The natural injection

TotB∗∗(C) → TotC∗∗

sending x ∈ TotB(C)pq = Cq−p to x ⊕ sN(x) ∈ Cq−p ⊕ Cq−p+1 = CC2p,q−p ⊕

CC2p−1,q−p+1 ⊂ TotCCp+q defines a quasi-isomorphism.

2.4 Periodic cyclic (co)homology of (co)cyclic

module

In this section we explain three equivalent methods to define the periodic cyclic

cohomology of a cocyclic module. We also provide two equivalent methods to define

periodic cyclic homology of a cyclic module.

19

2.4.1 First description of periodic cyclic cohomology

Definition 2.4.1. Let C = (Cn, δni , σni , τn), n ≥ 0, be a cocyclic module. The

periodic cyclic cohomology of C, denoted by HP ∗(C), can be defined as the cohomology

of the total complex of the following bicomplex, C∗∗(C):

......

......

......

· · ·C2 1−λ−−−→ C2 N−−−→ C2 1−λ−−−→ C2 1−λ−−−→ C2 N−−−→ · · ·x−b′ xb

xb

x−b′ xb

· · ·C1 1−λ−−−→ C1 N−−−→ C1 1−λ−−−→ C1 1−λ−−−→ C1 N−−−→ · · ·x−b′ xb

xb

x−b′ xb

· · ·C0 1−λ−−−→ C0 N−−−→ C0 1−λ−−−→ C0 1−λ−−−→ C0 N−−−→ · · ·

The total complex is:

· · ·⊕i≥0

Ci −−−→⊕i≥0

Ci −−−→⊕i≥0

Ci · · ·

Obviously, this total complex is Z2-periodic. Therefore, there exists only two

possible cases for cohomology, HP 0(C) and HP 1(C), which represents the even and

odd degrees respectively.

20

2.4.2 Second description of periodic cyclic cohomology

Alternatively, if we consider the following bicomplex, B∗∗(C)

......

......

...

· · ·C4 B−−−→ C3 B−−−→ C2 B−−−→ C1 B−−−→ C0

b

x b

x b

x b

x· · ·C3 B−−−→ C2 B−−−→ C1 B−−−→ C0

b

x b

x b

x· · ·C2 B−−−→ C1 B−−−→ C0

b

x b

x· · ·C1 B−−−→ C0

b

x· · ·C0

The total complex is:

· · ·⊕i≥0

C2i (b+B)−−−−→

⊕i≥0

C2i+1 (b+B)−−−−→

⊕i≥0

C2i (b+B)−−−−→

⊕i≥0

C2i+1 · · ·

The periodic cyclic cohomology HP ∗(C) can be defined as the cohomology of this

total complex. Note that this total complex is also Z2-periodic.

2.4.3 Third description of periodic cyclic cohomology

Let us first recall that the operator B in the (b, B)-bicomplex appears in the following

long exact sequence, called the Connes SBI sequence [7].

. . . HCn(C)I−−−→ HHn(C)

B−−−→ HCn−1(C) S−−−→ HCn+1(C)I−−−→ HHn+1(C) . . .

(2.4.5)

Here I is the map induced by the injection C∗λ −→ C∗HH , and S : HCn(C) →

HCn+2(C), called the Connes periodicity map, can be defined, at the level of com-

21

plexes, by

S[x] = n(n+ 1)bB−1[x] = n(n+ 1)[b(1− λ)−1b′N−1x].

For the third description of periodic cyclic cohomology, let C = (Cn, δni , σni , τn), n ≥

0, be a cocyclic module. The periodic cyclic cohomology of C can be defined by the

direct limit as follows:

HP i(C) := lim→S

HCi+2n(C), i = 0, 1,

where the direct limit, lim→S

, is with respect to the Connes’ periodicity operator S.

Remark 2.4.1. It is important to mention that this definition is based on the fact

that direct limit commutes with homology. More precisely, considering the following

identity:

lim→S

Tot2n+∗B∗∗(C) =⊕n=0

C2n+∗,

where ∗ = 0, 1, one has:

lim→S

HC2n+∗(C) = lim→S

H(Tot2n+∗B∗∗(C)) = H(lim→S

Tot2n+∗B∗∗(C)) =

H(⊕n=0C2n+∗) = HP ∗(C).

Note that in the second equality we commute the homology with direct limit.

2.4.4 First and second description of periodic cyclic

homology

One can define the periodic cyclic homology by dualizing the bicomplexes B∗∗(C) or

C∗∗(C) and replacing the direct sum by direct product in the definition of the total

22

complexes. The reason for this replacement is that if we consider the total complexes

with direct sum, then homology of the total complexes will vanish.

Definition 2.4.2. The periodic cyclic homology of the cyclic module C = (Cn, δni , sni , τn)

can be defined as the homology of the total complex of the following bicomplex B∗∗(C):

......

......

...

· · ·C4 B←−−− C3 B←−−− C2 B←−−− C1 B←−−− C0

b

y b

y b

y b

y· · ·C3 B←−−− C2 B←−−− C1 B←−−− C0

b

y b

y b

y· · ·C2 B←−−− C1 B←−−− C0

b

y b

y· · ·C1 B←−−− C0

b

y· · ·C0

The total complex of B∗∗(C) is:

· · ·∏i≥0

C2i+1 (b+B)−−−−→

∏i≥0

C2i (b+B)−−−−→

∏i≥0

C2i+1 (b+B)−−−−→

∏i≥0

C2i · · ·

Alternatively, one can use the bicomplex C∗∗(C) dual to C∗∗(C) to obtain the

periodic cyclic homology of C.

Remark 2.4.2. Recall that in the homology case, we have the following long exact

ISB-sequence:

. . . HHn(C)I−−−→ HCn(C)

S−−−→ HCn−2(C) B−−−→ HHn−1(C) I−−−→ HCn−1(C) . . .

23

However, in contrast to the cohomology case, the definition for periodic cyclic

homolog in terms of inverse limit fails. Because the inverse limit does not commute

with homology in general:

lim←−HC2n+∗(C) = HP∗(C).

A good example for this fact, using group algebra of cyclic groups, can be found in

[22].

In Chapter 5, we will provide a condition on the inverse system (HC(C)[−2m], S)m

which guarantees the commutativity of inverse limit and homology. We need this

property to generalize the Kunneth formula.

2.5 Normalized complex

In this section we introduce the notion of a normalized complex, which provides us

with a strong computational method for cyclic (co)homology.

Definition 2.5.1. [31, 34] Let C = (Cn, δni , s

ni , τn) be a cyclic module and Dn(C) be

the module generated by the images of all degeneracies sn−1i . We define

Nn(C) :=Cn

Dn,

with induced faces, degeneracies and cyclic maps on the quotient. One can prove that

Cn = Dn ⊕Nn, (2.5.6)

HH∗(C) = H∗(N), (2.5.7)

HC∗(C) = H∗(B(N)), (2.5.8)

HP∗(C) = H∗(B(N)). (2.5.9)

24

The complex Nn(C) :=CnDn

is known as the normalized complex for C = (Cn, δni , s

ni , τn).

The same notion can be defined for cocyclic modules.

Definition 2.5.2. Let (Cn, δni , sni , τn) be a cocyclic module. If we define

Nn(C) :=n−1∩i=0

ker(δni ),

then we have the similar relations as (2.5.6)- (2.5.9). The complex Nn(C) is known

as the normalized complex for (Cn, δni , sni , τn).

Example 2.5.1. For the cyclic module in Example (2.1.1), we have:

Nn(A) =A⊗(n+1)

D,

where D is generated by all elements (a0⊗...⊗an) such that ai = 1 for some 1 ≤ i ≤ n.

2.6 Hopf cyclic cohomology

Hopf cyclic cohomology which is, in fact, the right noncommutative analogue of both

group and Lie algebra homology, was introduced by Connes and Moscovici in [7, 8].

In this section we review the basics of this theory.

Definition 2.6.1. [7, 8] Let (H,∆, ε, S) be a Hopf algebra, δ : H → k a character

of H, and σ a group-like element of H. The pair (δ, σ) is called a modular pair if

δ(σ) = 1. One also can define a δ-twisted antipode, S, by

S(h) := δ(h(1))S(h(2)), ∀h ∈ H.

A modular pair (δ, σ) is called a modular pair in involution if:

σ−1S2(h)σ = h, ∀h ∈ H.

25

We usually use the abbreviation MPI to refer to a modular pair in involution.

Example 2.6.1. In any commutative or cocommutative Hopf algebra, the pair (ε, 1)

is a MPI for H.

Example 2.6.2. For the Hopf algebra A(SLq(2, k)) of the Example (1.1.3), one can

define a MPI for H by:

δ(x) = q, δ(u) = δ(v) = 0, δ(y) = q−1,

and σ = 1.

Example 2.6.3. If H and K are two Hopf algebras and (δ1, σ1) and (δ2, σ2) are two

MPI’s for H and K respectively, then (δ1 ⊗ δ2, σ1 ⊗ σ2) is a MPI for H ⊗K.

Now we proceed to the definition of Hopf cyclic cohomology. Let (H,∆, ε, S)

be a Hopf algebra, and (δ, σ) be a modular pair in involution for H. One can associate

a cocyclic module to (H, (δ, σ)) as follows. Define Cn(H) = H⊗n, n ≥ 0, and cofaces,

codegeneracies, and cocyclic maps by:

δi(h1 ⊗ ...⊗ hn−1) =

(1⊗ h1 ⊗ ...⊗ hn−1) i = 0

(h1 ⊗ ...⊗∆hi ⊗ ...⊗ hn−1) 1 ≤ i ≤ n− 1

(h1 ⊗ ...⊗ hn−1 ⊗ σ) i = n

σi(h1 ⊗ ...⊗ hn+1) = ε(hi+1)(h1 ⊗ ...⊗ hi ⊗ hi+2 ⊗ ...⊗ hn+1), 0 ≤ i ≤ n

τn(h1 ⊗ ...⊗ hn) =(S(h

(n)1 )h2 ⊗ ...⊗ S(h

(2)1 )hn ⊗ S(h

(1)1 )σ

).

Definition 2.6.2. The cohomology of the aforementioned cocyclic module is called

the Hopf cyclic cohomology of the Hopf algebra H endowed with a MPI (δ, σ). We

26

use the notation HC∗(δ,σ)

(H), to refer to the Hopf cyclic cohomology. We also denote

the periodic Hopf cyclic cohomology of H by HP ∗(δ,σ)

(H).

The following Lemma plays an important role in our future computations:

Lemma 2.6.1. If H is a Hopf algebra with MPI (δ, σ) then [8]

τkn(h1 ⊗ ...⊗ hn) = (S(h(n)k )hk+1 ⊗ S(h

(n−1)k )hk+2 ⊗ ...⊗ S(h

(k+1)k )hn ⊗

S(h(k)k )σ ⊗ S(h(k−1)k )σh1 ⊗ ...⊗ S(h

(2)k )σhk−2 ⊗ S(h

(1)k )σhk−1).

Example 2.6.4. [7, 8] Let H = CG, the group algebra of a discrete group G. Since

H is cocommutative, (ε, 1) is an MPI for CG. It can be shown that:

HHn(CG) =

C if n = 0

0 if n ≥ 1

Using the IBS sequence (2.4.5) we have

HCn(δ,σ)(CG) =

C if n = even

0 if n = odd.

Therefore:

HP 0(δ,σ)(CG) = C, HP 1

(δ,σ)(CG) = 0.

Example 2.6.5. The normalized complex Nn(U(g)) associated to the Hopf algebra

U(g) described in the Definition (2.5.2) is generated by all elements h1 ⊗ ... ⊗ hn,

where hi = 1U(g).

27

2.7 Dual Hopf cyclic cohomology

In Section (2.6) we introduced the Connes-Moscovici cocyclic module assigned to

Hopf algebras. There is also a cyclic module, as a dual to Connes-Moscovici cocyclic

module, for Hopf algebras. This dual theory was introduced in [28]. In this Section,

we will recall this dual Hopf cyclic theory. Let (H,m, η,∆, ε, S) be a Hopf algebra,

σ be a nonzero group-like element of H, and δ : H −→ k be a character for H. The

pair (δ, σ) is called a modular pair if δ(σ) = 1. Define a twisted antipode, in the dual

sense, by:

Sσ(h) = σ∑(h)

δ(h(2))S(h(1)). (2.7.10)

A modular pair (δ, σ) is called a modular pair in involution, in the dual sense, if

S2σ = id. (2.7.11)

A cyclic module can be associated to (H, δ, σ) as follows. Set C(δ,σ)n (H) = H⊗n, n ≥

0, with faces, degeneracies, and cyclic maps given by:

δ0(h1 ⊗ h2 ⊗ ...⊗ hn) = ε(h1)(h2 ⊗ h3, ...⊗ hn),

δi(h1 ⊗ h2 ⊗ ...⊗ hn) = (h1 ⊗ h2 ⊗ ...⊗ hihi+1 ⊗ ...⊗ hn)⊗ 1 ≤ i ≤ n− 1,

δn(h1 ⊗ h2 ⊗ ...⊗ hn) = δ(hn)(h1 ⊗ h2 ⊗ ...⊗ hn−1),

s0(h1 ⊗ h2 ⊗ ...⊗ hn) = (1⊗ h1 ⊗ ...⊗ hn),

si(h1 ⊗ h2 ⊗ ...⊗ hn) = (h1 ⊗ h2 ⊗ ...⊗ hi ⊗ 1⊗ hi+1 ⊗ ...⊗ hn), 1 ≤ i ≤ n− 1,

sn(h1 ⊗ h2 ⊗ ...⊗ hn) = (h1 ⊗ h2 ⊗ ...⊗ hn, 1)

τn(h1 ⊗ h2 ⊗ ...⊗ hn) = δ(h(2)n )

(Sσ(h

(1)1 h

(1)2 ...h

(1)n )⊗ h(2)1 ⊗ ...⊗ h

(2)n−1)

where, Sσ(h) = σS(h). The cyclic homology of this cyclic module is called the dual

Hopf cyclic homology of H and is denoted by HC(δ,σ)∗ (H) [28]. We also denote the

dual periodic cyclic homology by HP(δ,σ)∗ (H).

28

Example 2.7.1. [28] Let H = kG be the group algebra of a discrete group G. The

faces, degeneracies and cyclic maps defined before reduce to the following ones:

δi(g1 ⊗ ...⊗ gn) =

(g2 ⊗ ...⊗ gn) if i = 0

(g1 ⊗ ...⊗ gigi+1 ⊗ ...⊗ gn) 1 ≤ i < n

(g1 ⊗ ...⊗ gn−1) if i = n

si(g1 ⊗ ...⊗ gn) =

(1⊗ g1 ⊗ ...⊗ gn) if i = 0

(g1 ⊗ ...⊗ gi ⊗ 1⊗ gi+1 ⊗ ...⊗ gn) 1 ≤ i ≤ n− 1

(g1 ⊗ ...⊗ gn−1 ⊗ gn ⊗ 1) if i = n

τ(g1 ⊗ g2 ⊗ ...⊗ gn) = ((g1g2...gn)−1 ⊗ g1 ⊗ ...⊗ gn−1).

One also proves that

HP(ϵ,1)n (kG) =

∏i≥0

H2i(G; k) n = 0∏i≥0

H2i+1(G; k) n = 1

in which H∗(G; k) is the group homology with trivial coefficients. We will give more

details about the group (co)homology in Chapter 3.

Example 2.7.2. [28] By a lengthy computation, one can see that if q ∈ k is not a

root of unity then HC(δ,1)1 (A(SLq(2, k))) = k⊕ k and HC

(δ,1)n (A(SLq(2, k))) = 0 for

all n = 1.

In particular, HP(δ,1)0 (A(SLq(2, k))) = HP

(δ,1)1 (A(SLq(2, k))) = 0.

Example 2.7.3. Let g be a Lie algebra over k and U(g) be its enveloping algebra. It

can be shown that [8]

HC(ϵ,σ)n (U(g)) ∼=

⊕k≥0

HLien−2k(g; k).

29

Lemma 2.7.1. (Mac Lane isomorphism for Hopf algebras)

Let H be a Hopf algebra andM be a H-bimodule. One can define a new left H-module

[28], M =M , with the left action given by

h.m = h(2)mS(h1).

Let define Hn(H; M) to be the homology of the following simplicial module. Set

Cn(H; M) = H⊗n ⊗ M , with faces and degeneracies defined by

δ0(h1 ⊗ · · · ⊗ hn ⊗m) = (ϵ(h1)h2 · · · ⊗ hn ⊗m)

δi(h1 ⊗ · · · ⊗ hn ⊗m) = (h1 ⊗ · · · ⊗ hihi+1 ⊗ · · · ⊗ hn ⊗m) 1 ≤ i ≤ n− 1

δn(h1 ⊗ · · · ⊗ hn ⊗m) = (h1 ⊗ · · · ⊗ hn−1 ⊗ hn.m)

si(h1 ⊗ · · · ⊗ hn ⊗m) = (h1 ⊗ · · · ⊗ hi ⊗ 1⊗ hi+1 · · · ⊗ hn ⊗m) 0 ≤ i ≤ n.

Then there is a canonical isomorphism citekr1,

θ∗ : Hn(H,M) ∼= Hn(H; M),

called Mac Lane isomorphisms where in the left-hand side Hn(H,M) is the Hochshcild

homology of H, as an algebra, with coefficients in M .

Example 2.7.4. Let k be equipped with the trivial H-module structure via ε. Then

k is also a trivial H-module and we have,

Hn(H, k) = HHn(H; k).

Example 2.7.5. Using Mac Lane isomorphism one can prove that for any cocom-

30

mutative Hopf algebra H [28],

HC(ϵ,1)n (H) =

⊕k≥0

Hn−2k(H, k). (2.7.12)

Example 2.7.6. Let T (V ) be the tensor algebra of a vector space V . It is known that

HHn(T (V ), k) =

k n = 0

V n = 1

0 n = 0, 1

Therefore,

HC(ϵ,1)n (T (V )) =

k n is even

V n is odd

Example 2.7.7. Let H, A, δ and σ ba as in Definition (1.1.7) and Tr : A −→ k be a

δ-trace on A which is σ-invariant. Let C∗(A) denote the cyclic module of the algebra

A, and define the map

γ : Cn(A) −→ C(δ,σ)n (H),

γ(a0 ⊗ a1 ⊗ ...⊗ an) =∑

Tr(a0a(0)1 a

(0)2 ...a

(0)n )(a

(1)1 ⊗ ...⊗ a

(1)n ). (2.7.13)

Then γ is a map of cyclic modules, and induces a canonical map in the level of

homology:

γ∗ : HC∗(A) −→ HC(δ,σ)∗ (H). (2.7.14)

Example 2.7.8. One knows that any Hopf algebra H has a right coaction on itself

by comultiplication. Let H have a σ-invariant, δ-trace trace Tr. Then the map γ

(2.7.13) is reduced to

γ : Cn(H) −→ C(ϵ,σ)n (H),

31

by

γ(h0 ⊗ h1 ⊗ · · · ⊗ hn) =∑

Tr(h0h11 · · ·h

1n)(h

21 ⊗ · · · ⊗ h

2n).

It is possible to define a reverse map to γ in the following way.

θ : C(ϵ,σ)n (H) −→ Cn(H),

θ(h1 ⊗ h2 ⊗ · · · ⊗ hn) = (Sσ(h(1)1 h

(1)2 · · ·h

(1)n )⊗ h(2)1 ⊗ h

(2)2 ⊗ · · · ⊗ h

(2)n ).

Notice that the map θ is not an inverse for γ.

2.7.1 Non localized cyclic module for dual Hopf cyclic

homology

We will need the notion of non-localized cyclic module for dual Hopf cyclic homology,

defined bellow, to study the isomorphism (2.7.12) which we use later to get the

application of our coproducts in the dual theory.

Definition 2.7.1. [28] Let C = (Cn, δi, si) be a simplicial module. We define a new

simplicial module, called non-localized simplicial module, by

ECn := Cn+1,

with the same degeneracies as C and with the nth face being the (n+ 1)th face of C.

Example 2.7.9. If we apply the above procedure to the module C(δ,σ)∗ (H), the non

32

localized simplicial module EC reduces to the following one:

ECn(H) = H⊗n+1,

δi(h0 ⊗ h1 ⊗ ...⊗ hn) = h0 ⊗ h1 ⊗ ...⊗ hihi+1 ⊗ ...⊗ hn 0 ≤ i ≤ n− 1

δn(h0 ⊗ h1 ⊗ ...⊗ hn) = δ(hn)h0 ⊗ h1 ⊗ ...⊗ hn−1

si(h0 ⊗ h1 ⊗ ...⊗ hn) = h0 ⊗ h1...⊗ hi ⊗ 1⊗ hi+1...⊗ hn 0 ≤ i ≤ n− 1

sn(h0 ⊗ h1 ⊗ ...⊗ hn) = h0 ⊗ h1 ⊗ ...⊗ 1.

Lemma 2.7.2. If H is a cocommutative Hopf algebra, then ECn(H) is a cyclic

module with the cyclic maps defined by:

τn(h0 ⊗ · · · ⊗ hn) =∑

(h0h(1)1 . . . h

(1)n )⊗ S(h(2)1 . . . h

(2)n )⊗ h(3)1 ⊗ · · · ⊗ h

(3)n−1.

Lemma 2.7.3. Let define [28]

π : ECn(H) −→ C(ϵ,σ)(H),

π(h0 ⊗ · · · ⊗ hn) = (ϵ(h0)h1 ⊗ · · · ⊗ hn).

Then π is a simplicial map. If H is cocommutative, then π is a cyclic map.

Remark 2.7.1. It is obvious that ECn(H) is a H-module via

h(h0 ⊗ · · · ⊗ hn) := (hh0 ⊗ · · · ⊗ hn),

and also we have the following isomorphism [28]:

C(ϵ,σ)(H) ∼= k ⊗H ECn(H).

33

2.8 Hopf cyclic cohomology with coefficients

In order to define a system of coefficients for cyclic (co)homology, one needs the

notion of anti-Yetter-Drinfeld H-module. This notion was first defined in [16] and

Hopf cyclic cohomology with coefficients was introduced in [17]. In this section we

will recall the definition of a stable anti-Yetter Drienfeld module and review the

Hopf cyclic cohomology with coefficients. There are four possible cases of an anti-

Yetter-Drinfeld H-module, i.e., left-left, left-right, right-left and right-right module

comodule. Here we will mention only the two cases right-left and left-left.

Definition 2.8.1. A right-left anti-Yetter-Drinfeld H-module M is a right H-module

and a left H-comodule with the following compatibility condition

(mh)(−1) ⊗ (mh)(0) = S(h(3))m(−1)h(1) ⊗m(0)h

(2), (2.8.15)

for all h ∈ H and m ∈M. If the relation

m(0)m(−1) = m,

is satisfies for all m ∈M , we say M is stable.

Definition 2.8.2. A left-left anti-Yetter-Drinfeld H-module M is simultaneously a

left H-module and left H-comodule such that

(hm)(−1) ⊗ (hm)(0) = h(1)m(−1)S−1(h(3))⊗ h(2)m(0), (2.8.16)

for all h ∈ H and m ∈M. We say that M is stable if

m(−1)m(0) = m,

for all m ∈M .

34

Throughout this thesis we use the abbreviation SAYD to refer to a stable anti-

Yetter-Drinfeld module.

Example 2.8.1. Let H be a Hopf algebra. Given a character δ : H −→ k and a group-

like element σ ∈ H, one defines a module and comodule structure on k as follows.

The action of H is defined by the character δ, mh = δ(h)m, and the coaction is

defined via the group-like element σ, φ(m) = σ⊗m. Then this module and comodule,

denoted by M =σkδ, is stable if and only (δ, σ) is a modular pair, and is anti-Yetter-

Drinfeld, if and only if (δ, σ) is modular pair in involution.

Example 2.8.2. [17] For the group algebra kG, a left H-comodule M , is a G-graded

vector spaceM =⊕

g∈GMg, where the coaction is defined by m 7→ g⊗m for m ∈Mg.

Let G also act on M . Then M is an AYD kG-module if and only if hm ∈ Mhgh−1

for all g, h ∈ G and m ∈ Mg. Also M is stable if and only if gm = m for all g ∈ G,

m ∈Mg.

Example 2.8.3. Let H = U(g) be the universal enveloping algebra of a Lie algebra

g and M be a module over H. If we define a comodule structure on M by the trivial

coaction m 7−→ 1⊗m, one can easily check that M is a SAYD module over H.

In [17], to define Hopf cyclic cohomology with coefficients, the authors assign

a cocyclic module to any triple (H,C,M) where C is a H-module coalgebra and M

is a SAYD H-module . We do not go into that general theory in this thesis, but we

mention two important special cases. Let H be a Hopf algebra with multiplication

mH and (δ, σ) be a modular pair in involution for H. ConsiderM =σkδ as in Example

(2.8.1), and let C = H, equipped with a H-module coalgebra structure given by the

multiplicationmH . Then the theory provided in [17] reduces to the Connes-Moscovici

theory. We like to mention the following Lemma which is, in fact, the main ingredient

in the proof of the above statement.

35

Lemma 2.8.1. The map f : H⊗(n+1) → H⊗(n+1), defined by

f(h0 ⊗ · · · ⊗ hn) := h(1)0 ⊗ S(h

(n+1)0 )h1 ⊗ · · · ⊗ S(h

(2)0 )hn,

is an H-module isomorphism, where H⊗(n+1), on the left-hand side, is considered as

an H-module via diagonal action and H⊗(n+1), on the right-hand side, is considered

as an H-module via multiplication by the first term. The inverse of f is given by

f−1(h0 ⊗ · · · ⊗ hn) := h(1)0 ⊗ h

(2)0 h1 ⊗ · · · ⊗ h

(n+1)0 hn.

Now we proceed to the more general case of the Hopf cyclic cohomology with

coefficients.

Definition 2.8.3. LetM be a SAYD H-module. Set Cn(H,M) :=M⊗HH⊗n+1, n ≥

0, with the cofaces, codegeneracies and cocyclic maps as follows:

δi(m⊗ h0 ⊗ · · · ⊗ hn−1) = m⊗ h0 ⊗ · · · ⊗ h(1)i ⊗ h

(2)i ...⊗ hn−1, 0 ≤ i < n

δn(m⊗ h0 ⊗ · · · ⊗ hn−1) = m(0) ⊗ h(2)0 ⊗ h1 ⊗ · · · ⊗ hn−1 ⊗m(−1)h(1)0

si(m⊗ h0 ⊗ · · · ⊗ hn+1) = m⊗ h0 ⊗ · · · ⊗ ε(hi+1)⊗ · · · ⊗ hn+1, 0 ≤ i ≤ n

τn(m⊗ h0 ⊗ · · · ⊗ hn) = m(0) ⊗ h1 ⊗ · · · ⊗ hn ⊗m(−1)h0

It can be proved that this data defines a cocyclic module. The cohomology of this

cocyclic module is called Hopf cyclic cohomology of H with coefficients in M .

Now we recall the cyclic homology with coefficients for the dual Hopf cyclic

theory. Let us first recall the definition of cotensor. Let H be a Hopf algebra A be

a right H-comodule, and B left H-comodule. One defines the cotensor product of A

and B over H by:

A2HB = a⊗ b ∈ A⊗B | a(0) ⊗ a(1) ⊗ b = a⊗ b(−1) ⊗ b(0).

36

Definition 2.8.4. Let M be a left-left stable anti-Yetter Drinfeld H-module. We set

Cn(H,M) := H⊗n+12HM , n ∈ N, with faces, degeneracies and cyclic maps defined

by:

δi(h0 ⊗ . . .⊗ hn ⊗m) = h0 ⊗ . . .⊗ hihi+1 ⊗ . . .⊗ hn ⊗m, 0 ≤ i < n,

δn(h0 ⊗ . . .⊗ hn ⊗m) = h(0)n h0 ⊗ h1 . . .⊗ hn−1 ⊗ h

(1)n m,

si(h0 ⊗ . . .⊗ hn ⊗m) = h0 ⊗ . . .⊗ hi ⊗ 1⊗ . . .⊗ hn ⊗m, 0 ≤ i ≤ n,

τn(h0 ⊗ . . .⊗ hn ⊗m) = h(0)n ⊗ h0 ⊗ . . .⊗ hn−1 ⊗ h

(1)n m.

This structure defines a cyclic module. The homology of this cyclic module is called

the dual Hopf cyclic homology of H with coefficients in M .

Example 2.8.4. As a special case, whenM =σkδ, then the cyclic module in Definition

(2.8.4) reduces to the cyclic module of dual Hopf cyclic homology.

Remark 2.8.1. In the Definition (2.8.4) we have considered H as a left H-comodule

via comultiplication. Also H⊗(n+1) is a left H-comodule via diagonal coaction given

by

h0 ⊗ ...⊗ hn −→ h(0)0 ⊗ ...⊗ h

(0)n ⊗ h

(1)0 h

(1)1 ...h

(1)n .

Chapter 3

Preliminaries of cup product and

coproduct

Cup product is a method to define a graded product on (co)homology. Given two

(co)cycles of degrees p and q, the cup product forms a (co)cycle of degree p+ q. This

operation turns the (co)homology space into a graded ring, called the (co)homology

ring. The cup product was first introduced by J. W. Alexander, Eduard Cech and

Hassler Whitney between 1935 and 1938, and, in full generality, by Samuel Eilenberg

in 1944. In a dual manner one can define a graded coproduct on (co)homology. For

example, in de Rham cohomology, the cup product of differential forms is also known

as the wedge product. As we mentioned in Chapter (1), any Hopf algebra H contains

a group, the group of group-like elements, and a Lie algebra, the Lie algebra of

primitive elements. In this thesis we define appropriate cup products and coproducts

on Connes-Moscovici Hopf cyclic cohomology and the dual Hopf cyclic homology and

then we show the relations of these products to the ones for Lie algebra (co)homology

and group (co)homology. In this chapter after a short review of group (co)homology

and Lie algebra (co)homology we recall the cup products on group cohomology and Lie

algebra cohomology. We also recall the coproducts on group homology and Lie algebra

homology. Let us start with general definitions of cup products and coproducts.

Definition 3.0.5. Suppose H∗(C) is the cohomology of the cochain complex (C∗, d)

where d’s are coboundaries. A cup product for H∗(C) is a linear map

: Hp(C)⊗Hq(C) −→ Hp+q(C),

37

38

induced from a pairing:

Cp ⊗ Cq −→ Cp+q,

by

[cp]⊗ [cq] 7−→ [cp cq],

where cp ∈ Cp and cq ∈ Cq such that in the level of complexes we have:

d(cp cq) = d(cp) cq + (−1)pcq d(cq).

Dually, a coproduct for H∗(C) is a linear map

: Hn(C) −→⊕

p+q=n

Hp(C)⊗Hq(C),

induced from a coproduct

cn 7−→⊕

p+q=n

cp ⊗ cq,

such that in the level of complexes we have:

d(cn) = d(cp)⊗ cq + (−1)pcp ⊗ d(cq).

We will have some examples in the next two sections.

3.1 Group (co)homology

In this section we provide a quick review of group (co)homology. In the whole section,

G denotes a discrete group.

Definition 3.1.1. Let G be a discrete group andM a left kG-module. Set C0(G,M) =

M and let

Cn(G,M) = φ : Gn −→M, n ≥ 1

39

be all functions from Gn toM . We define coboundaries dn : Cn(G,M) −→ Cn+1(G,M)

given by

dn(φ)(g1, ..., gn+1) := g1.φ(g2, ..., gn+1)+

n∑i=0

(−1)iφ(gi, ..., gi−1, gigi+1, ..., gn+1) + (−1)n+1φ(g1, ..., gn).

The cohomology of C∗(G,M) with respect to d∗ is called the group cohomology of G

with coefficients in M .

Definition 3.1.2. Let G be a discrete group, and M be a left kG-module. Set

C0(G,M) = M and Cn(G,M) = M ⊗ kGn for n ≥ 1 and define dn : Cn(G,M) −→

Cn−1(G,M) by:

dn(m, g1, ..., gn) = (mg1, g2, ..., gn)

−n−1∑i=1

(−1)i(m, g1, ..., gigi+1, ..., gn) + (−1)n+1(m, g1, ..., gn−1).

The homology of C∗(G,M) with respect to d∗ is called the group homology of G with

coefficients in M .

Remark 3.1.1. Group cohomology with coefficients inM coincides with the Hochschild

cohomology of kG, as an algebra, with coefficients in M ,i.e,

HHn(kG,M) ∼= Hn(G,M).

Example 3.1.1. If M = kG where G acts by conjugation then

HHn(kG, kG) ∼= Hn(G, kG).

Example 3.1.2. Group homology with trivial coefficients coincides with the Hochschild

homology of kG as a Hopf algebra in dual theory:

HHn(KG) ∼= Hn(G, k).

40

Example 3.1.3. Let G be a discrete group. Let φ(g1, ..., gn) be a group n-cocycle on

G. Thus φ : Gn −→ k satisfies the cocyclic condition

g1.φ(g2, ..., gn+1)− φ(g1g2, ..., gn+1) + ...+ (−1)n+1φ(g1, ..., gn) = 0,

for all g1, ..., gn+1 ∈ G. It can be shown that any cocycle is cohomologous to a

normalized element f in the sense that

f(g1, ..., gn) = 0

if gi = e for some i, or if g1g2...gn = e.

Example 3.1.4. Let M = k be the trivial coefficient. There is a way to obtain a

cyclic cocycle in HC∗(kG) as an algebra, from a group cocycle φ in H∗(G, k). In fact

the following map is a map of complexes:

Φφ(g1, ..., gn) =

φ(g1, ..., gn) g0g1...gn = e,

0 otherwise

In this way we obtain a map from the group cohomology of G to the cyclic cohomology

of kG as an algebra,

Hn(G, k) −→ HCn(kG),

φ 7−→ Φφ.

Example 3.1.5. One can easily see

HH0(kG, kG) = H0(G, kG) =∏<G>

k

and

HH0(kG, kG) = H0(G, kG) =⊕<G>

k.

41

Now we generalize the previous example.

Example 3.1.6. Let Cg be the centralizer of g in G, i.e.,

Cg = h ∈ G;hg = gh.

One can show

HH∗(kG, kG) ∼=⊕<G>

H∗(Cg, k),

and

HH∗(kG, kG) ∼=⊕<G>

H∗(Cg, k).

The reason is

C∗(kG, kG) =⊕

c∈<G>

B∗(G, c),

where c is a conjugacy class in < G > and Bn(G, c) is the linear span of all (n+ 1)-

tuples (g0, g1, ..., gn) ∈ Gn+1 such that g0g1...gn ∈ c. Also one can show that

H∗(B(G, c)) = H∗(Cg, k).

3.2 Cup products and coproducts for group

(co)homology

In this section, we introduce a cup product for group cohomology and a coproduct

for group homology.

Lemma 3.2.1. Suppose G is a discrete group, M , N and P are G-modules. Let

× :M ×N −→ P be a bilinear map in the sense that

(g.m)× (g.n) = g.(m× n).

42

We define the following cup product

Gr: Hp(G,M)⊗Hq(G,N) −→ Hp+q(G,P ),

given by

(fp Gr fq)(g1, ..., gp+q) = (fp(g1, ..., gp).g1...gp)× fq(gp+1, ..., gp+q).

One can check that:

d(fp Gr fq) = dfp Gr fq + (−1)pfp Gr dfq.

Example 3.2.1. As a special case of the previous lemma, if M = N = P , then we

obtain the following cup product

Gr: Hp(G,M)⊗Hq(G,M) −→ Hp+q(G,M),

where M is a G-module algebra.

Lemma 3.2.2. For any discrete group G one can define the following coproduct for

group homology

Gr: Hn(G, k) −→⊕

p+q=n

Hp(G, k)⊗Hq(G, k),

given by

Gr (g1, ..., gn) = 1⊗ (g1, ..., gn) +n−1∑p=1

(g1, ..., gp)⊗ (g1, ..., gq) + (g1, ..., gn)⊗ 1,

43

which satisfies the following property

Gr dn = (∑

p+q=n

dp ⊗ idq + (−1)pidq ⊗ dq)Gr .

We check the last statement for n = 3.

Gr d(g1, g2, g3)

= Gr ((g2, g3)− (g1g2, g3) + (g1, g2g3)− (g1, g2))

= 1⊗ (g1, g3) + g2 ⊗ g3 + (g2, g3)⊗ 1

− 1⊗ (g1g2, g3)− (g1g2)⊗ g3 − (g1g2, g3)⊗ 1

+ 1⊗ (g1, g2g3) + g1 ⊗ (g2g3) + (g1, g2g3)⊗ 1

− 1⊗ (g1, g2)− g1 ⊗ g2 − (g1, g2)⊗ 1

On the other hand

(d⊗ id+ (−1)pid⊗ d)(Gr (g1, g2, g3))

+ 1⊗ ((g2, g3)− (g1g2, g3) + (g1, g2g3)− (g1, g2))

− g1 ⊗ (g3 − g2g3 + g2) + (g2 − g1g2 + g1)⊗ g3

+ ((g2, g3)− (g1g2, g3) + (g1, g2g3)− (g1, g2))⊗ 1.

3.3 Lie algebra (co)homology

In this section we recall the basic facts about Lie algebra homology and Lie algebra

cohomology. In fact Lie algebra (co)homology is an example of Hochschild cohomol-

ogy.

Definition 3.3.1. Let g be a Lie algebra and M a g-module, i.e.,

[g1, g2].m = g1(g2.m)− g2.(g1.m).

44

We define the Chevalley-Eilenberg complex by Cn(g,M) = Homk(∧n g,M), where

n = 0, 1, ..., dimg and C0(g,M) =M and coboundaries are

dLie : Cn(g,M) −→ Cn+1(g,M),

given by

dLie(f)(g1∧ ...∧gn+1) =∑

1≤i<j≤n+1

(−1)i+jf([gi, gj ]∧g1∧ ...∧ gi∧ ...gj ∧ ...∧gn+1)+

n+1∑i=1

(−1)igi.f(g1 ∧ ... ∧ gi ∧ ... ∧ gn+1).

The cohomology of the Chevalley-Eilenberg complex is called the Lie algebra cohomol-

ogy of g with coefficients in M .

Definition 3.3.2. Let g be a Lie algebra and M a g-module. Let C0(g,M) =M and

Cn(g,M) =∧n g⊗M . We define the boundaries as follows:

dLie : Cn(g,M) −→ Cn−1(g,M),

given by

dLie(g1 ∧ ... ∧ gn ⊗m) =∑

1≤i<j≤n(−1)i+j [gi, gj ] ∧ g1 ∧ ...gi ∧ ...gj ... ∧ gn ⊗m+

n∑i=1

(−1)i+1g1 ∧ ... ∧ gi ∧ ... ∧ gim.

The homology of C∗(g,M) with respect to dLie∗ is called the Lie algebra homology of

g with coefficients in M .

Example 3.3.1. Lie algebra homology is an example of Hochschild homology. In fact

45

citeL:

H∗(g,Mad) ∼= HH∗(U(g),M),

where Mad can be defined as follows. Let M be an U(g)-bimodule. One can define

a left g-module Mad, where as a set we have Mad = M and the left action can be

define as:

X.m = Xm−mX,

for all X ∈ g and m ∈M .

Example 3.3.2. Let g be a Lie algebra acting by derivations on an algebra A. For

each n ≥ 0 we define a map

HLien (g,C) −→ HCn(A)

given by

c = (g1 ∧ ... ∧ gn) 7−→ φc,

where

φc(a0, ..., an) =∑σ∈Sn

sgn(σ)τ(a0gσ(1)(a1)...gσ(n)(an)),

where τ : A −→ C is a trace which is invariant under the action of g,i.e.,

τ(g(a)) = 0, ∀g ∈ g, a ∈ A.

The following theorem plays an important role for the application of the co-

products in Hopf cyclic cohomology, which will be introduced in chapter (6).

Theorem 3.3.1. [7, 8] Let H = U(g) be the universal enveloping algebra of a Lie

algebra g. Let δ : U(g)→ C be a character of H. One has:

HP ∗(δ,1)(U(g)) =⊕i≥0

HLie2i+∗(g,Cδ), ∗ = 0, 1. (3.3.1)

46

Here HLie∗ (g,Cδ) is the Lie algebra homology with coefficients in Cδ.

Proof. [7, 8, 10] The anti-symmetrization map defined by

A :n∧g→ U(g)⊗n,

A(g1 ∧ ... ∧ gn) = (∑σ∈Sn

sign(σ)(gσ(1), ..., gσ(n)))/n!,

induces a quasi-isomorphism between the following two bicomplexes;

......

......

...

· · ·∧4 g

dLie−−−→∧3 g



dLie−−−→ C

0

x 0

x b

x 0

x· · ·

∧3 gdLie−−−→


∧1 gdLie−−−→ C

0

x 0

x 0

x· · ·



0

x 0

x· · ·


0

x· · ·C

and

47

......

......

...

· · ·U(g)⊗4 B−−−→ U(g)⊗3 B−−−→ U(g)⊗2 B−−−→ U(g)B−−−→ C

b

x b

x b

x b

x· · ·U(g)⊗3 B−−−→ U(g)⊗2 B−−−→ U(g)

B−−−→ C

b

x b

x b

x· · ·U(g)⊗2 B−−−→ U(g)

B−−−→ C

b

x b

x· · ·U(g) B−−−→ C

b

x· · ·C

The rows in the first bicomplex are the Chevalley-Eilenberg complex

3.4 Cup products and coproducts for Lie algebra

(co)homology

In this section we define a cup product for Lie algebra cohomology and a coproduct

for Lie algebra homology.

Lemma 3.4.1. Let g be a Lie algebra and M a g-module, which has an algebra

structure whose multiplication is compatible with its g-module structure in the sense

that

g.(ab) = (g.a)b+ a(g.b), ∀g ∈ g, a, b ∈ A.

Let f ∈ Cp(g,M) and h ∈ Cq(g,M). We define the following cup product

Lie: Cp(g,M)⊗ Cq(g,M) −→ Cp+q(g,M)

48

given by

(f Lie h)(g1∧...∧gp+q) =∑

σ∈Shp,qsgn(σ)f(gσ(1)∧...∧gσ(p))h(gσ(p+1)∧...∧gσ(p+q)

)

which induces the following cup product on the level of cohomology

Lie: Hp(g,M)⊗Hq(g,M) −→ Hp+q(g,M).

here Shp,q is a (p, q)-shuffle.

Proof. Here we check the following crucial condition:

dLie(f Lie g) = (dLief)Lie g − (f Lie (dLieg))

for f, g ∈ C1(g,M):

49

dLie(f Lie g)(x1 ∧ x2 ∧ x3) =

− x1(f Lie g)(x2 ∧ x3) + x2(f Lie g)(x1 ∧ x3)− x3.(f Lie g)(x1 ∧ x2)

+ (f Lie g)([x1, x2], x3)− (f Lie g)([x1, x3], x2) + (f Lie g)([x2, x3], x1) =

− x1((f(x2))(g(x3))) + x1((f(x3))(g(x2))) + x2((f(x1))(g(x3)))

− x2((f(x3))(g(x1)))− x3((f(x1))(g(x2))) + x3((f(x2))(g(x1)))

+ f([x1, x2])g(x3)− f([x1, x3])g(x2) + f([x2, x3])g(x1)

− f(x3)g([x1, x2]) + f(x2)g([x1, x3])− f(x1)g([x2, x3]) =

− (x1f(x2))g(x3)− f(x2)(x1g(x3))

+ (x1f(x3))g(x2) + f(x3)(x1g(x2))

+ (x2f(x1))g(x3) + f(x1)(x2g(x3))

− (x2f(x3))g(x1)− f(x3)(x2g(x1))

+ (x3f(x1))g(x2) + f(x1)(x3g(x2))

+ f([x1, x2])g(x3)− f([x1, x3])g(x2) + f([x2, x3])g(x1)

− f(x3)g([x1, x2]) + f(x2)g([x1, x3])− f(x1)g([x2, x3])

50

On the other hand

(dLief)Lie g(x1 ∧ x2 ∧ x3)− (f Lie (dLieg))(x1 ∧ x2 ∧ x3) =

+ dLie(x1 ∧ x2)g(x3)− dLie(x1 ∧ x3)g(x2) + dLie(x2 ∧ x3)g(x1)

− f(x1)dLieg(x2 ∧ x3) + f(x2)dLieg(x1 ∧ x3)− f(x3)dLieg(x1 ∧ x2) =

+ (f([x1, x2])− x1f(x2) + x2f(x1))g(x3)

− (f([x1, x3])− x1f(x3) + x3f(x1))g(x2)

+ (f([x2, x3])− x2f(x3) + x3f(x2))g(x1)

− f(x1)(g([x2, x3])− x2g(x3) + x3g(x2))

+ f(x2)(g([x1, x3])− x1g(x3) + x3g(x1))

− f(x3)(g([x1, x2])− x1g(x2) + x2g(x1)).

Example 3.4.1. (Poincare duality) If the Lie algebra g is finite dimensional and

dimg = n, we get dimCn(g, k) = 1. Suppose g is unitary which means if for some

basis e1, ..., en of g we have, [ei, ej ] =∑n

k=1 akijek, then

∑nk=1 a

jij = 0 for all

1 ≤ i ≤ n. For any nonzero cycle x ∈ Cn(g, k), one can define a nondegenerate

pairing

Cm(g, k)× Cn−m(g, k) −→ k,

given by

(f, g) 7−→ (f Lie g)(x).

This is a nice application of the cup product Lie of Lie algebra cohomology. A result

of the Poincare duality is Poincare isomorphism:

Cm(g, k) ∼= Cn−m(g, k).

51

Therefore we get the following map

Hm(g, k)×Hn−m(g, k) −→ k,

and we have:

Hm(g, k) ∼= Hn−m(g, k).

In the Chapter 8, we will define a cup product ⊔HC

for the dual Hopf cyclic

homology. A question is if H is a finite dimensional Hopf algebra, can we have the

Hopf cyclic version of the Poincare duality using this cup product? More precisely, If

dim(H) = n and f ∈ Cn(H) is an n-cocycle, then does the following map defines a

nondegenerate pairing:

TotBmC(H)× TotBn−mC(H) −→ k

given by

(a, b) 7−→ f(a ⊔HC

b).

Lemma 3.4.2. Let g be a Lie algebra and k a trivial g-module. Therefore we have

dLie(g1 ∧ ... ∧ gn) =∑

1≤i≤i≤n(−1)i+j+1[xi, xj ] ∧ (x1 ∧ ...xi ∧ ...xj ∧ ... ∧ xn).

Lie algebra homology with trivial coefficients has a coalgebra structure by the following

coproduct:

Lie: Cn(g, k) −→⊕

p+q=n

(Cp(g, k))⊗ (Cq(g, k)),

given by

Lie (g1 ∧ ...∧ gn) =∑σ

sign(σ)(gσ(1) ∧ ...∧ gσ(p))⊗ (gσ(p+1) ∧ ...∧ gσ(n)), (3.4.2)

52

where σ runs over all (p, q)-shuffles. Here we check this statement for n = 3.

(dLie ⊗ id+ (−1)pid⊗ dLie)(Lie (g1 ∧ g2 ∧ g3))

= (dLie ⊗ id+ (−1)pid⊗ dLie)(1⊗ (g1 ∧ g2 ∧ g3)

+ g1 ⊗ (g2 ∧ g3)− g2 ⊗ (g1 ∧ g3) + g3 ⊗ (g1 ∧ g2)

+ (g1 ∧ g2)⊗ g3 − (g1 ∧ g3)⊗ g2

+ (g2 ∧ g3)⊗ g1 + (g1 ∧ g2 ∧ g3)⊗ 1)

+ 1⊗ ([g1, g2] ∧ g3 − [g1, g3] ∧ g2 + [g2, g3] ∧ g1)

− g1 ⊗ [g2, g3] + g2 ⊗ [g1, g3]− g3 ⊗ [g1, g2]

+ [g1, g2]⊗ g3 − [g1, g3]⊗ g2 + [g2, g3]⊗ g1

+ ([g1, g2] ∧ g3 − [g1, g3] ∧ g2 + [g2, g3] ∧ g1)⊗ 1.

On the other hand

Lie d(g1 ∧ g2 ∧ g3))

= Lie ([g1, g2] ∧ g3 − [g1, g3] ∧ g2 + [g2, g3] ∧ g1)

= [g1, g2]⊗ g3 − g3 ⊗ [g1, g2] + 1⊗ ([g1, g2] ∧ g3) + ([g1, g2] ∧ g3)⊗ 1

− [g1, g3]⊗ g2 − g2 ⊗ [g1, g3]− 1⊗ ([g1, g3] ∧ g2)− ([g1, g3] ∧ g2)⊗ 1

+ [g2, g3]⊗ g1 − g1 ⊗ [g2, g3] + 1⊗ ([g2, g3] ∧ g1) + ([g2, g3] ∧ g1)⊗ 1.

Therefore

(dLie ⊗ id+ (−1)pid⊗ dLie)Lie=Lie dLie.

53

3.5 Cup products for cyclic cohomology of

algebras

In this section, we recall the Connes approach to define a cup product for cyclic

cohomology of algebras. This is based on a geometric presentation of cyclic cocycles.

Using this cup product one can define the Connes periodicity map S.

Theorem 3.5.1. We have a one-to-one correspondence between cyclic n-cocycles on a

algebra A and closed graded traces on subalgebra ΩnA of noncommutative differential

forms on A.

To explain the proof, let (Ω, d) be a differential graded algebra where d : Ω∗ −→

Ω∗+1 is a square zero graded derivation. An n-cycle over an algebra A is a triple

(Ω,∫, ρ) where

∫is an n-dimensional closed graded trace on (Ω, d),i.e., a linear map

∫: Ωn −→ k

which ∫dω = 0 and,

∫(ω1ω2 − (−1)deg(ω1)deg(ω2)) = 0

for all ω ∈ Ωn−1, ω1 ∈ Ωi, ω2 ∈ Ωj and i + j = n. The map ρ : A −→ Ω0 is an

algebra homomorphism. Given a n-cycle (Ω,∫, ρ) over A, the following character is

a cyclic n-cocycle on A:

φ(a0, ..., an) =

∫ρ(a0)dρ(a1)...dρ(an).

Conversely, if φ is a n-cocycle on A then the following linear map∫φ : ΩnA −→ k, is

a closed graded trace on ΩA:

∫φ

(a0 + λ1)da1...dan = φ(a0, ..., an).

54

M. Khalkhali and B. Rangipour have found the similar statement as the previous

theorem For Hopf algebras.

Now we can define a cup product for cyclic cohomology of algebras.

Example 3.5.1. (Cup product in cyclic cohomology of algebras) Let (Ω,∫, ρ) and

(Ω′,∫ ′, ρ′) be p and q-dimensional cycles on algebra A and B respectively. Let ω ∈ Ω

and ω′ ∈ Ω′ where deg(ω) = p and deg(ω′) = q. One can define a closed graded trace

of dimension p + q on the graded tensor product of the differential graded algebra

Ω⊗ Ω′ given by′′∫ω ⊗ ω′ =

∫ω

∫ω′,

where the the differentials in Ω⊗ Ω′ are of the forms

d(ω ⊗ ω′) = (dω)⊗ ω′ + (−1)deg(ω)ω ⊗ (dω′).

One can obtain a closed graded trace of dimension p+ q on (Ω(A⊗B), d). Therefore,

we get a cup product in cyclic cohomology of algebras:

⊔c : HCp(A)⊗HCp(B) −→ HCp+q(A⊗B).

A nice application of this cup product is explained in the following example

Example 3.5.2. In the previous cup product, let B = k and ϕ be a 2-cocycle on k

defined by

ϕ(1, 1, 1) = 1

One can define the Connes periodicity map

S : HCn(A) −→ HCn+2(A),

by

φ 7−→ φ ⊔c ϕ.

Chapter 4

Eilenberg-Zilber isomorphisms

In this chapter we state the Eilenberg-Zilber isomorphisms for Hochschild, cyclic and

periodic cyclic (co)homology for (co)cyclic modules. The Eilenberg-Zilber isomor-

phisms are important tools to understand the relation between the homology of the

cross product space X×Y and those of spaces X and Y . The theorem was published

in the American Journal of Mathematics in 1953 for the first time.

4.1 Diagonal and tensor product complex

Recall that given a (co)cyclic module (C, δ, s, τ), we denote its Hochschild differential

by b =∑n

i=0(−1)iδi, and Connes differential by B. Any (co)cyclic module defines a

mixed complex (C, b, B) in a natural way. Suppose (C, δ, s, τ) and (C ′, δ′, s′, τ ′) are

two cocyclic modules. The diagonal C × C ′ is the cocyclic module ((C × C ′)n, δni ⊗

δ′ni , sni ⊗ s

′ni , τn ⊗ τ

′n) where

(C × C ′)n = Cn ⊗ C ′n.

The tensor product complex C ⊗ C ′ where

(C ⊗ C ′)n =⊕

p+q=n

Cp ⊗ C ′q,

is not a cocyclic module but it has a mixed complex structure given by

bn =⊕

i+j=n

(bi ⊗ idj + (−1)jidi ⊗ b′j),

55

56

and similarly for B.

Example 4.1.1. Let H and K be two Hopf algebras. To compute Hochschild, cyclic

or periodic cyclic cohomology of H ⊗K, One has

Cn = (H ⊗K)⊗n,

and the following isomorphism of cocyclic modules:

(H ⊗K)⊗n ∼= H⊗n ⊗K⊗n.

Therefore the problem leads to compute the cohomology of the diagonal complex. Now

the Eilenberg-Zilber theorem gives us a nice relation between the diagonal complex and

the tensor product complex which seems to be more difficult to study, but surprisingly,

is easier and has more information regarding cohomology of H and K.

4.2 Shuffle and Alexander-Whitney maps

In this section, we introduce some morphisms which help us to find out the relation

between C × C ′ and C ⊗ C ′. We have two natural maps between these complexes.

The Alexander-Whitney map

AWn : (C ⊗ C ′)n −→ (C × C ′)n,

is given by

AWn =⊕

i+j=n

AWi,j , (4.2.1)

where

AWi,j : Ci ⊗ C ′j −→ Cn ⊗ C ′n

57

are defined by

AWi,j = δnnδn−1n−1...δ

i+1i+1 ⊗ δ

n0 δ

n−10 ...δi+1

0 . (4.2.2)

The Shuffle map

shn : (C × C ′)n −→ (C ⊗ C ′)n,

is given by

shn =⊕

i+j=n

shi,j , (4.2.3)

where

shi,j : (C × C ′)n −→ Ci ⊗ C ′j

are defined by

shi,j =∑σ

sign(σ)siσ(i+1)...sn−1σ(n)⊗ sj

σ(1)...sn−1

σ(i), (4.2.4)

and σ runs over all (i, j)-shuffles. By an (i, j)- shuffle we mean a permutation σ on

elements 1, 2, ..., n such that

σ(1) < σ(2) < ... < σ(i), σ(i+ 1) < σ(i+ 2) < ... < σ(n).

For example, all (2, 1)-shuffles are

(1, 2)(3), (1, 3)(2) (2, 3)(1).

The Shuffle map was found first by Eilenberg-Maclane. In the dual case of cyclic

modules, one can define

AWn : (C × C ′)n −→ (C ⊗ C ′)n

58

which is given by

AWn =⊕

i+j=n

AWi,j , (4.2.5)

where AWi,j : Cn ⊗ C ′n −→ Ci ⊗ C ′j are given by:

AWi,j = δi+1i+1δ

i+2i+2...δ

nn ⊗ δi+1

0 δi+20 ...δn0 . (4.2.6)

Also

shn : (C ⊗ C ′)n −→ (C × C ′)n

is given by

shn =⊕

i+j=n

shi,j , (4.2.7)

where shi,j : Ci ⊗ C ′j −→ Cn ⊗ C ′n are given by:

shi,j =∑σ

sign(σ)sn−1σ(n)

...siσ(i+1) ⊗ sn−1σ(i)

...sjσ(1)

. (4.2.8)

Theorem 4.2.1. Alexander-Whitney and shuffle maps are maps of Hochschild com-

plexes, i.e, in the normalized setting we have

[b, sh] = 0, [b, AW ] = 0

Example 4.2.1. Let H and K be two Hopf algebras with modular pair in involution

(δ, σ) and (δ′, σ′) respectively. For the Connes-Moscovici cocyclic module for Hopf

algebras, one has

AW :⊕

i+j=n

H⊗i ⊗K⊗j −→ H⊗n ⊗K⊗n,

59

where

AWi,j((h1⊗...⊗hi)⊗(k1⊗...⊗kj)) =n∑

i=0

(h1⊗...⊗hi⊗σ⊗...⊗σ)⊗(k1⊗...⊗kj⊗σ′⊗...⊗σ′).

Example 4.2.2. Let H and K be two Hopf algebras, (δ, σ) a modular pair in invo-

lution for H, and ε′ the counit of K. For the dual cyclic module for Hopf algebras,

one has

AWn : H⊗n ⊗K⊗n −→⊕

i+j=n

H⊗i ⊗K⊗j ,

where

AWi,j((h1⊗...⊗hn)⊗(k1⊗...⊗kn)) =n∑

i=0

ε′(k1...kn)δ(hi+1...hn)(h1⊗...⊗hn)⊗((ki+1⊗...⊗kn)).

Example 4.2.3. Let H and K be two Hopf algebras with counits ε and ε′ respectively.

For the Connes-Moscovici cocyclic module of Hopf algebras, one has

shn : H⊗n ⊗K⊗n −→⊕

i+j=n

H⊗i ⊗K⊗j ,

where

shi,j((h1 ⊗ ...⊗ hn)⊗ (k1 ⊗ ...⊗ kn)) =∑

ε′(kσ(1)...kσ(i))ε(hσ(i+1)...hσ(n))

(hσ(1) ⊗ ...⊗ hσ(i))⊗ (kσ(i+1) ⊗ ...⊗ hσ(n)).

where σ runs over all (i, j)-shuffles.

60

4.3 The Eilenberg-Zilber isomorphisms for

Hochschild (co)homology of (co)cyclic

modules

In this section, using Alexander-Whitney and shuffle maps, we state the Eilenberg-

Zilber isomorphisms for Hochschild (co)homology of (co)cyclic modules.

Theorem 4.3.1. For any cocyclic module, Alexander-Whitney and shuffle maps in-

duce an inverse isomorphism on Hochschild cohomology, called the Eilenberg-Zilber

isomorphism,i.e.,

sh AW = id,

Aw sh = bφ+ φb+ id

for some chain homotopy map φ : (C × C ′)n −→ (C × C ′)n−1 where

φ =∑

(−1)n−p−q+σ(α,β) (sβq+n−p−q...sβ1+n−p−qsn−p−q−1δn−q+1...δn)⊗

(sαp+1+n−p−q...sα1+n−p−qδn−p−q...δn−q−1).

The sum is taken over all 0 ≤ q ≤ n− 1, 0 ≤ p ≤ n− q − 1, and (α, β) runs over all

(p+ 1, q)-shuffles where σ(α, β) =∑

(αi − i+ 1). We have

HHn(C ⊗ C ′) ∼= HHn(C × C ′) ∀n ≥ 0.

Lemma 4.3.1. For any cocyclic module, one can see

shφ = 0, φAW = 0, φφ = 0.

Theorem 4.3.2. For any cyclic module, Alexander-Whitney and shuffle maps induce

an inverse isomorphism on Hochschild homology, called the Eilenberg-Zilber isomor-

61

phism, i.e.,

Aw sh = id,

sh Aw = bφ+ φb+ id

for some chain homotopy map φ : (C × C ′)n −→ (C × C ′)n+1. We have

HHn(C ⊗ C ′) ∼= HHn(C × C ′) ∀n ≥ 0.

Lemma 4.3.2. For any cyclic module, one has the following relations:

Awφ = 0, φsh = 0, φφ = 0.

4.4 The Eilenberg-Zilber isomorphisms for cyclic

(co)homology of (co)cyclic modules

Since Alexander-Whitney and shuffle maps do not commute with the Connes operator

B, we need to introduce another map for the case of cyclic (co)homology. Now we

state the Eilenberg-Zilber formula for cyclic cohomology. To this end, we define

another map namely the cyclic shuffle map as follows. First we define an (i, j)-cyclic

shuffle.

Definition 4.4.1. Let i, j, n ∈ N with n = i + j. Consider the permutation σ on

the n elements 1, ..., n obtained by first performing a cyclic permutation p times on

1, ..., i and a cyclic permutation q times on i+1, ..., i+ j and there after applying

an (i, j)-shuffle to the combined result. We call σ an (i, j)-cyclic shuffle if 1 appears

before i+ 1 in the resulting sequence.

Example 4.4.1. To find all (2, 1)-cyclic shuffles, i = 2 and j = 1. So p = 0, 1 and

q = 0. We have two cases: p = 0, q = 0 and p = 1, q = 0. In the first case we get

(1, 2)(3) and in the second one we get (2, 1)(3). Now we apply (2, 1)-shuffle for

62

these two cases. In the first case we get (1, 2)(3), (1, 3)(2) (2, 3)(1) and in the

second one (2, 1)(3), (2, 3)(1), (1, 3)(2). Now (2, 1)-cyclic shuffles are the ones

which 1 appears before 3. Thus all the (2, 1)-cyclic shuffles are:

(1, 2)(3), (1, 3)(2) if p=q=0, (2, 1)(3), (1, 3)(2) if p=1,q=0.

Now we define another map namely sh′ which is in fact a cyclic version of the

shuffle map:

sh′n : (C × C ′)n −→ (C ⊗ C ′)n−2,

where

sh′n =⊕

i+j=n

sh′i,j , (4.4.9)

and sh′i,j : Cn ⊗ C ′n −→ Ci−1 ⊗ C ′j−1 are given by

sh′i,j =∑σ

sign(σ)si−1i−p−1τp+1i siσ(i+1)...s

n−1σ(n)⊗ sj−1j−q−1τ

q+1j s

jσ(1)

...sn−1σ(i)

. (4.4.10)

Here σ runs over all (i, j)- cyclic shuffles related to permutations p and q. The map

sh′ commutes with Connes differential B, i.e., [31]

[B, sh′] = 0.

Lemma 4.4.1. One can show [1]

sh′ = shBφ.

A S-map between mixed complexes is a map of complexes f : TotB(C) −→

TotB(C ′) which commutes with Connes periodicity map S. In fact a S-map has a

matrix representation:

63

f0 f1 f2 . . .

f−1 f0 f1 . . .

f−2 f−1 f0 . . .

......

. . .

(4.4.11)

where fi : C∗−2i −→ C ′∗ and

[B, fi] + [b, fi+1] = 0. (4.4.12)

This map induces a map on the level of cyclic cohomology.

Lemma 4.4.2. The following map is a S-map.

Shn : TotBn(C × C ′) −→ TotBn(C ⊗ C ′)

given by

Shn =

sh sh′ 0 . . .

0 sh sh′ . . .

0 0 sh . . .

......

. . .

(4.4.13)

The Condition (4.4.12) reduces to the following relations in the normalized setting:

(1) [b, sh] = 0,

(2) [B, sh] + [b, sh′] = 0,

(3) [B, sh′] = 0.

Lemma 4.4.3. The following map is a S-map

AWn : TotBn(C ⊗ C ′) −→ TotBn(C × C ′)

given by

64

AWn =

Aw Aw′ 0 . . .

0 Aw Aw′ . . .

0 0 Aw . . .

......

. . .

(4.4.14)

where [1]

Aw′ = φBAw.

Theorem 4.4.1. We have [1]

Sh AW = id,

AW Sh = (b+B) h+ h (b+B) + id,

for some homotopy h : C ⊗ C ′ −→ C × C ′. Thus we have the Eilenberg-Zilber

isomorphism in cyclic cohomology:

HC∗(C × C ′) ∼= HC∗(C ⊗ C ′).

In the case of cyclic modules, one can define

sh′n : (C ⊗ C ′)n −→ (C × C ′)n+2 ∀n ≥ 0. (4.4.15)

Two S-maps

Shn : TotBn(C ⊗ C ′) −→ TotBn(C × C ′)

and

AWn : TotBn(C × C ′) −→ TotBn(C ⊗ C ′)

65

give the Eilenberg-Zilber isomorphism for cyclic homology.

HC∗(C × C ′) ∼= HC∗(C ⊗ C ′).

Proposition 4.4.1. For cyclic modules one can show [1]

sh′ = ϕBsh,

Aw′ = φBAw.

4.5 The Eilenberg-Zilber isomorphisms for

periodic cyclic (co)homology of (co)cyclic

modules

In this section, we state the Eilenberg-Zilber isomorphism for periodic cyclic cohomol-

ogy as follows. One knows lim−→Tot2n+∗B(C) = ⊕n=0C

2n+∗ where ∗ = 0, 1 and direct

limit is with respect to the Connes periodicity map S. Since direct limit commutes

with homology we have

lim−→HC2n+∗(C) = lim−→H(Tot2n+∗B(C)) = H(lim−→Tot

2n+∗B(C)) =

H(⊕n=0C2n+∗) = HP ∗(C).

where ∗ = 0, 1 . Therefore we have the Eilenberg-Zilber isomorphism for periodic

cyclic cohomology:

HP ∗(C ⊗ C ′) ∼= HP ∗(C × C ′),

because

HP ∗(C ⊗ C ′) = lim−→HC2n+∗(C ⊗ C ′) ∼= lim−→HC

2n+∗(C × C ′) = HP ∗(C × C ′).

66

The explicit isomorphism maps are infinite matrix versions of Sh and AW .

For periodic cyclic homology we have some obstacles. The problem comes from

the fact that homology does not commute with inverse limit in general, i.e.,

lim←−HC2n+∗(C) = HP∗(C).

Definition 4.5.1. The inverse system X = (Xi, fij) where fij : Xj −→ Xi sat-

isfies the Mittag-Leffler condition if for every k there exist a j ≥ k such that for all

i ≥ j we have:

Im(fkj : Xj −→ Xk) = Im(fki : Xi −→ Xk).

The inverse limit functor for any cyclic or cocyclic module, in general in all

abelian categories, is left exact. In fact Mittag-Leffler condition on morphisms fij

insures the exactness of lim−→. The Mittag-Leffler condition on inverse system

(HC(C)[−2m], S)m,

guarantees commutativity of homology and inverse limit [12]. If we have this condi-

tion, with a similar argument we have

HPn(C ⊗ C ′) ∼= HPn(C × C ′).

Example 4.5.1. If the morphisms fij of an inverse system X = (Xi, fij) are

surjective then the system satisfies Mittag-Leffeler condition.

Example 4.5.2. Let cyclic module X be a flat module over a field k and HC∗(X)

be the direct sum of an extended comodule k[u] ⊗ U and of a trivial comodule V

where U is a finitely generated projective k-module and V is a flat k-module. One

can show X with these conditions satisfies Mittag-Leffeler condition and furthermore

HP∗(X) = U .

67

Example 4.5.3. If A is a commutative algebra of finite type on a field with charac-

teristic zero then (HC(A)[−2m], S)m satisfies the Mittag-Leffler condition.

Chapter 5

Kunneth formulas

In this section we recall the Kunneth formulas for Hochschild and cyclic (co)homology

of (co)cyclic modules. Also we introduce the theorem for the periodic cyclic (co)homology

of (co)cyclic modules. This last case was known only for algebras [11]. Recall that

the Kunneth formulas is about the homology of tensor product complex C ⊗ C ′ in

terms of the homology of (co)cyclic modules C and C ′.

5.1 Kunneth formulas for Hochschild


Let C and C ′ be cocyclic modules. For all n ≥ 0, we have

HHn(C ⊗ C ′) ∼=⊕

p+q=n

HHp(C)⊗HHq(C ′). (5.1.1)

This isomorphism is induced by a natural map [9]

[α]⊗ [β] −→ [α⊗ β]. (5.1.2)

We have the same isomorphism with the same natural map for Hochschild homology

of cyclic modules. This follows from the fact that for any chain complexes C and C ′,

by Universal Coefficient Theorems, we have the following short exact sequence:

0 −→⊕

p+q=nHp(C)⊗Hq(C′)

ζ−−−→ Hn(C ⊗ C ′) −−−→

68

69⊕p+q=n−1

Tork1(Hp(C), Hq(C′)) −→ 0. (5.1.3)

Example 5.1.1. Let H and K be two Hopf algebras. Let C and C ′ be the Connes-

Moscovici cocyclic modules for H and K respectively. We have

(C ⊗ C ′)n =⊕

i+j=n

H⊗i ⊗K⊗j .

Also

(C × C ′)n = H⊗n ⊗K⊗n ∼= (H ⊗K)n = CnHH(H ⊗K).

The isomorphism H⊗n ⊗K⊗n ∼= (H ⊗K)n is an isomorphism of cocyclic modules.

Now by the Eilenberg-Zilber isomorphism and Kunneth formula we have:

HHn(H ⊗K) ∼=⊕

i+j=n

HH(H)i ⊗HH(K)j .

5.2 Kunneth formulas for cyclic (co)homology of

(co)cyclic modules

In the case of cyclic homology we do not have the isomorphism (5.1.1). Instead, there

is the following short exact sequence [24, 31]

0 −→ TotnB(C ⊗ C ′)I−−−→

⊕i+j=n

TotiB(C)⊗ TotjB(C ′)S⊗id−id⊗S−−−−−−−−→

⊕i+j=n−2

TotiB(C)⊗ TotjB(C ′) −→ 0. (5.2.4)

The map I is called the Kunneth map which can be defined as follows.

Let

x = (xp, xp−2, xp−4, ...) ∈ TotpB(C),

y = (yq, yq−2, yq−4, ...) ∈ TotqB(C ′),

70

z = (zr, zr−2, zr−4, ...) ∈ TotrB(C ⊗ C ′).

Since TotB(C), TotB(C ′) and TotB(C ⊗ C ′) have k[u], k[v] and k[U ]-comodule

structures respectively, where u ,v and U are polynomials of degree two, we have

x =∑

k=0 xp−2kuk, y =

∑k=0 yq−2kv

k and z =∑

k=0 zr−2kUk. One can define :

I(xm ⊗ ynUq) =

q∑s=0

(xmus)⊗ (ynv

s−1). (5.2.5)

From the Kunneth short exact sequence (5.2.4), we get the following long exact

sequence in homology:

... −→ HCn(C ⊗ C ′)I−−−→

⊕i+j=n

HCi(C)⊗HCj(C′) S⊗id−id⊗S−−−−−−−−→

⊕i+j=n−2

HCi(C)⊗HCj(C′) ∂−−−→ HCn−1(C ⊗ C ′) −→ ...

For cyclic cohomology, we have the following short exact sequence:

0 −→⊕

i+j=n−2TotiB(C)⊗ TotjB(C ′) S⊗id−id⊗S−−−−−−−−→

⊕i+j=n

TotiB(C)⊗ TotjB(C ′)µ−−−→ TotnB(C ⊗ C ′) −→ 0.

Therefore we get the following long exact sequence in cohomology [15]:

... −→ HCn−1(C ⊗ C ′) ∂−−−→⊕

p+q=n−2HCp(C)⊗HCq(C ′) S⊗id−id⊗S−−−−−−−−→

⊕r+s=n

HCr(C)⊗HCs(C ′)µ−−−→ HCn(C ⊗ C ′) ∂−−−→

71⊕i+j=n−1

HCi(C)⊗HCj(C ′) S⊗id−id⊗S−−−−−−−−→ ...

where ∂ is the boundary map. In this case, the Kunneth map is :

I : HCn(C ⊗ C ′) −→⊕

i+j=n

HCi(C)⊗HCj(C ′)

where

I = ∂µ(S ⊗ id− id⊗ S)∂. (5.2.6)

We notice that we had the Kunneth map I for cyclic homology at the level of chains

but for cyclic cohomology the map exists at the level of cohomology.

5.3 Kunneth formulas for periodic cyclic


In this section, we state our results about the Kunneth formulas for periodic cyclic

(co)homology. One can find the formula for algebras in [11]. Here we mention a

similar argument in general case of (co)cyclic modules.

Definition 5.3.1. A supercomplex is a Z2-graded vector space V = V0⊕ V1 endowed

with linear maps ∂0 : V0 −→ V1 and ∂1 : V1 −→ V0 such that ∂0∂1 = 0 and ∂1∂0 = 0.

We denote the corresponding chain complex V0∂0∂1

V1, where its homology is a Z2-

graded vector space given by:

H0 = Ker∂0/Im∂1, H1 = Ker∂1/Im∂0.

Example 5.3.1. The inverse limit lim←−TotB(C) is a supercomplex. In fact

V = TotBn(C) = Cn ⊕ Cn−2 ⊕ ..., T = B + b

72

We have

lim←−TotB(C) = (∏

C2n)n≥0 ⊕ (∏

C2n+1)n≥0.

For cohomology case

lim−→TotB(C) = (⊕C2n)n≥0 ⊕ (⊕C2n+1)n≥0.

Definition 5.3.2. A map of supercomplexes is a linear map

f : V = V0 ⊕ V1 −→W = W0 ⊕W1,

which sends V0 to W0 and V1 to W1.

Proposition 5.3.1. If we have two supercomplexes V and W , then V ⊗W is a su-

percomplex where

(V ⊗W )0 = (V0 ⊗W0)⊕ (V1 ⊗W1),

(V ⊗W )1 = (V0 ⊗W1)⊕ (V1 ⊗W0).

We define

∂V ⊗W0 :=

1⊗ ∂W0 ∂V1 ⊗ 1

∂V0 ⊗ 1 −1⊗ ∂W1

and ∂V ⊗W1 :=

1⊗ ∂W1 ∂V1 ⊗ 1

∂V0 ⊗ 1 −1⊗ ∂W0

(5.3.7)

The Kunneth formula for supercomplexes hold [18]:

H(X⊗Y ) ≃ H(X)⊗H(Y ).

We define the supercomplex map:

I : (lim←−TotB(C))⊗(lim←−TotB(C′)) −→ lim←−TotB(C ⊗ C

′). (5.3.8)

73

The restriction of I on (∏

n≥0C2n) ⊗ (

∏n≥0C

′2n) sends ξnn ⊗ ξ′nn onto

∑

i+j=n ξi ⊗ ξjn. Similarly one can define restriction of I on (∏

n≥0C2n+1) ⊗

(∏

n≥0C′2n+1), (

∏n≥0C

2n) ⊗ (∏

n≥0C′2n+1) and (

∏n≥0C

2n+1) ⊗ (∏

n≥0C′2n).

To prove I is an isomorphism, we have two major problems. The first one is the fact

that generally homology does not commute with inverse limit,i.e.,

H∗(lim←−Zm) = lim←−H∗(Zm),

where Zm is an inverse system. To solve this problem we need the Mittag-Leffler

condition which guarantees the commutativity. The second problem is in general

(∞∏i=1

Vi)⊗W =∞∏i=1

(Vi ⊗W ),

where Vi and W are some vector spaces . If W is finite dimensional then we have the

equality. In our case we can think about Vi and W as homology of a cyclic module.

Now we are ready to have the following theorem :

Theorem 5.3.1. Suppose C is a cyclic module which has the following two properties:

(i) The inverse system (HC(C)[−2m], S)m satisfies the Mittag-Leffler condi-

tion.

(ii) The periodic cyclic homology HP∗(C) is finite dimensional.

the map

I∗ : HP∗(C)⊗HP∗(C ′) −→ HP∗(C ⊗ C ′) ∼= HP∗(C × C ′), (5.3.9)

where I is as (5.3.8), is an isomorphism for any cyclic module C ′.

Proof. The proof is similar to the one in [11] for algebras. For any cyclic

module C we define

Xm = TotmB∗∗(C)[−2m], X ′m = TotmB∗∗(C ′)[−2m],

74

Ym = TotmB∗∗(C ⊗ C ′)[−2m], Zm = (TotB∗∗(C)⊗ TotmB∗∗(C))m[−2m].

By tensoring the following short exact sequence

0 −→ lim→S

X ′m −−−→∏

mX ′mid−S−−−→

∏mX ′m −→ 0.

with lim→S

Xm we get

0 −→ (lim→S

Xm)⊗(lim→S

X ′m) −−−→ (lim→S

Xm)⊗(∏

mX ′m)1−S−−−→ (lim

→S

Xm)⊗(∏

mX ′m) −→ 0.

One knows that Zmm is a surjective inverse system of complexes with respect to

S ⊗ id. By passing the inverse limits lim←− to the short exact sequence (5.2.4), we get

0 −→ lim→S

YmI−−−→

∏m Zm

S⊗id−id⊗S−−−−−−−−→∏

m Zm −→ 0..

One can see that the following diagram of supercomplexes commutes:

0 −→ (lim→S

Xm)⊗(lim→S

X ′m) −→ (lim→S

Xm)⊗(lim→S

∏mX ′m) −→ (lim

→S

Xm)⊗(lim→S

∏mX ′m)

↓I ↓I ′ ↓I ′

0 −→ lim→S

Ym −→ lim→S

Zm −→ lim→S

Zm

Here I ′ is the restriction of the map

(lim→S

Xm)⊗(lim→S

X ′m) −→∏m

Zm,

which sends xmm ⊗ x′mm to ∑

i+j=n xi ⊗ x′jm. We have

H∗(Zm) =⊕

i+j=m

HCi(C)⊗HCj(C′).

75

Since Zmm is a surjective inverse system and by (i), the inverse system (HC(C)[−2m], S)m

satisfies the Mittag-Leffler condition, the inverse system (H∗(Zm), S ⊗ id)m satisfies

the Mittag-Leffler condition. This implies that:

H∗(lim←−Zm) = lim←−H∗(Zm) = lim←−(HC(C)⊗HC(C′))m.

The map I ′ induces an isomorphism because:

H1((lim→S

Xm)⊗(lim→S

∏m

X ′m))

= (HP (C)⊗∏m

HC(C ′)[−2m])1 =

= HP1(C)⊗∏n=0

HC2n(C′) +HP0(C)⊗

∏n=0

HC2n+1(C′)

∼=∏n=0

(HP1(C)⊗HC2n(C′))⊕

∏n=0

(HP0(C)⊗HC2n+1(C′))

∼=∏n=0

((lim←−HC2n+1(C))⊗HC2n(C′))⊕

∏n=0

((lim←−HC2n(C))⊗HC2n+1(C′))

∼=∏n=0

(lim←−HC2n+1(C)⊗HC2n(C′))⊕

∏n=0

(lim←−HC2n(C)⊗HC2n+1(C′))

= lim←−(HC(C)⊗HC(C′)2n+1)

= H∗(lim←−Zm)

Now using the five-lemma, we conclude that I is an isomorphism in homology as well.

Here we notice that the condition (i) implies

HP (C) = lim←−HC(C)[−2m],

and condition (ii) leads to

(HP (C)⊗∏

HC(C ′)[−2m])∗ ∼=∏

(HP∗−[m](C)⊗HCm(C ′)).

76

Thus having conditions (i) and (ii) we have the Kunneth formula for periodic

cyclic homology:

HP0(C)⊗HP0(C ′)⊕HP1(C)⊗HP1(C ′) ∼= HP0(C ⊗ C ′) (5.3.10)

and

HP0(C)⊗HP1(C ′)⊕HP1(C)⊗HP0(C ′) ∼= HP1(C ⊗ C ′). (5.3.11)

For periodic cyclic cohomology, since direct limit commutes with cohomology,

we do not need (i) and also since

lim−→TotB(C) = (⊕C2n)n≥0 ⊕ (⊕C2n+1)n≥0)

and

⊕i≥0(Vi ⊗W ) ≃ (⊕i≥0Vi)⊗W

for all vector spaces Vi and W , we do not need (ii). Therefore we have the following

theorem:

Theorem 5.3.2. Suppose C and C ′ are two cocyclic modules. Then, the map

I∗ : HP ∗(C)⊗HP ∗(C ′) −→ HP ∗(C ⊗ C ′) ∼= HP ∗(C × C ′), (5.3.12)

is an isomorphism and we have the relations (5.3.10) and (5.3.11) in cohomology.

Chapter 6

Coproducts for Hopf cyclic cohomology

with coefficients

In this chapter, we define the coproducts for Hochschild, cyclic and periodic cyclic

cohomology of cocommutative Hopf algebras. Throughout the chapter we assume

that H is a Hopf algebra with a bijective antipode. The coproduct, counit and

antipode of H are denoted by ∆, ε and S, respectively. For the coproduct we use

the Sweedler notation in the form ∆(h) = h(1) ⊗ h(2); for a left coaction of H on M ,

φ :M −→ H ⊗M , we write φ(m) = m(−1)⊗m(0), and for a right coaction we write

φ(m) = m(0) ⊗m(1).

6.1 Eilenberg-Zilber isomorphisms and Kunneth

formulas for Hopf cyclic cohomology with

coefficients

In this section, we state the Eilenberg-Zilber isomorphisms and Kunneth formulas for

Hochshcild, cyclic and periodic cyclic cohomology of a Hopf algebra with coefficients

in a SAYD module.

Lemma 6.1.1. Let H and K be two Hopf algebras and M and N, SAYD modules

over H and K respectively. Then M ⊗N is a SAYD module over H⊗K in a natural

way.

77

78

Proof. We define the right action by:

(m⊗ n)(h⊗ k) = mh⊗ nk, (6.1.1)

and the left coaction by:

φ(m⊗ n) = m(−1) ⊗ n(−1) ⊗m(0) ⊗ n(0). (6.1.2)

We check the compatibility of the action and coaction:

φ((m⊗ n)(h⊗ k)) = φ(mh⊗ nk) = (mh⊗ nk)(−1) ⊗ (mh⊗ nk)(0)

= (mh)(−1) ⊗ (nk)(−1) ⊗ (mh)(0) ⊗ (nk)(0)

= S(h(3))m(−1)h(1) ⊗ S(k(3))n(−1)k(1) ⊗m(0)h(2) ⊗ n(0)k(2)

= SH⊗H(h(3) ⊗ k(3))(m⊗ n)(−1)(h(1) ⊗ (k(1))⊗ ((m⊗ n)(0))(h(2) ⊗ k(2)),

and check the stability:

(m(0) ⊗ n(0))(m(−1) ⊗ n(−1)) = (m(0)m(−1), n(0)n(−1)) = (m,n).

We need the following statement later.

Proposition 6.1.1. If H and K are two Hopf algebras and M and N are SAYD

modules over H and K, respectively, then the map

Ωr : (M ⊗N)⊗H⊗K (H ⊗K)⊗r+1 −→ (M ⊗H H⊗r+1)⊗ (N ⊗K K⊗r+1)

given by

((m⊗ n)⊗ (h0 ⊗ k0)⊗ ...⊗ (hr ⊗ kr)) 7−→ (m⊗ h0 ⊗ ...⊗ hr)⊗ (n⊗ k0 ⊗ ...⊗ kr),

79

defines an isomorphism of cocyclic modules:

C(H ⊗K,M ⊗N) ∼= C(H,M)× C(K,N).

Proof. We prove that Ωr commutes with δi and τr where 0 ≤ i < r. The

commutativity of Ωr with δr and si is left to the reader. For δi:

Ωrδi((m⊗ n)⊗ (h0 ⊗ k0)⊗ ...⊗ (hr−1 ⊗ kr−1))

= Ωr((m(0) ⊗ n(0))⊗ (h0 ⊗ k0)⊗ ...(hi ⊗ ki)(1) ⊗ (hi ⊗ ki)(2) ⊗ ...⊗ (hr−1 ⊗ kr−1)

= Ωr((m(0) ⊗ n(0))⊗ (h0 ⊗ k0)⊗ ...(h

(1)i ⊗ k

(1)i )⊗ (h

(2)i ⊗ k

(2)i )⊗ ...(hr−1 ⊗ kr−1))

= (m(0) ⊗ h0 ⊗ ...⊗ h(1)i ⊗ h

(2)i ⊗ ...hr−1)⊗ ((n(0) ⊗ k0 ⊗ ...⊗ k

(1)i ⊗ k(2)i ⊗ ...kr−1)

= (δi ⊗ δi)(m⊗ h0 ⊗ ...⊗ hr−1)⊗ (n⊗ k0 ⊗ ...⊗ kr−1)

= (δi ⊗ δi)Ωr((m⊗ n)⊗ (h0 ⊗ k0)⊗ ...⊗ (hr−1 ⊗ kr−1)).

For τr:

Ωrτr((m⊗ n)⊗ (h0 ⊗ k0)⊗ ...⊗ (hr ⊗ kr))

= Ωr((m(0) ⊗ n(0))⊗ (h0 ⊗ k0)⊗ ...⊗ (hr ⊗ kr)⊗ ((m(−1) ⊗ n(−1))(h0 ⊗ k0))

= (m(0) ⊗ h1 ⊗ ...⊗ hr ⊗m(−1)h0)⊗ (n(0) ⊗ k1 ⊗ ...⊗ kr ⊗ n(−1)k0)

= (τr ⊗ τr)((m⊗ h0 ⊗ ...⊗ hr)⊗ (n⊗ k0 ⊗ ...⊗ kr)

= (τr ⊗ τr)Ωr(((m⊗ n)⊗ (h0 ⊗ k0)⊗ ...⊗ (hr ⊗ kr)).

The bijectivity of Ωr is obvious.

Using previous Lemma and Proposition, we are ready to introduce the Eilenberg-

Zilber isomorphisms in Hopf cyclic cohomology with coefficients.

Theorem 6.1.1. Let H and K be two Hopf algebras and M and N SAYD modules

80

over H and K, respectively. One has the Eilenberg-Zilber isomorphisms:

HH∗(H ⊗K,M ⊗N) ∼= HH∗(C(H,M)× C(K,N)) ∼= HH∗(C(H,M)⊗ C(K,N)),

(6.1.3)

for Hochschild cohomology;

HC∗(H ⊗K,M ⊗N) ∼= HC∗(C(H,M)× C(K,N)) ∼= HC∗(C(H,M)⊗ C(K,N)),

(6.1.4)

for cyclic cohomology; and

HP ∗(H ⊗K,M ⊗N) ∼= HP ∗(C(H,M)× C(K,N)) ∼= HP ∗(C(H,M)⊗ C(K,N)).

(6.1.5)

for periodic cyclic cohomology with coefficients.

Now we use the Eilenberg-Zilber isomorphisms and (6.1.1) to obtain the follow-

ing Kunneth formulas:

Theorem 6.1.2. Let H and K be two Hopf algebras and M and N be SAYD modules

over H and K, respectively. One has the Kunneth formula for Hochschild cohomology:

HHn(C(H,M)⊗ C(K,N)) ∼=⊕

i+j=n

HHi(H,M)⊗HHj(K,N), (6.1.6)

and for cyclic cohomology, one obtains:

... −→ HCn−1(C(H,M)⊗ C(K,N))∂−−−→

⊕p+q=n−2

HCp(H,M)⊗HCq(K,N)S⊗id−id⊗S−−−−−−−−→

⊕r+s=n

HCr(H,M)⊗HCs(K,N)µ−−−→ HCn(C(H,M)⊗ C(K,N))

∂−−−→

⊕i+j=n−1

HCi(H,M)⊗HCj(K,N)S⊗id−id⊗S−−−−−−−−→ ...

81

For the periodic case we have:

HP 0(C(H,M)⊗C(K,N) ∼= HP 0(H,M)⊗HP 0(K,N)⊕HP 1(H,M)⊗HP 1(K,N),

and similarly for the odd case.

6.2 Coproducts for Hochschild, cyclic and

periodic cyclic cohomology of Hopf algebras

In this section, we define the desired coproducts for Hochschild, cyclic and periodic

cyclic cohomology of cocommutative Hopf algebras. The following proposition plays

an important role in the definition of coproducts.

Proposition 6.2.1. Let H be a cocommutative Hopf algebra and M a H-SAYD mod-

ule. Let Φn = ψ ⊗∆⊗n+1 be a linear map, where ψ : M −→ M ⊗M satisfying the

following condition:

(∆⊗ ψ) φM = φM⊗M ψ. (6.2.7)

The map

ρn = ΩnΦn : Cn(H,M) −→ Cn(H ⊗H,M ⊗M)

is a map of cyclic modules. Recall that φ denotes the coaction.

Proof. Since H is cocommutative, one can easily see that

Ωn (ψ ⊗∆⊗n)δi = (δi ⊗ δi)Ωn (ψ ⊗∆⊗n),

where 0 ≤ i ≤ n − 1. We prove Ωn (ψ ⊗ ∆⊗n) commutes with δn. We use the

summation notation ψ(m) = m(1) ⊗m(2). The condition (6.2.7) is equivalent to:

(m(1))(−1)⊗(m(2))

(−1)⊗(m(1))(0)⊗(m(2))

(0) = m(−1)(1)⊗m(−1)(2)⊗(m(0))(1)⊗(m(0))(2).

82

By cocommutativity of H we have:

(δn ⊗ δn)Ωn (ψ ⊗∆⊗n)(m⊗ h0 ⊗ ...⊗ hn−1)

= (δn ⊗ δn)(ψ(m)⊗∆(h0)⊗ ...⊗∆(hn−1))

= (δn ⊗ δn)(m(1) ⊗ h(1)1 ⊗ ...h

(1)n )⊗ (m(2) ⊗ h

(2)1 ⊗ ...h

(2)n )

= ((m(1))(0) ⊗ h(1)(2)0 ⊗ h(1)2 ⊗ ...⊗ h

(1)n−1 ⊗ (m(1))

(−1)h(1)(1)0 )⊗ ((m(2))(0) ⊗ h(2)(2)0 ⊗

h(2)1 ⊗ ...⊗ h

(2)n−1 ⊗ (m(2))

(−1)h(2)(1)0 )

= (((m(1))(0) ⊗ (m(2))

(0))⊗ (h(1)(2)0 ⊗ h(2)(2)0 )⊗ (h

(1)1 ⊗ h

(2)1 )⊗ ...⊗

(h(1)n−1 ⊗ h

(2)n−1)⊗ (m(1))

(−1)h(1)(1)0 ⊗ (m(2))(−1)h(2)(1)0 ))

= (ψ(m)(0) ⊗ (∆(h1))(2) ⊗∆(h2)⊗ ...⊗∆(hn)⊗ ψ(m)(−1)(∆(h1))

(1))

= (ψ(m(0))⊗∆(h(2)0 )⊗∆(h1)⊗ ...⊗∆(hn−1)⊗∆(m(−1))∆(h

(1)0 ))

= (ψ ⊗∆⊗n)(m(0) ⊗ h(2)0 ⊗ h1 ⊗ ...⊗ hn−1 ⊗m(−1)h(1)0 )

= (ψ ⊗∆⊗n)δn(m⊗ h0 ⊗ ...⊗ hn−1).

Now we show Ωn(ψ ⊗∆⊗n) commutes with τn:

Ωn (ψ ⊗∆⊗n+1)τn(m⊗ h0 ⊗ h1...⊗ hn)

= Ωn (ψ ⊗∆⊗n+1)(m(0) ⊗ h1 ⊗ ...⊗ hn ⊗m(−1)h0)

= Ωn (ψ(m0))⊗∆(h1)⊗ ...⊗∆(hn)⊗∆(m(−1)∆(h0))

= Ωn(((m(0))(1) ⊗ (m(0))(2))⊗ (h

(1)1 ⊗ h

(2)1 )⊗ ...⊗ (h

(1)n ⊗ h

(2)n )⊗

(m(−1)(1)h(1)0 ⊗m(−1)(2)h(2)0 ))

= ((m(0))(1) ⊗ h(1)1 ⊗ ...⊗ h

(1)n ⊗m(−1)(1)h(1)0 )⊗ ((m(0))(2) ⊗ h

(2)1 ⊗ ...⊗ h

(2)n

⊗m(−1)(2)h(2)0 )

= ((m(1))(0) ⊗ h(1)1 ⊗ ...⊗ h

(1)n ⊗m

(−1)(1)

h(1)0 )⊗ ((m(2))

(0) ⊗ h(2)1 ⊗ ...⊗ h(2)n ⊗m

(−1)(2)

h(2)0 )

= τn(m(1) ⊗ h(1)0 ⊗ ...⊗ h

(1)n )⊗ τn(m(2) ⊗ h

(2)0 ⊗ ...⊗ h

(2)n )

= Ωn(τn ⊗ τn)(ψ(m)⊗∆(h0)⊗ ...⊗∆(hn)).

83

Therefore we have:

Ωn (ψ ⊗∆⊗(n+1))τn = (τn ⊗ τn)Ωn (ψ ⊗∆⊗(n+1)).

Now we have all the needed tools to define the desired coproducts. For Hochschild

cohomology, using the Eilenberg-Zilber isomorphism and the Kunneth formula we

have:

HHn(H ⊗K,M ⊗N) ∼=⊕

i+j=n

HHi(H,M)⊗HHj(K,N). (6.2.8)

The following theorem provides a coproduct for Hochschild cohomology of a Hopf

algebra with coefficients in a SAYD module.

Theorem 6.2.1. Suppose H is a cocommutative Hopf algebra, M a SAYD module

over H, and ψ :M −→M ⊗M a linear map satisfying condition (6.2.7). The map

⊔HH = I∗(Shρ)∗ : HHn(H,M) −→⊕

i+j=n

HHi(H,M)⊗HHj(H,M), (6.2.9)

defines a coproduct for Hochschild cohomology with coefficients in M , where I is the

Kunneth map for Hochschild cohomology.

Proof. It is obvious because the shuffle map Sh, the Kunneth map I and ρ

are maps of complexes.

Proposition 6.2.2. In the Connes-Moscovici case where M =1kδ, we will have an

explicit formula for Shρ:

Shnρn(h1 ⊗ ...⊗ hn) =∑σ

sign(σ)(hσ(1) ⊗ ...⊗ hσ(p))⊗ (hσ(p+1) ⊗ ...⊗ hσ(p+q)),

(6.2.10)

84

where σ runs over all (p, q)-shuffles.

Proof. Using ϵ(h(1))h(2) = ϵ(h(2))h(1) = h, we have:

ShnΩnΦn(h1 ⊗ ...⊗ hn)

= Shn((h(1)1 ⊗ ...⊗ h

(1)n )⊗ (h

(2)1 ⊗ ...⊗ h

(2)n ))

=∑σ

sign(σ)ϵ(h(2)σ(1)

)...ϵ(h(2)σ(p)

)ϵ(h(1)σ(p+1)

)...ϵ(h(1)σ(n)

)(h(1)σ(1)⊗ ...h(1)

σ(p))⊗ (h

(2)σ(p+1)

⊗ ...h(2)σ(n)

)

=∑σ

sign(σ)(ϵ(h(2)σ(1)

)h(1)σ(1)⊗ ...⊗ ϵ(h(2)

σ(p))h

(1)σ(p)

)⊗ (ϵ(h(1)σ(p+1)

)h(2)σ(p+1)

⊗ ...⊗ ϵ(h(1)σ(n)

)h(2)σ(n)

)

=∑σ

sign(σ)(hσ(1) ⊗ ...⊗ hσ(p))⊗ (hσ(p+1) ⊗ ...⊗ hσ(n)).

One knows that Hochschild homology of a commutative algebra is a graded

commutative and associative algebra [31]. Analogous to this classic result we have:

Theorem 6.2.2. The Hochschild cohomology of a Hopf algebra H with coefficients in

a SAYD module M is a graded cocommutative and coassociative colagebras by (6.2.9).

Proof. The cocommutativity and coassociativity can be verified by series of

long, but straight forward, computation which we omit here.

We obtain the Kunneth long exact sequence for cyclic cohomology with coeffi-

cients in a SAYD module:

... −→ HCn−1(H ⊗K,M ⊗N)∂−−−→

⊕p+q=n−2

HCp(H,M)⊗HCq(K,N)S⊗id−id⊗S−−−−−−−−→

⊕r+s=n

HCr(H,M)⊗HCs(K,N)µ−−−→ HCn(H ⊗K,M ⊗N)

∂−−−→⊕

i+j=n−1HCi(H,M)⊗HCj(K,N)

∂−−−→ ...

85

The Kunneth map in this case is

I∗ = ∂µ(S ⊗ id− id⊗ S)∂. (6.2.11)


over H, and ψ : M −→ M ⊗M a linear map satisfying condition (6.2.7). Then the

map

⊔HC = I∗(Shϱ)∗ : HCn(H,M) −→⊕

p+q=n

HCp(H,M)⊗HCq(H,M),

defines a coproduct for Hopf cyclic cohomology with coefficients in M , where ϱ =⊕i≥0Φn−2i and I is as (6.2.11).

Proof. This is the consequence of the fact that ϱ and Sh are maps of (b, B)-

complexes.

In the case of periodic cyclic cohomology, we have the following Kunneth iso-

morphism:

HP 0(H ⊗K,M ⊗N) = HP 0(H,M)⊗HP 0(K,N)⊕HP 1(H,M)⊗HP 1(K,N),

(6.2.12)

and similarly for the odd case. Now we have all the ingredients for the following

theorem:


over H, and ψ : M −→ M ⊗M a linear map satisfying condition (6.2.7). Then the

map

⊔HP = I∗(Shϱ)∗ : HP 0(H,M) −→ HP 0(H,M)⊗HP 0(H,M)⊕HP 1(H,M)⊗HP 1(H,M),

86

defines a coproduct for periodic cyclic cohomology with coefficients in M , where ϱ =

⊕i≥0Φ2i+∗ and I is as (5.3.8). There is a similar map in the odd case.

Now we provide some examples of the map ψ satisfying condition (6.2.7).

Example 6.2.1. Let G be a discrete group and H = kG the group algebra of G.

Suppose M is a SAYD module over H. One can check that M is a G-graded vector

space M = ⊕gϵGMg, where the coaction φ is defined by φ(m) = g ⊗m. The stability

condition implies gm = m for all m ∈ Mg, and anti-Yetter-Drinfeld is equivalent to

hm ∈ Mhgh−1 for all g and h ∈ G. Since g is a group-like element, any linear map

ψ ⊗∆⊗n+1 with ψ(Mg) ⊆Mg ⊗Mg satisfies the condition (6.2.7).

Example 6.2.2. Let H = U(g) be the universal enveloping algebra of a Lie algebra

g and M an arbitrary module over H. We can define a comodule structure on M by

trivial coaction: m 7−→ 1 ⊗ m. Now M is a SAYD module over H and any linear

map ψ ⊗∆⊗n+1 satisfies (6.2.7).

Example 6.2.3. Let H be any cocommutative Hopf algebra and M =1kδ. Then any

linear map ψ ⊗∆⊗n+1 satisfies the condition (6.2.7). We use this example to get a

coproduct in Connes-Moscovici setting in the following corollary.

Corollary 6.2.1. If M =1kδ then HH∗(H,M), HC∗(H,M) and HP ∗(H,M) reduce

to Connes-Moscovici Hochschild, cyclic and periodic cyclic cohomology of H and we

get the coproducts for them as described.

One has a cocyclic structure for bialgebras , by Atabey Kaygun, and coalge-

bras. Similarly, we can define coproducts for Hochschild, cyclic and periodic cyclic

cohomology for cocommutative bialgebras and cocommutative coalgebras .

87

6.3 Applications of the coproducts in Hopf

cyclic cohomology

In this section, we show that the coproduct which we have defined for periodic cyclic

cohomology, when H is the universal enveloping algebra U(g), agrees with the co-

product that we have in Lie algebra homology.

One knows [10]

HP ∗(δ,1)(U(g))∼=

⊕k≥0

H2k+∗(g, kδ), (6.3.13)

where ∗ = 0, 1, and kδ is a g-module via the character δ. The right hand side of

(6.3.13) is the homology of the mixed complex:

∧0 gdLie0

∧1 gdLie0

∧2 gdLie0

... , (6.3.14)

where dLie is the Chevalley-Eilenberg differentials for Lie algebra homology. The

quasi-isomorphism (6.3.13) is induced by the anti-symmetrization map An :∧n g −→

U(g)⊗n where

(g1 ∧ ... ∧ gn) 7−→1

n!(∑σ

sign(σ)(gσ(1) ⊗ ...⊗ gσ(n)).

Here σ runs over all permutations of the set 1, 2, ..., n.

If δ = ε, then kδ will be the trivial coefficient and

dLie(g1 ∧ ... ∧ gn) =∑

1≤i≤i≤n(−1)i+j+1[gi, gj ] ∧ (g1 ∧ ...gi ∧ ...gj ∧ ... ∧ gn).

We denote Cn(g) =∧n g. The Lie algebra homology with trivial coefficients is a

coalgebra by the coproduct

88

Lie: Cn(g) −→⊕

p+q=n

(Cp(g))⊗ (Cq(g)),

given by

Lie (g1∧ ...∧gn) =∑σ

sign(σ)(gσ(1)∧ ...∧gσ(p))⊗ (gσ(p+1)∧ ...∧gσ(n)), (6.3.15)

where σ runs over all (p, q)-shuffles. In fact Lie= ∆Λg. Since ε(gi) = 0, for all

gi ∈ g, the image of the anti-symmetrization map is in the normalized complex of

Cn = U(g)⊗n. Since anti symmetrization map Ak is a map complexes then

A =∑

i+j=n

Ai ⊗ Aj :⊕

i+j=n

(Λig)⊗ (Λjg) −→⊕

i+j=n

(U i(g))⊗ (U j(g)), (6.3.16)

is a map of mixed complexes and induces the following map

A∗ :⊕k≥0

H2k+∗(C(g)⊗ C(g)) −→ HP ∗(C(U(g)⊗ C(U(g))

Now using the Kunneth formula we have the map:

I∗(A∪Lie)∗ :⊕n≥0

H2n+1(g, kε) −→ HP 1(U(g))⊗HP 0(U(g))+HP 0(U(g))⊗HP 1(U(g)),

and similarly for the even case where I is the Kunneth map (6.2.11).

Theorem 6.3.1. The coproduct for the periodic cyclic cohomology of the universal

enveloping algebra U(g) with the trivial coefficients, agrees with the coproduct of the

Lie algebra homology. Equivalently the following diagram commutes:

89

Cn(g)A−−−→ Cn(U(g))y∪Lie y∪′

(C(g)⊗ C(g))nA−−−→ (C(U(g))⊗ C(U(g)))n

(6.3.17)

where ∪′ = ShΩ.

Proof. The commutativity of the diagram is equivalent to

shnΩn∆⊗nU(g)

An ⊕ sh′n+2Ωn+2∆⊗n+2U(g)

An+2 = A∆Λ(g). (6.3.18)

One can easily see

shnΩn∆⊗nU(g)

An = A∆Λ(g),

where

A∆Λ(g)(g1 ∧ ... ∧ gn) =n∑

p=0

∑σ∈Sn

sign(σ)(gσ(1) ⊗ ...⊗ gσ(p))⊗ (gσ(p+1) ⊗ ...⊗ gσ(n)).

Now it is enough to show:

sh′n+2Ωn+2∆⊗n+2U(g)

An+2 = AdLie,

which means it is zero in homology. To do this, first we compute Ωn+2∆⊗(n+2)U(g)

An+2 :

90

Ωn+2(∆⊗(n+2)U(g)

)An+2(g1 ∧ ... ∧ gn+2)

= Ωn+2

∑σ∈Sn+2

∆⊗(n+2)U(g)

(gσ(1) ⊗ ...⊗ gσ(n+2))

= Ωn+2

∑σ∈Sn+2

(∆(gσ(1))∆(gσ(2))...∆(gσ(n+2)))

= Ωn+2

∑σ∈Sn+2

((1⊗ gσ(1) + gσ(1) ⊗ 1)...(1⊗ gσ(n+2) + gσ(n+2) ⊗ 1))

=n+2∑p=0

∑σ∈Sn+2

(gσ(1) ⊗ ...⊗ gσ(p))⊗ (gσ(p+1) ⊗ ...⊗ gσ(n+2))

where gσ(1) ⊗ ...⊗ gσ(p) is an element in U(g)⊗(n+2) such that everywhere we

miss σ(i) ∈ 1, ..., n, instead we have 1U(g). For example for n = 1, we have:

Ω3∆⊗3U(g)

A3(g1 ∧ g2 ∧ g3) = (1⊗ 1⊗ 1)⊗ (gσ(1) ⊗ gσ(2) ⊗ gσ(3))

+ (1⊗ 1⊗ gσ(3))⊗ (gσ(1) ⊗ gσ(2) ⊗ 1) + (1⊗ gσ(2) ⊗ 1)⊗ (gσ(1) ⊗ 1⊗ gσ(3))

+ (1⊗ gσ(2) ⊗ gσ(3))⊗ (gσ(1) ⊗ 1⊗ 1) + (gσ(1) ⊗ gσ(2) ⊗ 1)⊗ (1⊗ 1⊗ gσ(3))

+ (gσ(1) ⊗ gσ(2) ⊗ gσ(3))⊗ (1⊗ 1⊗ 1) + (gσ(1) ⊗ 1⊗ 1)⊗ (1⊗ gσ(2) ⊗ gσ(3))

+ (gσ(1) ⊗ 1⊗ gσ(3))⊗ (1⊗ gσ(2) ⊗ 1).

Since sh′n = ⊕i+j=n+2sh′i,j , it is enough to show sh′i,j = 0, for all 1 ≤ i, j ≤ n + 1.

For sh′i,j , we apply degeneracies and therefore counit ϵ, n+ 2 times on the elements:

gσ(1) ⊗ ...⊗ gσ(p)⊗ gσ(p+1)⊗...⊗ gσ(n).

If these terms, after applying (σσ′(1) ⊗ ...σσ′(q)) ⊗ (σσ′(q+1)...σσ′(n)), are zero, then

there is nothing more to prove. If they are not zero, they should be in the form of

(gσ(1) ⊗ ...⊗ gσ(i))⊗ (gσ(i+1) ⊗ ...⊗ gσ(n+2))

91

for some i and j. Here σ runs over all permutations in Sn+2. Now we shall compute

σi−1i−p−1τp+1(gσ(1) ⊗ ...⊗ gσ(i))⊗ σ

j−1j−q−1τ

q+1(gσ(i+1) ⊗ ...⊗ gσ(n+2))

For this let σ(i+ 1), ..., σ(n) be fixed. For any k we have:

σi−1i−kτk(gσ(1) ⊗ ...⊗ gσ(i))

= σi−1i−k(∑σ

S(g(i)σ(k)

)gσ(k+1) ⊗ S(g(i−1)σ(k)

)gσ(k+2) ⊗ ...⊗ S(g(k)σ(k)

)σ

⊗S(g(k−1)σ(k)

)gσ(1) ⊗ ...⊗ S(g(2)σk )gσ(2) ⊗ S(g

(1)σ(k)

)σgσ(i−1))

= σi−1i−k(∑σ∈Sp

i−1∑r=k+1

gσ(k+1) ⊗ ...gσ(r−1) ⊗

gσ(k)gσ(r) ⊗ gσ(r+1) ⊗ ...⊗ gσ(i) ⊗ 1⊗ gσ(1) ⊗ ...⊗ gσ(i−1)

+ gσ(k+1) ⊗ ...⊗ gσ(i) ⊗ gσ(k) ⊗ gσ(1) ⊗ ...⊗ gσ(i−1))

=∑σ∈Si

p−1∑r=k+1

gσ(k+1) ⊗ ...⊗ gσ(r−1) ⊗

gσ(k)gσ(r) ⊗ gσ(r+1) ⊗ ...⊗ gσ(i) ⊗ gσ(1) ⊗ ...⊗ gσ(i−1)

=∑

σ∈S′i,σ(v)>σ(w)

gσ(1) ⊗ ...⊗ gσ(v)gσ(w) ⊗ ...⊗ gσ(i)

+∑

σ∈S”i,σ′(v)>σ(w)

gσ(1) ⊗ ...⊗ gσ(w)gσ(v) ⊗ ...⊗ gσ(i)

=∑

σ∈Si,σ(v)>σ(w)

gσ(1) ⊗ ...⊗ gσ(v)gσ(w) − gσ(w)gσ(v) ⊗ ...⊗ gσ(i)

=∑

σ∈Si,σ(v)>σ(w)

gσ(1) ⊗ ...⊗ [gσ(v), gσ(w)]⊗ ...⊗ gσ(i)

= A([gσ(v), gσ(w)] ∧ gσ(1) ∧ ... ∧ gσ(z) ∧ ... ∧ gσ(w) ∧ ... ∧ gσ(i))

= Ad(g1 ∧ ... ∧ gi)

Chapter 7

Coproducts for the dual Hopf cyclic

homology

One has the cyclic module for Hopf algebras [17] which in a special case is the dual

of Connes-Moscovici cocyclic module for Hopf algebras. We assume M is a left-left

stable anti-Yetter Drinfeld module on a Hopf algebra H. Recall that: Cn(H,M) =

H⊗n+12HM , n ∈ N. The following homomorphisms establish, respectively, the

desired faces, degeneracies and cyclic operators on Cn(H,M)n∈N.

δi(h0 ⊗ . . .⊗ hn ⊗m) = h0 ⊗ . . .⊗ hihi+1 ⊗ . . .⊗ hn ⊗m, 0 ≤ i < n,

δn(h0 ⊗ . . .⊗ hn ⊗m) = h(0)n h0 ⊗ h1 . . .⊗ hn−1 ⊗ h

(1)n m,

σi(h0 ⊗ . . .⊗ hn ⊗m) = h0 ⊗ . . .⊗ hi ⊗ 1⊗ . . .⊗ hn ⊗m, 0 ≤ i ≤ n,

τn(h0 ⊗ . . .⊗ hn ⊗m) = h(0)n ⊗ h0 ⊗ . . .⊗ hn−1 ⊗ h

(1)n m.

In this section, we define similar coproducts for HH∗(H,M), HC∗(H,M) and HP ∗(H,M)

where H is a cocommutative Hopf algebra.

Lemma 7.0.1. If H and K are two Hopf algebras and M and N two SAYD modules

over H and K, respectively, then the following map is an isomorphism of cyclic

modules:

Ωn : (H ⊗K)n+12H(M ⊗N) −→ (H⊗n+12HM)⊗ (K⊗n+12KN)

92

93

by

((h1 ⊗ k1)⊗ ...⊗ (hn ⊗ kn)⊗ (m⊗ n)) 7−→ (h1 ⊗ ...⊗ hn ⊗m)⊗ (k1 ⊗ ...⊗ kn ⊗ n).

Proof. We prove Ω commutes with δi and τn, where 1 ≤ i ≤ n. One can easily

check this for δn and degeneracies.

(δi ⊗ δi)Ωn((h0 ⊗ k0)⊗ ...⊗ (hn ⊗ kn)⊗ (m⊗ n))

= (δi ⊗ δi)((h0 ⊗ ...⊗ hn ⊗m), (k0 ⊗ ...⊗ kn ⊗ n))

= δi(h0 ⊗ ...⊗ hn ⊗m)⊗ δi(k0 ⊗ ...⊗ kn ⊗ n)

= (h0 ⊗ ...hihi+1 ⊗ ...⊗ hn ⊗m)⊗ (k0 ⊗ ...kiki+1 ⊗ ...⊗ kn ⊗ n)

= Ωn((h0 ⊗ k0)⊗ ...⊗ (hihi+1 ⊗ kiki+1)⊗ ...⊗ ((hn ⊗ kn)

= Ωnδi((h0 ⊗ k0)⊗ ...⊗ (hn ⊗ kn)⊗ (m⊗ n)).

Since H is a H-comodule algebra by comultiplication, we have:

(τn ⊗ τn)Ωn((h0 ⊗ k0)⊗ ...⊗ (hn ⊗ kn)⊗ (m⊗ n))

= τn(h0 ⊗ ...⊗ hn ⊗m)⊗ τn(k0 ⊗ ...⊗ kn ⊗ n)

= (h(1)n ⊗ h0 ⊗ ...⊗ hn−1 ⊗ h

(2)n m)⊗ (k

(1)n ⊗ k0 ⊗ ...⊗ kn−1 ⊗ k

(2)n n).

= Ωn(h(1)n ⊗ k

(1)n )⊗ (h0 ⊗ k0)⊗ ...⊗ (hn−1 ⊗ hn−1)⊗ (h

(2)n m⊗ k(2)n n)

= Ωn(h(1)n ⊗ k

(1)n )⊗ (h0 ⊗ k0)⊗ ...⊗ (hn−1 ⊗ hn−1)⊗ (h

(2)n m⊗ k(2)n n)

= Ωτn((h0 ⊗ k0)⊗ ...(hn ⊗ kn)⊗ (m⊗ n)).

Applying the previous lemma to the Eilenberg-Zilber isomorphism and Kunneth

formula, we obtain the following isomorphism for Hochschild homology with coeffi-

94

cients:

HHn(H ⊗K,M ⊗N) ∼=⊕

i+j=n

HHi(H,M)⊗ HHj(K,N).

Also we have the following long exact sequence for Hopf cyclic homology with coeffi-

cients:

... −→ HCn(H ⊗K,M ⊗N)I−−−→

⊕i+j=n HCi(H,M)⊗ HCj(K,N)

S⊗id−id⊗S−−−−−−−−→

⊕i+j=n−2 HCi(H,M)⊗ HCj(K,N)

∂−−−→ HCn−1(H ⊗K,M ⊗N) −→ ...

(7.0.1)

If conditions (i) and (ii) in theorem (5.3.1) hold then we have:

HP 0(H ⊗K,M ⊗N) ≃ HP 0(H,M)⊗ HP 0(K,N)⊕ HP 1(H,M)⊗ HP 1(K,N),

and similarly for the odd case.

Proposition 7.0.1. Let H be a cocommutative Hopf algebra , M a SAYD module

over H and ψ ⊗ ∆⊗(n+1) a linear map where ψ : M −→ M ⊗ M , satisfies the

condition:

ψ(h.m) = ∆(h).ψ(m), (7.0.2)

then the map ρn = ΩnΦn : Cn(H,M) −→ (C(H,M)⊗ C(H,M))n is a map of cyclic

modules where Φn = ψ ⊗∆⊗n+1.

Proof. We just check the commutativity of Φ with δn. The condition (7.0.2)

is equivalent to:

(h.m)(1) ⊗ (h.m)(2) = h(1).m(1) ⊗ h(2).m(2).

95

By cocommutativity of H we have

(δn ⊗ δn)Ω(∆⊗n+1 ⊗ ψ)(h0 ⊗ ...⊗ hn ⊗m)

= δn(h(1)0 ⊗ ...⊗ h

(1)n ⊗m(1))⊗ δn(h

(2)0 ⊗ ...⊗ h

(2)n ⊗m(2))

= (h(1)(1)n h

(1)0 ⊗ h

(1)1 ⊗ ...h

(1)n−1 ⊗ h

(1)(2)n m(1))

⊗(h(2)(1)n h(2)0 ⊗ h

(2)1 ⊗ ...h

(2)n−1 ⊗ h

(2)(2)n m(2))

= Ωn((h(1)(1)n h

(1)0 ⊗ (h

(2)(1)n h

(2)0 )⊗ (h

(1)1 ⊗ h

(2)1 )

⊗...⊗ (h(1)n−1 ⊗ h

(2)n−1)⊗ (h

(1)(2)n m(1) ⊗ h

(2)(2)n m(2))

= Ωn((h(1)(1)n h

(1)0 ⊗ (h

(1)(2)n h

(2)0 )⊗ (h

(1)1 ⊗ h

(2)1 )

⊗...⊗ (h(1)n−1 ⊗ h

(2)n−1)⊗ (h

(2)n m)(1) ⊗ (h

(2)n m)(2))

= Ωn(∆⊗n+1 ⊗ ψ)(h(1)n h0 ⊗ h1 ⊗ ...⊗ hn−1 ⊗ h

(2)n m)

= Ωn(∆⊗n+1 ⊗ ψ)δn(h0 ⊗ ...⊗ hn ⊗m).

Now we are ready to define the coproducts.

Theorem 7.0.2. Let H be a cocommutative Hopf algebra, M a SAYD module over

H, and ψ : M −→ M ⊗M a linear map satisfying condition (7.0.2). The following

maps define coproducts for HH∗(H,M), HC∗(H,M) and HP ∗(H,M):

⊔HH

: HHn(H,M) −→ ⊕p+q=nHHp(H,M)⊗ HHq(H,M)

⊔ = I∗(AWnΩnΦn)∗, (7.0.3)

for dual Hochschild homology; and

⊔HC

: HCn(H,M) −→ ⊕p+q=nHCp(H,M)⊗ HCq(H,M)

⊔ = I∗(AWΩΦ)∗, (7.0.4)

96

for dual cyclic homology, where Φ = ⊕i≥0Φn−2i and Ω = ⊕i≥0Ωn−2i. Also if

HC(H,M) satisfies condition (i) and (ii) in theorem (5.3.1) then for dual periodic

cyclic homology we have

⊔HP

: HP 1(H,M) −→ HP 0(H,M)⊗ HP 1(H,M)⊕ HP 1(H,M)⊗ HP 0(H,M)

⊔ = I∗(AWΩΦ)∗, (7.0.5)

where Φ = ⊕i≥0Φ2i and Ω = ⊕i≥0Ω2i and similarly for the even case.

Proof. The statements follows from the fact that the morphisms Φ’s are maps

of complexes. We notice that our coproducts for HCn(H,M) and HPn(H,M) are

defined in (b, B)-complexes.

7.1 Applications of the coproducts for the dual

Hopf cyclic homology

In this section, we show the coproducts that we have defined for the dual Hochschild

and cyclic homology, in the special case of group algebra kG with trivial coefficients,

agree with the coproduct that we have for group homology.

Theorem 7.1.1. The coproduct for dual Hochschild homology of group algebra kG

with trivial coefficients agrees with the coproduct of group homology, i.e.,

⊔HH

(g1 ⊗ ...⊗ gn) =Gr (g1, ..., gn) =n∑

k=0

(g1 ⊗ ...⊗ gk)⊗ (gk+1 ⊗ ...⊗ gn).

97

Proof. This is obvious because of the MacLane isomorphism [28].

By [16], since kG is a cocommutative Hopf algebra, then

HC(ϵ,1)n (KG)

ν∼=n⊕

i=0

Hn−2i(G, k). (7.1.6)

Recall that we have a coproduct for group homology H∗(G, k):

Gr: Hn(G,K) −→ ⊕p+q=nHp(G,K)⊗Hq(G,K)

by

(g1 ⊗ ...⊗ gn) 7−→n∑

k=0

(g1 ⊗ ...⊗ gk)⊗ (gk+1 ⊗ ...⊗ gn).

Theorem 7.1.2. The coproduct for the dual cyclic homology of H = kG with the

trivial coefficients agrees with the coproduct that we have for group homology. More

precisely, based on the isomorphism (7.1.6), the following diagram commutes:

C∗∗(G)ν−−−→ C

(δ,1)n (KG)y∪G y∪′

(C∗∗(G)⊗ C∗∗(G))nν⊗ν−−−→ (C(δ,1)(KG)⊗ C(δ,1)(KG))n

where ∪′ = AWΩΦ.

Chapter 8

Cup products for the dual Hopf cyclic

homology

In this chapter we define cup prodcuts for the dual Hochschild and cyclic homology of

commutative Hopf algebras. Also at the end of the section we define a new coproduct

for dual Hopf cyclic homology.

Definition 8.0.1. Let H and K be two Hopf algebras. One can define the map

× = shp,q : Cp(H)⊗ Cq(K) −→ Cp+q(H ⊗K), (8.0.1)

given by:

(h1 ⊗ ...⊗ hp)× (k1 ⊗ ...⊗ kq) =∑σ

sgn(σ)((hσ−1(1) ⊗ 1)⊗ ...(hσ−1(p) ⊗ 1)⊗

⊗(1⊗ kσ−1(1))⊗ ...⊗ (1⊗ kσ−1(q)))

where σ runs over all (p, q)-shuffles.

Since the Hochschild boundary is a graded derivation for the shuffle map, we

have:

b(x× y) = b(x)× y + (−1)deg(x)x× b(y).

Therefore we get the following statement:

98

99

Theorem 8.0.3. Let H be a commutative Hopf algebra. The shuffle product

× : HHp(H)⊗ HHq(H) −→ HHp+q(H)

induces a graded commutative algebra structure on HH∗(H).

Proof. Since H is commutative, the multiplication map m : H ⊗H −→ H is

a Hopf algebra map. By composing the shuffle product × with the induced map by

m, for H = K, one obtains an inner shuffle product map:

× : Cp(H)⊗ Cq(H) −→ Cp+q(H)

given by

(h1 ⊗ ...⊗ hp)× (hp+1 ⊗ ...⊗ hp+q) =∑σ

sgn(σ)hσ−1(1) ⊗ ...⊗ hσ−1(p+q)

where σ runs over all (p, q)-shuffles. Therefore × induces a commutative graded

algebra structure on HH∗(H).

We propose a project here: Let H be a commutative Hopf algebra. The

Hochschild complex CHH(H) is a CDG-algebra which is augmented over H. Its

product is the shuffle product. Let I = ⊕n>0CHH(H) be the augmentation ideal.

For any H- SAYD module M , C(H,M) is a CHH(H)- module. Since [sh, b] = 0, the

quotient C(H,M)/I.C(H,M) is a well defined complex. We define its homology as

Harrison Hopf homology and we denote it by:

HHarrn(H,M) := Hn(C(H,M)/I.C(H,M)).

If there exists an analogue of Andre-Quillen homology for Hopf algebras, then a

question is if Harrison Hopf homology is equivalent to the Andre-Quillen homology.

100

Now we define a cup product for the dual Hopf cyclic homology of a commuta-

tive Hopf algebra.

Proposition 8.0.1. Let H and K be two Hopf algebras. The following map is a map

of chain complexes:

∗ : TotBCp(H)⊗ TotBCq(K) −→ TotB(C(H)× C(K))p+q+1,

given by

(xp, xp−2, ...) ∗ (yp, yp−2, ...) = (Bxp × yq, Bxp × yq−2, ...). (8.0.2)

Proof. It is enough to prove:

b(Bxp × yq) +B(Bxp × yq−2) = B(bxp +Bxp−2)× yq + (−1)pBxp × (byq +Byq−2),

and similarly for the other terms. Since [B, sh] + [b, sh′] = 0, we have:

B(x×By) = Bx×By.

The argument is completed by [b, sh] = 0.

The previous proposition implies:

Theorem 8.0.4. For any Hopf algebras H and K the map ∗ induces the following

product:

HCp(H)⊗ HCq(K) −→ HCp+q+1(H ⊗K).

This product is associative and graded commutative and

x ∗ y = (−1)(p+1)(q+1)T∗(y ∗ x),

for x ∈ HCp(H) and y ∈ HCq(K) where T : H ⊗K ∼= K ⊗H is the twisting map.

101

Proof. The proof is similar to the one in [31] for algebras.

Proposition 8.0.2. The boundary map ∂ in the long exact sequence for dual Hopf

cyclic homology with trivial coefficient is the product ∗ for dual cyclic homology, i.e.,

∂(x⊗ y) = x ∗ y.

Proof. The proof is similar to the one in [31] for algebras.

Now we are ready to define a cup product for cyclic homology:

Theorem 8.0.5. Let H be a commutative Hopf algebra. The chain product ∗ induces

a graded algebra structure on the dual Hopf cyclic homology:

HCp(H)⊗ HCq(H) −→ HCp+q+1(H).

Proof. This is a consequence of the fact that if H is commutative, then the

multiplication map m is a Hopf algebra map.

Chapter 9

Coproducts for cocyclic modules

In this chapter, we define a coproduct for any cocyclic module with certain conditions.

For the Connes-Moscovici cocyclic module for algebras, this will lead to a coalgebra

structure for cyclic cohomology of any algebra endowed with a trace map. Since

the comultiplication map ∆ played an important role for defining our coproducts in

the previous chapters, this will introduce different coproducts for Hopf algebras. By

[31], if A is a commutative algebra, then multiplication map is an algebra map and

therefore the map

∇ : Cn −→ Cn ⊗ Cn

by

∇(f)((a0, b0), ..., (an, bn)) = f(a0b0, ..., anbn)

where f ∈ Cn = Hom(A⊗(n+1), k) is a map of cocyclic modules . Therefore

Ish∇, ISh(⊕i≥0∇n−2i) and ISh(⊕i≥0∇2i+∗), where ∗ = 0, 1, define coproducts

for HHn(A), HCn(A) and HP ∗(A) respectively. Here we are going to define dif-

ferent coproducts for an algebra endowed with a trace. To do this, first we define a

coproduct for cocyclic modules.

9.1 Coalgebra structure for cocyclic modules

In this section, we define a coproduct for cocyclic modules satisfying certain condi-

tions.

102

103

Lemma 9.1.1. Let (Cn, δni , sni , τn) be a cocyclic module and a ∈ C0. The following

map

∇ : Cn −→ Cn ⊗ Cn

defined by

∇n(x) = x⊗ δn0 δn−10 ...δ10(a) + δn0 δ

n−10 ...δ10(a)⊗ x,

where

δ10(a) = δ11(a),

is a map of cocyclic modules.

Proof. First we prove that ∇ commutes with face δn+1j for any 0 ≤ j ≤ n+1.

∇(δn+1j (x)) = δn+1

j (x)⊗ δn+10 ...δ10(a) + δn+1

0 ...δ10(a)⊗ δn+1j (x).

On the other hand,

(δn+1j ⊗δn+1

j )∇(x) = δn+1j δn0 (x)...δ

10(a)⊗δ

n+1j (x)+δn+1

j (x)⊗δn+1j δn0 (x)...δ

10(a)⊗δ

n+1j (x).

Now there are three cases. If j = 0, then the equality ∇(δn+1j (x)) = (δn+1

j ⊗

δn+1j )∇(x) is obvious. If 1 ≤ j ≤ n+ 1, since

δnj δn−10 = δn0 δ

n−1j−1

we have:

104

δn+1j δn0 (x)...δ

10(a)

= δn+10 δnj−1δ

n−10 ...δ10(a)

= δn+10 δn0 δ

n−1j−2 ...δ

10(a)

= δn+10 δn0 ...δ

10(a).

So we get ∇(δn+1j (x)) = (δn+1

j ⊗ δn+1j )∇(x). If j = n+ 1, similarly, one can see

δn+1j δn0 (x)...δ

10(a) = δn+1

0 δn0 (x)...δ11(a).

From the condition δ11(a) = δ10(a), the equality follows. Now we prove that ∇ com-

mutes with the degeneracy sn−1j for any 0 ≤ j ≤ n − 1. We know that snj δn+10 =

δn0 sn−1j−1 and sn0δ

n+10 = idn. Thus:

sn−1j δn0 ...δ10

= δn−10 sn−2j−1 δn−10 ...δ10

= δn−10 δn−20 ...δn−k0 sn−k−10 δn−k0 ...δ10

= δn−10 δn−20 ...δn−k0 idn−k−1δn−k−10 ...δ10

= δn−10 ...δ10.

So we have ∇sj = (sj⊗sj)∇. Now we prove that ∇τn = (τn⊗ τn)∇. We know

that τnδn0 = δnn. So

τnδn0 ...δ

10 = δnnδ

n−10 ...δ10 = δn0 δ

n−1n−1δ

n−20 ...δ10 = δn0 ...δ

20δ

11.

Again from the condition δ10(a) = δ11(a) the statement is complete. Therefore

∇ is a map of cocyclic modules.

105

Theorem 9.1.1. Let (Cn, δni , sni , τn) be a cocyclic module and a ∈ C0 which δ10(a) =

δ11(a). The following maps define coproducts for HH∗(C), HC∗(C) and HP ∗(C).

For Hochschild cohomology we define the following coproduct:

⊔ : HHn(C) −→ ⊕p+q=nHHp(C)⊗HHq(C)

⊔ = Ish∇ (9.1.1)

where I is the Kunneth map for Hochschild cohomology. For cyclic cohomology we

define the following coproduct:

⊔ : HCn −→ ⊕p+q=nHCp(C)⊗HCq(C)

⊔ = ISh (9.1.2)

where = ⊕∇n−2i. Here I is the Kunneth map for cyclic cohomology. For periodic

cyclic cohomology we define the following coproduct:

⊔ : HP 1(C) −→−→ HP 0(C)⊗HP 1(C)⊕HP 1(C)⊗HP 0(C)

⊔ = ISh (9.1.3)

where = ⊕∇2i+∗,∗ = 0, 1 . Here I is the Kunneth map for periodic cyclic coho-

mology .

106

9.2 Coproducts for cyclic cohomology of

algebras endowed with trace

In this section, we apply the coproduct of the previous section to cocyclic module for

algebras. Therefore we obtain a coproduct for cyclic cohomology of algebras endowed

with a trace map.

Theorem 9.2.1. Suppose A is an algebra endowed with a trace map Tr. The follow-

ing map will define a coproduct for cyclic cohomology of an algebra

sh∇(f)((a0, ..., ap), (b0, ..., bq)) =∑σ

f(a0, ..., ap, 1, ..., 1)⊗ Tr(b0...bq)+

+Tr(a0...ap)⊗ f((b0, ..., bq), 1..., 1)

Where 1′s happen in the place of (p, q)-shuffles.

Proof. The condition δ10(a) = δ11(a) in lemma (9.1.1) for Connes- Moscovici

cocyclic module for algebras is equivalent to say a ∈ C0 = Hom(A,K) is a trace map.

Set a = Tr, then we get the desired result.

Example 9.2.1. This coproduct is different from the one that we have defined for co-

commutative Hopf algebras. In fact, condition δ0(a) = δ1(a), means that the modular

pair in involution (δ, σ) should be (δ, 1). So we could define coproducts for any Hopf

algebra with modular pair in involution (δ, 1). In the case of Hochschild cohomology

we have:

107

sh∇(h1 ⊗ ...⊗ hn)

= sh((h1 ⊗ ...⊗ hn)⊗ (1⊗ ...⊗ 1) + (1⊗ ...⊗ 1)⊗ (h1 ⊗ ...⊗ hn)

=∑σ

ϵ(hσ(p+1)...hσ(n))(hσ(1) ⊗ ...⊗ hσ(p))⊗ (1⊗ ...⊗ 1) +

ϵ(hσ(1)...hσ(p))(1⊗ ...⊗ 1)⊗ (hσ(p+1) ⊗ ...⊗ hσ(n)).

Chapter 10

The headlines of the new results in this

thesis

In this chapter we quickly mention all the main new results of this thesis.

R1. A Kunneth formula for periodic cyclic cohomology of cocyclic modules.

R2. A Kunneth formula for periodic cyclic homology of cyclic modules.

R3. A Kunneth formula for Hochschild cohomology of Hopf algebras with

coefficients in a SAYD-module.

R4. A Kunneth formula for cyclic cohomology of Hopf algebras with coefficients

in a SAYD-module.

R5. A Kunneth formula for periodic cyclic cohomology of Hopf algebras with


R6. The Eilenberg-Zilber isomorphisms for Hochschild cohomology of Hopf

algebras with coefficients in a SAYD-module.

R7. The Eilenberg-Zilber isomorphisms for cyclic cohomology of Hopf algebras

with coefficients in a SAYD-module.

R8. The Eilenberg-Zilber isomorphisms for periodic cyclic cohomology of Hopf


R9. A coproduct for Hochschild cohomology of cocommutative Hopf algebras


R10. A coproduct for cyclic cohomology of cocommutative Hopf algebras with


108

109

R11. A coproduct for periodic cyclic cohomology of cocommutative Hopf alge-

bras with coefficients in a SAYD-module.

R12. The coproduct for periodic cyclic cohomology of cocommutative Hopf

algebras with trivial coefficients agrees with coproduct of Lie algebra homology in a

sense of a commutative diagram.

R13. A coproduct for dual Hochschild homology of cocommutative Hopf alge-

bras with coefficients in a SAYD-module.

R14. A coproduct for dual cyclic homology of cocommutative Hopf algebras


R15. A coproduct for dual periodic cyclic homology of cocommutative Hopf


R16. The coproduct for dual Hochschild homology of cocommutative Hopf

algebras with trivial coefficients agrees with coproduct of group homology with trivial

coefficients.

R17. The coproduct for dual cyclic homology of cocommutative Hopf alge-

bras with trivial coefficients coincide with coproduct of group homology with trivial

coefficients.

R18. A graded commutative algebra structure on dual Hochschild homology

of commutative Hopf algebras with trivial coefficients.

R19. An associative and graded algebra structure on dual cyclic homology of

commutative Hopf algebras with trivial coefficients.

R20. The cup product of the dual cyclic homology of a commutative Hopf

algebra is the same as the boundary map of the Kunneth long exact sequence.

R21. A coalgebra structure on cyclic cohomology of cocyclic modules satisfying

a special condition.

R22. A coalgebra structure on cyclic cohomology of algebras endowed with a

trace map.

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113

Vita

Name: Mohammad Hassanzadeh

Post-secondary University of Tehran, IranEducation and University of Western Ontario, CanadaDegrees: 1995-2000, B. Sc.

University of Western OntarioLondon, Canada2005-2006, M.Sc.

The University of Western OntarioLondon, Ontario, Canada2006-2010, Ph.D.

Honors and Ontario Graduate Scholarship for Science and Technology(OGSST)Awards Western Graduate Scholarship

, 2005-2010

Related Work Teaching AssistantExperience The University of Western Ontario, 2005-2010

InstructorFanshawe CollegeLondon, Canada, 2009-2010

Publications:

[1] Masoud Khalkhali and Mohammad Hassanzadeh, “Cup products and coprod-ucts in Hopf cyclic cohomology”, ( preprint )

Documents

Operations On Hopf Cyclic Cohomology