Hopf Algebras Associated to Transitive Pseudogroups in Codimension 2

Home , Hopf algebra

HOPF ALGEBRAS ASSOCIATED TO TRANSITIVE PSEUDOGROUPS IN CODIMENSION 2

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of

Philosophy in the Graduate School of the Ohio State University

Jos´eRodrigo Cervantes Polanco , MS in Mathematics

Graduate Program in Mathematics

The Ohio State University

2016

Dissertation Committee:

Henri Moscovici, Advisor

James Cogdell

Thomas Kerler c Copyright by

Jos´eRodrigo Cervantes Polanco

2016 ABSTRACT

We associate two different Hopf algebras to the same transitive but not primitive pseudogrup of local diffeomorphisms on R2 leaving invariant the trivial foliation where we identify R2 as a product of lines R1 × R1. Their construction is based on ideas used to build the Hopf algebras associated to primitive Lie pseudogroups by Connes-Moscovici and Moscovici-Rangipour. Each of the two Hopf algebras is first defined via its action on the respective crossed product algebra associated to the pseudogroup, and then it is realized as a bicrossed product of a universal enveloping algebra of a Lie algebra and a Hopf algebra of regular functions on a formal group. Using the bicrossed product structure we prove that, although the two Hopf algebras are not isomorphic, they have the same periodic Hopf cyclic cohomology. More precisely, for each of them the periodic Hopf cyclic cohomology is canonically isomorphic to the Gelfand-Fuks cohomology of the infinite dimensional

Lie algebra related with the pseudogroup.

ii To my family and friends

iii ACKNOWLEDGMENTS

I would like to thank my advisor Henri Moscovici for his continuous support and advice during the completion of my Ph.D. degree. He has been a great academic men- tor and a wonderful friend for me. This dissertation grew out of numerous conversa- tions with him and all the team members from the research group of noncommutative geometry. I would like to thank my Ph.D. committee members, James Cogdell and

Thomas Kerler for their help during my time in the Department of Mathematics.

I also want to express my thanks to my friends who helped me many times during this process: Weitao Chen, Yang Liu, Angelo Nasca, Donald Robertson, Xiaoyue Xia,

Tao Yang and Kun Wang.

Finally, I wish to thank my parents Jos´eJavier Cervantes and Nilsa Concepci´on

Polanco Trujeque, and my girlfriend Evelyn Rodriguez for their support. It is their love and encouragement that lead me to this point of my life.

This project would have been impossible without the support of The Ohio State

University and CONACyT.

iv VITA

2005 ...... B.S. in Mathematics, University of Guanajuato, M´exico.

2007 ...... M.S. in Mathematics, CIMAT, M´exico.

2008-Present ...... Graduate Teaching Associate, The Ohio State University, USA.

FIELDS OF STUDY

Major Field: Mathematics

Specialization: Noncommutative Geometry

v TABLE OF CONTENTS

Abstract...... ii

Dedication...... ii

Acknowledgments...... iv

Vita...... v

CHAPTER PAGE

1 Introduction...... 1

2 Preliminaries...... 5

2.1 Basic concepts...... 5

3 Construction via Hopf actions...... 19

3.1 The Hopf algebra HH ...... 19 4 Bicrossed product realization...... 46

4.1 The bicrossed product Hopf algebra FH ICU(gh)...... 46 5 Hopf cyclic cohomology...... 64

∗ 5.1 HP (HH )...... 64 Bibliography...... 80

vi CHAPTER 1

INTRODUCTION

Hopf algebras were found in algebraic topology in 1941. The ﬁrst example was constructed in the paper of Heinz Hopf in his computation of the rational cohomology of compact connected Lie groups [18]. The ﬁrst book on Hopf algebras was written by Moss E. Sweedler in 1969 [30]. A large class of Hopf algebras are the

“quantum groups”. This term was coined by Vladimir Drinfel’d in his address to the

International Congress of Mathematicians in Berkeley on 1986 [10]. It stands for certain special noncommutative noncocommutative Hopf algebras which are non-trivial deformations of the universal enveloping algebras of classical Lie algebras or of the algebra of regular functions on the corresponding Lie groups.

Hopf algebras appear in many mathematical ﬁelds such as algebraic geometry, Lie theory, quantum mechanics, etc.

On the other hand, cyclic cohomology was discovered by Alain Connes in 1981; one of his main motivations came from index theory on foliated spaces [4]. Independently, cyclic homology was shown to be the primitive part of the Lie algebra homology of matrices by Boris Tsygan [31] and also by Jean-Louis Loday and Daniel Quillen [21].

This (co)homology is closely related to K-theory and has many interesting rela- tionships with several branches of mathematics. It can be seen as an extension of de

Rham theory, Hochschild (co)homology, group (co)homology.

1 The Hopf algebras Hn associated to the pseudogroup of local diﬀeomorphisms

n of R were found by Alain Connes and Henri Moscovici in their work on the local index formula for transversely hypoelliptic operators on foliations [5]. Extending the construction of Hn, Henri Moscovici and Bahram Rangipour deﬁned a Hopf algebra

HΠ for each inﬁnite primitive Lie-Cartan pseudogroup Π of local diﬀeomorphisms of

n R [25]. As reference for primitive pseudogroups we are using Singer and Sternberg [29], and Guillemin [13].

In this dissertation we are implementing a similar construction for the transitive

2 but not primitive pseudogrup of local diffeomorphisms on R leaving invariant the 2 1 1 trivial foliation where we identify R as a product of lines R × R , but in this case we are able to define two different Hopf algebras.

The reason for which one obtains two different Hopf algebras is because the construction relies on splitting the group G of globally defined diffeomorphims of the pseudogroup as a set-theoretical product of two groups

G = G · N,G ∩ N = { Id } .

For example, in each ﬂat primitive pseudogroup, there is a canonical splitting

n where G is the subgroup consisting of the aﬃne transformations of R that are in G, while N is the subgroup consisting of those diﬀeomorphisms in G that preserve the origin and its tangent map at zero is the identity matrix.

For our case, G is the group of globally deﬁned diﬀeomorphism of the form

1 2 ϕ(x1, x2) = (ϕ (x1, x2), ϕ (x2)) and one can consider two diﬀerent splittings

G = GH · NH ,GH ∩ NH = { Id } . where H is either the subgroup of upper triangular matrices or the subgroup of diagonal matrices of GL2(R). For each of this splittings, GH is the subgroup of 2 “aﬃne H-motions” transformations of the form Lx ◦ y where Lx are translations by

2 x ∈ R and y ∈ H, and NH is the subgroup of elements of G preserving the origin and with “H-tangent map” at zero the identity matrix. We will denoted by HH the Hopf algebra which depends on the group H.

For all primitive pseudogroups Π, it is proved that there is a quasi-isomorphism between the periodic Hopf cyclic cohomology of HΠ and the continuous cohomology of the Lie algebra of formal vector ﬁelds aΠ related to the pseudogroup [5, 26]. This type of cohomology is also called Gelfand-Fuks cohomology [11, 12].

The inﬁnite dimensional Lie algebra a related to the pseudogrop discussed on this dissertation is given by all formal vector ﬁelds v of the form

1 ∂ 2 ∂ v = p (x1, x2) + p (x2) ∂x1 ∂x2

1 2 where p (x1, x2) and p (x2) are formal polynomials in their respective variables. In this dissertation, it is shown that the periodic Hopf cyclic cohomology for each

Hop algebra HH is canonically isomorphic to the same continuous cohomology of the inﬁnite dimensional Lie algebra a. In this way, we have obtained two non-isomorphic

Hop algebras with the same periodic Hopf cyclic cohomology.

This dissertation is organized as follows.

In chapter §2, we introduce some background material on Hopf algebras, cohomology and pseudogroups.

In chapter §3, the algebra HH is deﬁned via its natural action on the crossed

∞ product algebra AH = Cc (GH ) o G. It is shown that every element h ∈ HH satisﬁes a “Leibniz rule” of the form

h(ab) = h(1)(a)h(2)(b) , for all a, b ∈ AH .

This property together with an “invariant” trace τ gives us the coproduct structure and the antipode map. Hence, we equipp HH with a Hopf algebra structure. 3 In chapter §4, the Hop algebra HH is realized as a bicrossed product Hopf algebra. The components of the bicrossed product realization are the universal enveloping algebra of the Lie algebra gh corresponding to GH , and the Hopf algebra of functions on NH given by the non-trivial coeﬃcients of the Taylor expansion of ψ ∈ NH at zero. It is worth mentioning that in this chapter in order to prove the compatibility conditions of the two Hopf algebras, we employed the Lie-Hopf algebra techniques developed in [27], instead of the more geometric techniques as in [25]. The reason for this is to provide a detailed account for both proofs in written form.

In chapter §5, by using the bicrossed product structure of HH , we prove that the periodic Hopf cyclic cohomology of HH is the same for both H, either the subgroup of upper triangular matrices or the subgroup of diagonal matrices. In fact, for each of them it is canonically isomorphic to the continuous cohomology of the inﬁnite dimensional algebra a. We use the techniques developed in [5, 26] to prove this isomorphism.

4 CHAPTER 2

PRELIMINARIES

The basic concepts related to Hopf algebras, cohomology, and pseudogroups are given through deﬁnitions and examples.

2.1 Basic concepts.

We begin this chapter with a resume of concepts related to Hopf algebras [30,1,8].

Let F be a ﬁeld and G a ﬁnite group. All unadorned tensor products are over F. An algebra is a triple (A, µ, η) with A a vector space, µ : A ⊗ A → A a linear map called the product, η : F → A a linear map called the unit, and such that the following diagrams commute

µ ⊗ Id η ⊗ Id Id ⊗ η A ⊗ A ⊗ A A ⊗ A F ⊗ A A ⊗ A A ⊗ F

Id ⊗ µ µ =∼ µ =∼

µ A ⊗ A A A

We have denoted by Id the identity map of A, and the unnamed arrows from the second diagram are the canonical isomorphisms.

op Let (A, µ, η) be an algebra an the map µ := µ ◦ τ where τ(a1 ⊗ a2) = a2 ⊗ a1, then (Aop = A, µop, η) is an algebra which we call the opposite algebra.

5 Let C(G) denote the set of all functions deﬁned on G with values in F. It is a vector space via

(λf1)(g) = λf1(g) , (f1 + f2)(g) = f1(g) + f2(g) , f1, f2 ∈ C(G) , λ ∈ F .

It becomes an algebra when we deﬁne the product and the unit of C(G) by

µ(f1, f2)(g) = f1(g)f2(g) , η(1) = 1 , f1, f2 ∈ C(G) , where 1 ∈ C(G) is the function deﬁned by 1(g) = 1 ∈ F.

A coalgebra is a triple (C, ∆, ε) with C a vector space, ∆ : C → C ⊗ C a linear map called the coproduct, ε : C → C a linear map called the counit, and such that the following diagrams commute

∆ ε ⊗ Id Id ⊗ ε C C ⊗ C F ⊗ C C ⊗ C C ⊗ F

∆ Id ⊗ ∆ =∼ ∆ =∼

∆ ⊗ Id C ⊗ C C ⊗ C ⊗ C C

cop Let (C, ∆, ε) be a coalgebra an the map ∆ := τ ◦ ∆ where τ(c1 ⊗ c2) = c2 ⊗ c1, then (Ccop = C, ∆cop, ε) is a coalgebra which we call the coopposite coalgebra.

If we identify C(G) ⊗ C(G) with C(G × G), which is ﬁne since G is ﬁnite, then

C(G) becomes a coalgebra when we deﬁne the coproduct and the counit via

∆(f)(g1 ⊗ g2) = f(g1g2) ε(f) = f(e) , f ∈ C(G) , where e is the unit element in G.

We will use Sweedler notation [30] and omit the summation to denote the comul- tiplication by

∆(c) = c(1) ⊗ c(2) .

6 Using this notation we can write the commutative diagrams for a coalgebra as follows

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

ε(c(1))c(2) = c = c(1)ε(c(2)) .

The ﬁrst element is usually written in the literature as c(1) ⊗ c(2) ⊗ c(3).

Let (A, µA, ηA), (B, µB, ηB) be two algebras. The linear map i : A → B is a morphism of algebras if the following diagrams are commutative

i ⊗ i i A ⊗ A B ⊗ B A B

µA µB ηA ηB

i A B F

The tensor product A ⊗ B is an algebra with µ :(A ⊗ B) ⊗ (A ⊗ B) → A ⊗ B and

η : F → A ⊗ B deﬁned by

µ = (µA ⊗ µB)(Id ⊗ τ ⊗ Id) , where τ(b ⊗ a) = (a ⊗ b) ,

η = (ηA ⊗ ηB) .

Let (C, ∆C , εC ), (D, ∆D, εD) be two coalgebras. The linear map j : C → D is a morphism of coalgebras if the following diagrams are commutative

j j C D C D

∆C ∆D εC εD

j ⊗ j C ⊗ C D ⊗ D F

The commutativity of the ﬁrst diagram may be written in the Sweedler notation as follows

j(c)(1) ⊗ j(c)(2) = j(c(1)) ⊗ j(c(2)) . 7 The tensor product C ⊗ D is a coalgebra with ∆ : C ⊗ D → (C ⊗ D) ⊗ C ⊗ D and

ε : C ⊗ D → F deﬁned by

∆ = (Id ⊗ τ ⊗ Id)(∆A ⊗ ∆B) , where τ(c ⊗ d) = (d ⊗ c) ,

ε = εC · εD .

A bialgebra is a quintuple (H, µ, η, ∆, ε), with (H, µ, η) an algebra, (H, ∆, ε) a coalgebra and the linear maps µ and η are morphisms of coalgebras (or equivalently, the maps ∆ and ε are morphisms of algebras), i.e., the diagrams below commute

µ µ H ⊗ H H H ⊗ H H

(Id ⊗ τ ⊗ Id)(∆ ⊗ ∆) ∆ ε · ε ε

µ ⊗ µ (H ⊗ H) ⊗ (H ⊗ H) H ⊗ H F

η η F H F H

ε ∆F ∆ F ε η ⊗ η F ⊗ F H ⊗ H F

where F is a Hopf algebra via ∆F(1) = 1 ⊗ 1 and εF(1) = 1.

Let (A, µ, η) be an algebra and (C, ∆, ε) a coalgebra. We deﬁne on the set

HomF(C,A) an algebra structure in which the product, denoted by ? and called the convolution product in Hom(C,A), is given as follows

α ? β(c) = α(c(1))β(c(2)) , α, β ∈ Hom(C,A) , and the unit is given by 1 → η ◦ ε ∈ Hom(C,A).

Let H be a bialgebra. A linear map S : H → H is called an antipode of the bialgebra H if S is the inverse of the identity map Id : H → H with respect to the convolution product in Hom(H,H). 8 Deﬁnition 2.1.1. A Hopf algebra is a sixtuple (H, µ, η, ∆, ε, S) with (H, µ, η, ∆, ε) a bialgebra and S is the antipode of H.

The linear maps µ and η for C(G) are morphisms of coalgebras, then C(G) is a bialgebra. Furthermore, it has an antipode which is deﬁned by

S(f)(g) = f(g−1) , f ∈ C(G) , since

−1 −1 f(1)S(f(2)) (g) = f(1)(g)S(f(2))(g) = f(1)(g)f(2)(g ) = f(gg ) = f(e) = ε(f)η(1)(g) ,

similarly S(f(1))f(2) = ε(h)η(1). In conclusion, C(G) is a Hopf algebra.

In a Hopf algebra, the antipode is unique, being the inverse of the element Id in the algebra Hom(H, H). The fact that S is the antipode is written as S∗Id = η◦ε = Id ∗S, and using the Sweedler notation

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

Let H1 and H2 be two Hopf algebras. A map α : H1 → H2 is called a morphism of Hopf algebras if it is a morphism of bialgebras, i.e. , a morphism of algebras and a morphism of coalgebras.

Proposition 2.1.2. Let H1 and H2 be two Hopf algebras with antipodes S1 and S2.

If α : H1 → H2 is a morphism of bialgebras, then S2 ◦ α = α ◦ S1.

Proof. It follows by the uniqueness of the antipode on a Hopf algebra.

Let A be an algebra. A left A-module is a pair (M,.) with M a vector space, and . : A ⊗ M → M a morphism of vector spaces such that the following diagrams are commutative

9 Id ⊗ . η ⊗ Id A ⊗ A ⊗ M A ⊗ M F ⊗ M A ⊗ M

µ ⊗ Id . =∼ .

. A ⊗ M M M

Similarly, one can deﬁne right A-modules, the only diﬀerence being that the structure map of the right module M is of the form / : M ⊗ A → M. An A-bimodule M is a vector space with both a left and a right module structure.

We will use the following notation for left and right modules, respectively,

a . m := .(a ⊗ m) , and m / a := /(m ⊗ a) , a ∈ A , m ∈ M.

Let C be a coalgebra. A left C-comodule is a pair (N, H) with N a vector space, and H : N → C ⊗ N a morphism of vector spaces such that the following diagrams are commutative

H ε ⊗ Id N C ⊗ N C ⊗ N F ⊗ N

∼ H ∆ ⊗ Id H =

Id ⊗ C ⊗ N H C ⊗ C ⊗ N N

Simarly, one can deﬁne right C-comodules, the diﬀerence being that the structure of a right C-comodule N is of the form H : N → N ⊗ C. An C-bicomodule N is a vector space with both a left and a right comodule structure.

We will also use Sweedler notation for left and right comodules, respectively,

H(n) := n<−1> ⊗ n<0> , and H(n) := n<0> ⊗ n<1>, n ∈ N. where n<0> always stay in the same vector space N while n<−1>, n<1> are in the coalgebra C and the −/+ sign tell the left/right direction of the coaction, respectively. 10 Next, we introduce a quick summary on complexes and bicomplexes [3, 23, 32].

Let k be an associative ring.

∗ n A cochain complex { C , d } of k-modules, is a family { C }n∈Z of k-modules, together with k-module homomorphism dn : Cn → Cn+1 such that each composite dn−1 ◦ dn = 0. The maps dn are called the diﬀerentials of C. The kernel of dn is the module of n-cocycles of C, denoted by Zn(C). The image of dn−1 is the module of n-coboundaries of C, denoted by Bn(C). Because dn−1 ◦ dn = 0, we have

n n n B ⊂ Z ⊂ C , n ∈ Z .

The nth-cohomology module of { C∗, d } is the subquotient of Cn given by

Hn(C) = Zn(C)/Bn(C) .

A map of complexes f : C → D is a family of k-module homomorphism f n : Cn →

Dn commuting with dn in the sense that f n+1 ◦ dn = dn ◦ f n. It induces a map fˆn : Hn(C) → Hn(D). The map of complexes f is called a quasi-isomorphism if fˆn is an isomorphism for all n.

Let A be a algebra over F and M an A-bimodule. Consider the family of F-modules n ⊗n C (A, M) = HomF(A ,M) and deﬁne the face operators

δi : Cn(A, M) → Cn+1(A, M) , 0 ≤ i ≤ n ,

0 δ (f)(a0 ⊗ · · · ⊗ an) = a0 . f(a1 ⊗ · · · ⊗ an) ,

i δ (f)(a0 ⊗ · · · ⊗ an) = f(a1 ⊗ · · · ⊗ ai−1ai ⊗ · · · ⊗ an) , 1 ≤ i ≤ n − 1 ,

n δ (f)(a0 ⊗ · · · ⊗ an) = f(a0 ⊗ · · · ⊗ an−1) / an .

Definition 2.1.3. The Hochschild cohomology H∗(A, M) of A with coefficients in M is defined to be the nth-cohomology modules of { C∗(A, M), b }, where n X bn = (−1)iδi . i=0 11 Let C be a coalgebra over F and N a C-bicomodule. Consider the family of n ⊗n F-modules C (C,N) = N ⊗ C and define the face operators

δi : Cn(C,M) → Cn+1(C,M) , 0 ≤ i ≤ n

0 δ (m ⊗ c1 ⊗ · · · ⊗ cn) = m<0> ⊗ m<1> ⊗ c1 ⊗ · · · ⊗ cn ,

i δ (m ⊗ c1 ⊗ · · · ⊗ cn) = m ⊗ c1 ⊗ · · · ⊗ ∆(ci) ⊗ · · · ⊗ cn , 1 ≤ i ≤ n − 1 ,

n δ (m ⊗ c1 ⊗ · · · ⊗ cn) = m<0> ⊗ c1 ⊗ · · · ⊗ cn ⊗ m<−1> .

Definition 2.1.4. The Hochschild cohomology H∗(C,N) of C with coefficients in N is defined to be the nth-cohomology modules of { C∗(C,N), b }, where

n X bn = (−1)iδi . i=0

Let g a Lie algebra over F and V a ﬁnite dimensional left g-module. Consider the n family of F-modules C (g,V ) of antisymmetric n-linear maps on g with values in V , n n i.e. , C (g,V ) = HomF(∧ g,V )

Definition 2.1.5. The cohomology H∗(g,V ) of the algebra g with coefficients in V is defined to be the nth-cohomology modules of { C∗(g,V ), d }, where

X s+t−1 d(ω)(x1, . . . , xn+1) = (−1) ω([xs, xt], x1,..., xˆs,..., xˆt, . . . , xn+1) 1≤s

X s + (−1) xi ω(x1,..., xˆs, . . . , xn+1) , 1≤s≤n+1 where the sign ˆ indicates that the argument below it must be omitted.

If g is a topological Lie algebra and the vector space V is a topological space, the action will be continuous and Cn(g,V ) is restricted to continuous maps. The

∗ cohomology is denoted by Hcont(g,V ).

12 Examples of Lie algebras:

• Finite dimensional. Let gln(R) be the algebra of n × n matrices with coeﬃ-

cients in R with Lie bracket given by

[X,Y ] = XY − YX,X,Y ∈ gln(R) .

n • Infinite dimensional. Let an(R) be the space of formal vector fields in R with Lie bracket defined by " # ! X ∂ X ∂ X X ∂gi ∂fi ∂ f , g = f − g , i ∂x i ∂x j ∂x j ∂x ∂x i i i i i j j j i

in which f and g denote formal power series in x1, . . . , xn. It is a topological Lie algebra when it is equipped with the projective limit topology.

A cochain bicomplex is a family of k-modules Cp,q, together with maps

dh : Cp,q → Cp+1,q , and dv : Cp,q → Cp,q+1 , such that dh ◦ dh = dv ◦ dv = dv ◦ dh + dh ◦ dv = 0. It is useful to picture the bicomplex

C•,• as a lattice ......

dh dh ··· Cp−1,q+1 Cp,q+1 Cp+1,q+1 ···

dv dv dv

dh dh ··· Cp−1,q Cp,q Cp+1,q ···

dv dv δv

dh dh ··· Cp−1,q−1 Cp,q−1 Cp+1,q−1 ···

......

13 in which the maps dh go horizontally, the maps dv go vertically, and each square anticommutes. Each row C•,q and each column Cp,• is a chain complex. Deﬁne the total complexes

Y M TotΠ(C)n = Cp,q and Tot⊕(C)n = Cp,q . p+q=n p+q=n

The formula d = dh + dv deﬁnes maps such that d ◦ d = 0, making TotΠ(C) and

Tot⊕(C) into chain complexes.

Note that in some special cases they are equal, for example if C is a ﬁrst quadrant double complex.

n ⊗(n+1) Let A be an algebra over F. Let C (A) = HomF(A , F) and the face, degen- eracy and cyclic operators

n−1 n δi : C (A) → C (A) , 0 ≤ i ≤ n ,

n+1 n σi : C (A) → C (A) , 0 ≤ i ≤ n .

n n τn : C (A) → C (A) . deﬁned by

δi(c)(a0 ⊗ · · · ⊗ an) = c(a0 ⊗ aiai+1 ⊗ an) , 0 ≤ i ≤ n − 1 ,

δn(c)(a0 ⊗ · · · ⊗ an) = c(ana0 ⊗ · · · ⊗ an−1) ,

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

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

Consider the family of F-modules   q−p C (A), if q ≥ p, Cp,q(A) =  0, if q < p,

14 Definition 2.1.6. The cyclic cohomology HC∗(A) of A is defined to be the nth cohomology modules of the first quadrant bicomplex Cp,q(A)

......

b b b

B B C2(A) C1(A) C0(A)

b b

B C1(A) C0(A)

C0(A) where

b : Cn(A) → Cn+1(A),B : Cn+1(A) → Cn(A) , n n X i X ni i n b = (−1) δi ,B = ( (−1) τn)σnτn+1(1 − (−1) τn+1) . i=0 i=0 A variant of the cyclic cohomology is the periodic cyclic cohomology HP ∗(A) which is Z2-graded. It is deﬁned as the even and odd cohomology of the total complex shown above, but extended indeﬁnitely to the left and down.

For completeness, we also deﬁne the concept of pseudogroup [20, 29].

Deﬁnition 2.1.7. A pseudogroup of transformations on a topological space S is a set

Γ of transformations satisfying the following axioms:

1. Each ψ ∈ Γ is a homeomorphism of an open set (called the domain of ψ) of S

onto another open set (called the range of φ) of S;

2. If ψ ∈ Γ, then the restriction of ψ to an arbitrary open subset of the domain of

ψ is in Γ; 15 [ 3. Let U = Ui where each Ui is an open set of S. A homeomorphism ψ of U i onto an open set of S belongs to Γ if the restrictions of ψ to Ui is in Γ for all i;

4. For every open set U of S, the identity transformation of U is in Γ;

5. If ψ ∈ Γ, then ψ−1 ∈ Γ;

6. If ψ ∈ Γ is a homeomorphism of U onto V and ψ0 ∈ Γ is a homeomorphism

of U 0 onto V 0 and if V ∩ U 0 is non-empty, then the homeomorphism ψ0 ◦ ψ of

ψ−1(V ∩ U 0) onto ψ0(V ∩ U 0) is in Γ.

n n A diffeomorphism ψ of an open set U ⊂ R onto an open set V ∈ R is called n a local diffeomorphism of R . We will be interested in certain pseudogroups defined n n on R and all pseudogroups appearing in this dissertation will be defined on R and n the elements in Γ will be local diffeomorphisms on R , not only homeomorphisms. A pseudogroup Γ of local diffeomorphisms is Lie if its elements ψ form the general solution of a set of PDE’s in the jet of ψ of some finite order.

It is clear from the deﬁnition that if Γα is a collection of pseudogroups deﬁned on

n \ n R then Γα is also a pseudogroup on R . It therefore make sense to speak of the α n pseudogroup generated by a family of diﬀeomorphisms deﬁned locally on R .

Let L = { Xr }r∈R be a collection of differentiable vector fields each defined on

n some Ur ⊂ R . We then get a collection of diffeomorphisms defined as follows: each x ∈ Ur has a neighborhood over which we can integrate Xr for sufficiently small t to obtain a (local) one parameter group ψt. The set of all such ψt is a family of diffeomorphisms and thus generate a pseudogroup. We shall called it the pseudogroup generated by L and denote by Γ(L).

On the other hand, the vector ﬁelds whose local ﬂows are in a pseudogroup Γ (for small time) form a Lie algebra L(Γ). If dim L(Γ) < ∞ then Γ comes from an action of a Lie group. 16 A pseudogroup Γ is called transitive if for every pair of points x and y in S there exists a ψ ∈ Γ with ψ(x) = y. It is called locally transitive if every point x in S has a neighborhood U such that for any point y in U there is a ψ ∈ Γ with ψ(x) = y.

A pseudogroup Γ is called primitive if it leaves no nontrivial foliation invariant.

Note that if Γ is primitive then it is automatically transitive.

E. Cartan [2] has classiﬁed the complex primitive inﬁnite dimensional pseudogroups,

ﬁnding 6 classes. Gaps in his proof rectiﬁed by Singer-Sternberg [29]; complete proof by Guillemin-Quillen-Sternberg [14] in the complex case and Shnider [28] in the real case. Here is the list for the real case:

n • The pseudogroup of local diﬀeormorphism of R .

n • The pseudogroup of local diﬀeormorphism preserving a volume form of R .

2n • The pseudogroup of local diﬀeormorphism preserving a symplectic form of R .

2n+1 • The pseudogroup of local diﬀeormorphism preserving a contact form of R .

n • The pseudogroup of local diﬀeormorphism preserving the volume form of R up to a constant.

2n • The pseudogroup of local diﬀeormorphism preserving a symplectic form of R up to a constant.

Finally, let us recall the Generalized Fa´adi Bruno’s formula [9, 17, 22].

n Let α = (α1, α2, . . . , αn) ∈ N , where N is the set of all nonnegative integers. For n every x = (x1, x2, . . . , xn) ∈ R , we set

n n N α Y αi X Y x = xi , |α| = αi , α! = αi! . i=1 i=1 i=1

17 n m Let us consider two open subsets U ⊂ R , V ⊂ R , and two diﬀerentiable maps of class greater or equal than r,

f g U −→ V −→ R .

If (x1, . . . , xn), (y1, . . . , ym) are the standard coordinates in U, V , respectively, then we set fi = yi ◦ f and y0 = f(x0), and x0 ∈ U.

Theorem 2.1.8. With the previous notation and assumptions, the following formula holds:

e |α| r |σ| m |αi| ! iαi ∂ (g ◦ f) X ∂ (g) X Y Y 1 1 ∂ (fi) (x ) = α! (y ) (x ) , α 0 σ 0 i αi 0 ∂x ∂y eiαi ! α ! ∂x |σ|=1 Eσ i=1 Aα where r = |α|,

( r ) i X Eσ = (e1α1 , . . . , emαm ): eiαi ∈ N, 1 ≤ |α | ≤ r, eiαi = σi, 1 ≤ i ≤ m , |αi|=1 and

( m r ) 1 m i X X i Aα = (α , . . . , α ) : 1 ≤ |α | ≤ r, 1 ≤ i ≤ m, eiαi · α = α . i=1 |αi|=1

18 CHAPTER 3

CONSTRUCTION VIA HOPF ACTIONS

The Hopf algebra HH associated to the group G of globally deﬁned diﬀeomor-

2 2 phisms ϕ : R → R of the form

1 2 ϕ(x1, x2) = (ϕ (x1, x2), ϕ (x2))

is deﬁned as an algebra of linear operators acting over an algebra AH . The coalgebra structure and antipode of HH are obtained by using the properties of these operators.

3.1 The Hopf algebra HH .

2 2 2 Let F R be the frame bundle on R , which we identify to R × GL2(R) in the 2 2 usual way: the ﬁrst jet at 0 ∈ R of a germ of a local diﬀeomorphism φ on R viewed as the pair 2 x := φ(0), y := (φ)∗,0 ∈ R × GL2(R).

2 i The ﬂat connection on F R is given by the matrix-valued 1-form ω = (ωj) where, with the usual summation convention,

i −1 i −1 i µ ωj := (y dy)j = (y )µdyj , i, j = 1, 2 , and the canonical form is the vector-valued 1-form θ = (θk) where

k −1 k −1 k µ µ θ := (y dx) = (y )µd , k = 1, 2 , where d := dxµ . 19 j Let Yi be the fundamental vertical vector ﬁelds associated to the standard basis of j gl2(R) formed by the elementary matrices Ei . Thus we have

j µ j j ∂ Yi = yi ∂µ , i, j = 1, 2 , where ∂µ := µ . ∂yj

Let Xk be the basic horizontal vector ﬁelds. They are given by

µ ∂ Xk = yk ∂µ , k = 1, 2 , where ∂µ := . ∂xµ 2 2 Besides the usual tangent space TxR for x ∈ R , it will be convenient to introduce 2 the graded version gr TxR given by the following gradation

2 2 gr−1 TxR := spanh ∂1 i gr−2 TxR := spanh ∂2 i .

2 2 2 2 If φ ∈ G, then its tangent map at x ∈ R ,(φ)∗,x : TxR → Tφ(x)R , leaves gr−1 TxR 2 2 invariant, and hence induces a graded tangent map gr(φ)∗,x : gr TxR → gr Tφ(x)R . In fact, we have that     1 1 1 ∂1 φ ∂2 φ ∂1 φ 0 (φ)∗,x =   ∈ P and gr(φ)∗,x =   ∈ D  2   2  0 ∂2 φ 0 ∂2 φ where P and D are the subgroup of upper triangular matrices and the subgroup of diagonal matrices of GL2(R), respectively. To simplify notation, we deﬁne  2 Tx if H = P. H 2  R Tx R :=  2 gr TxR if H = D.  (φ)∗,x if H = P. H  (φ)∗,x := gr(φ)∗,x if H = D. Throughout the rest of this dissertation H will be either the subgroup P or D.

2 2 Definition 3.1.1. Let FH R be the sub-bundle consisting of the “H-frames” on R . 2 It consists of “first H-jet” at 0 ∈ R of the germs of local diffeomorphism φ ∈ G H 2 x = φ(0), y := (φ)∗,0 ∈ R × H.

We denote this identiﬁcation by (x, y) 'H φ. 20 2 2 The ﬂat connection on F R restricts to a connection form on FH R with values 2 in the lie algebra h ⊂ gl2(R ) of H,

ω = y−1dy ∈ h, y ∈ H, and the canonical form restricts to a vector-valued 1-form:

θ := (y−1dx) y ∈ H.

2 The fundamental vertical vector ﬁelds are restrictions of those on F R :

j µ j Yi = yi ∂µ , y ∈ H, and the basic horizontal vector ﬁelds are:

µ Xk = yk ∂µ , y ∈ H.

j j Note that when Yi are assembled into a matrix-valued vector ﬁeld, Y = (Yi ) takes

j i value in the Lie subalgebra h. Therefore, the indexes of Yi (ωj) must be chosen depending of the subgroup H to be non-zero vector ﬁelds (non-zero 1-forms).

2 2 Let GH := R o H denote the group of “aﬃne H-motions” of R , which are a subgroup of G. Note that

2 ∼ FH R = GH via (x, y) 'H Lx ◦ y .

In this section, we will use the notation (x, y) for elements in GH .

Proposition 3.1.2. The vector ﬁelds

j { Yi ,Xk | { (i, j) } ∈ S and k = 1, 2 } where S is a set of indexes chosen depending of the subgroup H, form a basis of left-

2 invariant vector ﬁelds on the group GH associated to the standard basis of gh = R oh.

21 Proof. Represent GH as the subgroup of GL3(R) consisting of the matrices   y x   a =   0 1

2 j with y ∈ H and x ∈ R . Let { Ei , ek | { i, j } ∈ IH and k = 1, 2 } be the standard 2 j basis of the Lie algebra gh = R oh and denote by { Eei , ek | { i, j } ∈ IH and k = 1, 2 } j the corresponding left-invariant vector ﬁelds. By deﬁnition, at the point a, Eei is tangent to the curve       tEj tEj y x e i 0 ye i 0       t →     =   0 1 0 1 0 1

j µ j and therefore coincides with Yi = yi ∂µ with y ∈ H, while the ek is tangent to the curve       y x 1 te y tye + x    k   k  t →     =   0 1 0 1 0 1

µ which is precisely Xk = yk ∂µ with y ∈ H.

The group G acts on GH by the following formula

H ϕe(x, y) := ( ϕ(x), (ϕ)∗,x · y ) , for ϕ ∈ G which is derived by computing the “ﬁrst H-jet” of the composition of ϕ ◦ φ where

(x, y) 'H φ , i.e. ,

ϕe(x, y) 'H ϕ ◦ φ .

Deﬁnition 3.1.3. Viewing here G as a discrete group, we form the crossed product algebra

∞ AH := Cc (GH ) o G.

As a vector space is spanned by monomials of the form

∗ ∞ ∗ ∗ fUϕ, where f ∈ Cc (GH ) and Uϕ stands for ( ϕe ) , 22 while the product is given by the multiplicative rule

f U ∗ · f U ∗ := f (f ◦ ϕ )U ∗ . 1 ϕ1 2 ϕ2 1 2 e1 ϕ2ϕ1

∗ We will also use Uϕ := Uϕ−1 to simplify notation afterwards.

Alternatively, AH can be regarded as the subalgebra of the endomorphism algebra

∞ End(Cc (GH )), generated by the multiplication and the translation operators

∞ Mf (ξ) = fξ, f, ξ ∈ Cc (GH ),

∗ ∗ ∞ Uϕ(ξ) = ( ϕe ) ξ = ξ ◦ ϕ,e ϕ ∈ G, ξ ∈ Cc (GH ).

We proceed to associate to the pseudogroup G a Hopf algebra HH , realized via its Hopf action on the crossed product algebra AH following the construction given by Moscovici and Rangipour [25].

One starts with a ﬁxed basis { Zr }r∈R for the Lie algebra gH of GH . Each Z ∈ gH is a left invariant vector ﬁeld Z on GH , which is then extended to a linear operator on AH ,

∗ ∗ ∞ Z(fUϕ) = Z(f)Uϕ, f ∈ Cc (GH ), ϕ ∈ G.

One has

∗ p Uϕ Zq Uϕ = Γ q(ϕ) Zp , p, q ∈ R.

p ∞ p with Γ q(ϕ) ∈ C (GH ). The matrix of functions Γ(ϕ) = Γ q(ϕ) automatically p,q∈R satisﬁes the cocycle identity

Γ(ϕ1 ◦ ϕ2) = Γ(ϕ1) ◦ ϕe2 · Γ(ϕ2) , ϕ1, ϕ2 ∈ G.

p We next denote by Ξ q the following multiplication operator on AH

p ∗ p ∗ Ξ q(fUϕ) = Γq(ϕ)fUϕ , p, q ∈ R.

p p −1 where Γq(ϕ) := Γ(ϕ) . q 23 Deﬁnition 3.1.4. Let HH be the unital subalgebra of linear operators on AH gener-

p ated by the operators Zr’s and Ξ q’s with p, q, r ∈ R.

In particular, HH contains all iterated commutators

p p Ξ q;r1···rs = [ Zrs , ··· [ Zr1 , Ξ q ] ··· ], which act as multiplication operators by the function

p p −1 Γq;r ...r (ϕ) := Zrs ··· Zr1 Γ(ϕ) ϕ ∈ G. 1 s q

To understand better the algebra HH , i.e. relations between elements, and to provide it with a Hopf algebra structure, ﬁrst we are going to give a precise expression for

• the operators Ξ •;•···•.

Note that for Z ∈ hH

−1 ∗ ( ϕg )∗(Z) = Uϕ ZUϕ , this implies that the matrix function Γ(ϕ) expressed in terms of a basis of gH is equal

−1 to the matrix function ( ϕg )∗ expressed in the same basis. Since the deﬁnition of

• −1 Ξ •;•···• depends on Γ(ϕ) we need to ﬁnd ( ϕe )∗ which is the transpose of the matrix ∗ ∗ function ( ϕe ) expressed in terms of a basis of gH . j Let us consider the basis { Z j = Y ,Zk = Xk } given by Proposition 3.1.2 as well (i ) i i k j as the canonical and connection forms { ωj , θ }. At each point of GH , { Yi ,Xk }

i k and { ωj , θ } form basis of the tangent and cotangent space, respectively, dual to each other:

j r j r r h Yi , ωs i = δs δi , h Xk , ωs i = 0,

j t t t h Yi , θ i = 0, h Xk , θ i = δk.

24 H If ( xe , ye ) = ( ϕ(x), (ϕ)∗,x · y ), then

−1 −1 H H ye d ye = (ϕ)∗,ϕ(x) · y d (ϕ)∗,x · y

−1 −1 H H H = y · (ϕ )∗,ϕ(x) d(ϕ)∗,x · y + (ϕ)∗,x · dy

−1 −1 H H k −1 = y · (ϕ )∗,ϕ(x) · Xk((ϕ)∗,x) · y θ + y d y , and

−1 −1 H ye d xe = (ϕ)∗,x · y d ϕ(x)

−1 −1 H = y · (ϕ )∗,ϕ(x) · (ϕ)∗,x dx

−1 −1 H −1 = y · (ϕ )∗,ϕ(x) · (ϕ)∗,x · y y dx .

−1 −1 H H −1 −1 H Note that the matrices y ·(ϕ )∗,ϕ(x)·Xk((ϕ)∗,x)·y and y ·(ϕ )∗,ϕ(x)·(ϕ)∗,x·y are upper triangular matrix with the latest having ones on its diagonal, i.e., unipotent.

i k This can be written more precisely in terms of the basis { ωj , θ } as  ( ϕ )∗(ω1) = ω1 + γ1 (ϕ) θ1 + γ1 (ϕ) θ2  e 1 1 11 12   ∗ 1 1 1 1 1 2  ( ϕ ) (ω ) = ω + γ (ϕ) θ + γ (ϕ) θ  e 2 2 21 22   ∗ 2 2 2 2 for H = P ( ϕe ) (ω2) = ω2 + γ22(ϕ) θ  ∗ 1 1  ( ϕ ) (θ ) = θ  e   ∗ 2 2  ( ϕe ) (θ ) = θ  and  ∗ 1 1 1 1 1 2 ( ϕe ) (ω1) = ω1 + γ11(ϕ) θ + γ12(ϕ) θ    ∗ 2 2 2 2  ( ϕe ) (ω2) = ω2 + γ22(ϕ) θ  for H = D ( ϕ )∗(θ1) = θ1 + γ1(ϕ) θ2  e 2   ∗ 2 2  ( ϕe ) (θ ) = θ 

25 with

i i −1 −1 H H γjk(ϕ)(x, y) := y · (ϕ )∗,ϕ(x) · Xk((ϕ)∗,x) · y j i λ −1 −1 H ρ = y (ϕ )∗,ϕ(x) XjXk(ϕ ) , λ ρ 1 1 −1 −1 H γ2 (ϕ)(x, y) := y · (ϕ )∗,ϕ(x) · (ϕ)∗,x · y 2 1 1 −1 −1 H 1 = y (ϕ )∗,ϕ(x) X2(ϕ ) , 1 1

j i (i ) 1 1 where γjk(ϕ) := Γ k (ϕ) and γ2 (ϕ) := Γ2(ϕ).

Remark 3.1.5. Therefore, we obtain

j • (Deﬁnition 3.1.4) The algebra HH is generated by the operators Z j = Y , (i ) i

Zk = Xk and

i ∗ i ∗ ξjk(fUϕ) := γjk(ϕ)fUϕ ,

1 ∗ 1 ∗ ξ2 (fUϕ) := γ2 (ϕ)fUϕ ,

j i (i ) 1 1 where ξjk := Ξ k and ξ2 := Ξ2.

• Note that for the other combination of valid indexes (depending on H) we have

j (i ) j i k k k Ξ (s) = δsδr Id , Ξ j = 0 Id , Ξ l = δl Id , r (i )

where Id is the identity operator.

i i i i • Furthermore, we have that ξjk = ξkj because γjk(ϕ) = γjk(ϕ).

j The change of notation was made to simplify the indexes of the form (i ) and because the order of the indexes j, k is immaterial.

With the notation of Deﬁnition 3.1.3 we have the following “Leibniz rules” which are important to prove relations between the iterated commutators. 26 Lemma 3.1.6. For any a, b ∈ A, one has

p Zq(ab) = Zq(a)b + Ξ q(a)Zp(b) ,

p p µ Ξ q(ab) = Ξ µ(a)Ξ q (b) .

∗ ∗ Proof. Let a = f1Uϕ1 and b = f2Uϕ2 , one has Z (ab) = Z (f U ∗ f U ∗ ) = Z f (f ◦ ϕ ) U ∗ q q 1 ϕ1 2 ϕ2 q 1 2 e1 ϕ2◦ϕ1 = Z (f )(f ◦ ϕ )U ∗ + f Z (f ◦ ϕ )U ∗ q 1 2 e1 ϕ2◦ϕ1 1 q 2 e1 ϕ2◦ϕ1 = Z (f )U ∗ f U ∗ + f U ∗ Z (f ◦ ϕ ) ◦ ϕg−1 U ∗ q 1 ϕ1 2 ϕ2 1 ϕ1 q 2 e1 1 ϕ2

∗ ∗ ∗ ∗ ∗ = Zq(f1)Uϕ1 f2Uϕ2 + f1Uϕ1 Uϕ1 ZqUϕ1 (f2) Uϕ2

∗ ∗ ∗ p −1 ∗ = Zq(f1)Uϕ1 f2Uϕ2 + f1Uϕ1 Γq(ϕ1 )Zp(f2) Uϕ2

= Z (f )U ∗ f U ∗ + f Γp(ϕ−1) ◦ ϕ U ∗ Z (f )U ∗ q 1 ϕ1 2 ϕ2 1 q 1 e1 ϕ1 p 2 ϕ2 p ∗ ∗ −1 ∗ ∗ = Zq(f1)Uϕ f2Uϕ + f1 Γ(ϕ1) Uϕ Zp(f2)Uϕ 1 2 q 1 2

p = Zq(a)b + Ξ q(a)Zp(b) .

p by using the deﬁnition of Γq(ϕ) in line 5 and the cocycle identity in line 7. We also have

p p p ∗ ∗ −1 ∗ Ξ q(ab) = Ξ q(f1Uϕ f2Uϕ ) = Γ(ϕ2 ◦ ϕ1) f1(f2 ◦ ϕ1)Uϕ ◦ϕ 1 2 q e 2 1 p −1 ∗ = Γ(ϕ2) ◦ ϕ1 · Γ(ϕ1) f1(f2 ◦ ϕ1)Uϕ ◦ϕ e q e 2 1 p −1 ∗ = Γ(ϕ1) · Γ(ϕ2) ◦ ϕ1 f1(f2 ◦ ϕ1)Uϕ ◦ϕ e q e 2 1 p µ −1 ∗ = Γ(ϕ1) f1 Γ(ϕ2) ◦ ϕ1 (f2 ◦ ϕ1)Uϕ ◦ϕ µ e q e 2 1 p µ −1 ∗ ∗ = Γ(ϕ1) f1Uϕ Γ(ϕ2) f2Uϕ µ 1 q 2

p µ = Ξ µ(a)Ξ q (b) . where we are using the cocycle identity in line 2.

27 • Proposition 3.1.7. The operators Ξ •;• satisfy the structure constants identities:

p p p λ µ ν p Ξ q;r − Ξ r;q = cλµ Ξ q Ξ r − cqr Ξ ν ,

ν where cqr are the structure constants of g,

ν [ Zq,Zr ] = cqr Zν .

Proof. Applying the ﬁrst “Leibniz rule” from Lemma 3.1.6, one has for any a, b ∈ AH ,

µ ZqZr(ab) = Zq Zr(a)b + Ξ r (a)Zµ(b)

λ µ λ µ = Zq(Zr(a))b + Ξ q (Zr(a))Zλ(b) + Zq(Ξ r (a))Zµ(b) + Ξ q (Ξ r (a))Zλ(Zµ(b)) , and thus the commutator can be expressed as follows

λ µ [ Zq,Zr ](ab) = [ Zq,Zr ](a)b + Ξ q (Zr(a))Zλ(b) − Ξ r (Zq(a))Zµ(b)

µ λ + Zq(Ξ r (a))Zµ(b) − Zr(Ξ q (a))Zλ(b)

λ µ µ λ + Ξ q (Ξ r (a))Zλ(Zµ(b)) − Ξ r (Ξ q (a))Zµ(Zλ(b))

p p λ µ p = [ Zq,Zr ](a)b − Ξ q,r(a) − Ξ r,q(a) Zp(b) + (Ξ q Ξ r )(a)cλµZp(b) .

ν On the other hand, by using [ Zq,Zr ] = cqr Zν, the left hand side equals,

ν ν ν p ν p cqr Zν = cqr Zν(a)b + cqr Ξ ν(a)Zp(b) = [ Zq,Zr ](a)b + cqr Ξ ν(a)Zp(b) .

Equating the two expressions one obtains after cancelation

p p p λ µ ν p − Ξ q,r(a) − Ξ r,q(a) Zp(b) + cλµ(Ξ q Ξ r )(a)Zp(b) = cqr Ξ ν(a)Zp(b) .

Since a, b ∈ AH are arbitrary and the Zp’s are linearly independent, this gives the claimed identity.

j Now, using the basis { Z j = Y ,Zk = Xk } and the notation from Remark 3.1.5, (i ) i we rewrite the previous Lemma and Proposition.

28 (Lemma 3.1.6) For any a, b ∈ AH we have the following expressions:

 Y 1(ab) = Y 1(a)b + aY 1(b)  1 1 1    Y 2(ab) = Y 2(a)b + aY 2(b)  1 1 1   2 2 2  Y2 (ab) = Y2 (a)b + aY2 (b)    1 2 1 1  X1(ab) = X1(a)b + aX1(b) + ξ (a)Y (b) + ξ (a)Y (b)  21 1 11 1   2 2 1 2 1 1  X2(ab) = X2(a)b + aX2(b) + ξ22(a)Y2 (b) + ξ22(a)Y1 (b) + ξ12(a)Y1 (b)  for H = P 1 1 1  ξ11(ab) = ξ11(a)b + aξ11(b)    ξ1 (ab) = ξ1 (a)b + aξ1 (b)  12 12 12    ξ1 (ab) = ξ1 (a)b + aξ1 (b)  21 21 21   1 1 1  ξ22(ab) = ξ22(a)b + aξ22(b)    2 2 2  ξ22(ab) = ξ22(a)b + aξ22(b) 

 Y 1(ab) = Y 1(a)b + aY 1(b)  1 1 1   2 2 2  Y2 (ab) = Y2 (a)b + aY2 (b)    1 1  X1(ab) = X1(a)b + aX1(b) + ξ (a)Y (b)  11 1   1 2 2 1 1  X2(ab) = X2(a)b + aX2(b) + ξ2 (a)X1(b) + ξ22(a)Y2 (b) + ξ12(a)Y1 (b)  for H = D 1 1 1  ξ11(ab) = ξ11(a)b + aξ11(b)    ξ1 (ab) = ξ1 (a)b + aξ1 (b) + ξ1 (a)ξ1(b)  12 12 12 11 2    ξ2 (ab) = ξ2 (a)b + aξ2 (b)  22 22 22   1 1 1  ξ2 (ab) = ξ2 (a)b + aξ2 (b) 

29 j (Proposition 3.1.7) The basis { Z j = Y ,Zk = Xk } satisfy the following com- (i ) i mutation relations

j s j s s j j j [Yi ,Yr ] = δrYi − δi Yr , [Yi ,Xk] = δk Xi , [Xk,Xl] = 0 , where we are using just valid indexes depending on H. Therefore, the structure constant identities are given explicitly by

µ w q Case 1: Top index (λ) and bottom indexes (v ) and (p).

µ µ µ µ (λ) (λ) (λ) r s l (λ) Ξ w q − Ξ q w = crs Ξ w Ξ q − c w q Ξ , (v );(p) (p);(v ) (v ) (p) (v )(p) l reduce to µ µ (λ) (λ) 0 = c w q − c w q . (v )(p) (v )(p)

µ q Case 2: Top index (λ) and bottom indexes v and (p).

µ µ µ µ (λ) (λ) (λ) r s l (λ) Ξ q − Ξ q = crs Ξ Ξ q − c q Ξ , v;(p) (p);v v (p) v(p) l reduce to

q λ q λ λ q q λ [Yp , ξµv] = δµ ξpv − δp ξµv + δv ξµp .

µ Case 3: Top index (λ) and bottom indexes v and p.

µ µ µ µ (λ) (λ) (λ) r s l (λ) Ξ v;p − Ξ p;v = crs Ξ v Ξ p − cvp Ξ l , reduce to

λ λ s λ s λ [Xp, ξµv] − [Xv, ξµp] = ξµp ξsv − ξµv ξsp .

w q Case 4: Top index λ and bottom indexes (v ) and (p).

λ λ λ r s l λ Ξ w q − Ξ q w = c Ξ w Ξ q − c w q Ξ , (v );(p) (p);(v ) rs (v ) (p) (v )(p) l reduce to

0 = 0 30 q Case 5: Top index λ and bottom indexes v and (p).

λ λ λ r s l λ Ξ q − Ξ q = c Ξ Ξ q − c q Ξ , v;(p) (p);v rs v (p) v(p) l reduce to

0 = 0 for H = P,

q 1 q p 1 q p 1 [Yp , ξ2 ] = −δ1 δ1 ξ2 + δ2 δ2 ξ2 for H = D.

Case 6: Top index λ and bottom indexes v and p.

λ λ λ r s l λ Ξ v;p − Ξ p;v = crs Ξ v Ξ p − cvp Ξ l , reduce to

0 = 0 for H = P,

1 1 1 1 [X1, ξ2 ] = ξ12 − ξ2 ξ11 and 0 = 0 for H = D.

Remark 3.1.8. The previous non-trivial relations give us the following information:

q µ 1 • Since the commutators of Yp with ξλv and ξ2 can be written as a linear combination of generators (Case 2 and 5) and because of the Jacobi identity is true

for the commutator of operators, the only possible new operators on HH are

(j ) ξi := Ξ i = [X ,... [X , ξi ] ... ] , jkr1···rs k;r1···rs rs r1 jk

1 1 1 ξ2r1···rs := Ξ 2;r1···rs = [Xrs ,... [Xr1 , ξ2 ] ... ] ,

which are the multiplication operators by the function

i i γjkr1··· ,rs := Xrs ...Xr1 (γjk) ,

1 1 γ2r1··· ,rs := Xrs ...Xr1 (γ2 ) .

1 Note that γ2•···• is deﬁned in a diﬀerent way depending on the subgroup H. 31 • The order of the last s-indexes is immaterial for these operators because the

Xk’s commute and the Jacobi identity.

λ λ 1 • The operators ξµvp − ξµpv and ξ21 can be written as a sum of products of generators (Case 3 and 6).

In view of Remark 3.1.5 and 3.1.8, the algebra HH admits a basis as a vector space (similar to the Poincar´e-Birkhoﬀ-Witt basis of a universal enveloping algebra).

The notation needed to specify such a basis involves two kind of multi-indices. The

ﬁrst kind are of the form ( ) j1 jq I = k1 ≤ · · · ≤ kp; ≤ · · · ≤ , i1 iq and the second kind are of the form ( ) it K = { κ1 ≤ · · · ≤ κs } , where κt = . t t jt ≤ kt ≤ r1 ≤ · · · ≤ rpt

In both cases, the indexes jλ and iλ depend on the group H and the inner multi- indices are ordered lexicographically top to bottom. We then denote

Z := X ··· X Y j1 ··· Y jq , I k1 kp i1 iq

i1 is ξ := ξ ··· ξ s s . K j k r1···r1 jsksr ···r 1 1 1 p1 1 ps

Proposition 3.1.9. The monomials ξK ZI form a linear basis of HH as a vector space.

Before we go to the proof of this Proposition, we are going to provide some deﬁnitions and lemmas which will be useful not just for this proof but also for next chapter.

H Deﬁnition 3.1.10. Let NH := {ψ ∈ G | ψ(0) = 0 and (ψ)∗,0 = Id} and let

p p αq ···q (ψ) := ∂xq ··· ∂xq (ψ ) for ψ ∈ NH , 1 s s 1 x=0 32 • where the indexes are taken such that α•···• are non-trivial functions on NH . The algebra of functions on NH generated by these functions will be denoted by FH . Fur-

p thermore, we say that the length of αq1···qs is s.

Note that these functions are multiples of the non-trivial coeﬃcients of the Taylor expansion of ψ at 0 and they are symmetric in the lower indexes but otherwise arbitrary.

Lemma 3.1.11. The monomials αK form a basis for FH as a vector space.

Proof. Suppose that we have cK ∈ C such that

X cK αK (ψ) = 0 , for all ψ ∈ NH , K

• we need to prove that cK = 0 for any K. Let l be the largest length of the α•···• on all αK ’s. For this l consider all ψ ∈ NH such that the taylor coeﬃcients with size greater than l are zero, and let

α p := αp (ψ) ∈ , e q1···qs q1···qs C for all s ≤ l.

For example, if l = 2 and H = D then

1 1 1 1 1 ψ (x1, x2) = x1 + αe 2 x2 + αe 11 x1x1 + αe 12 x1x2 + αe 22 x2x2

2 2 ψ (x1, x2) = x2 + αe 22 x2x2 ,

By evaluating at this ψ, one obtains

X cK αeK = 0 K where the left hand side is a polynomial with variables α p . This implies that e q1···qs cK = 0 for any K, because it is the only way to obtain the zero polynomial.

33 Deﬁnition 3.1.12. For any ψ ∈ NH , deﬁne the functions

p p ηq1···qs (ψ) := γq1q2q3···qs (ψ)(0, Id) ,

• where the indexes are taken such that η•···• are non-trivial functions on NH .

Lemma 3.1.13. The monomials ηK form a basis for FH as a vector space.

i 1 Proof. First, by evaluating the explicit formulas for γjk(ψ) and γ2 (ψ) for ψ ∈ NH at the point (0, Id) ∈ GH , we obtain

i i −1 H ρ γjk(ψ)(0, Id) = (ψ )∗,ψ(x) ∂xj ∂xk ψ , ρ x=0 1 1 −1 H 1 γ2 (ψ)(0, Id) = (ψ )∗,ψ(x) ∂x2 ψ . 1 x=0

H −1 H Furthermore, since ψ(0) = 0 and (ψ)∗,0 = Id we get (ψ )∗,0 = Id, thus

i i i ηjk(ψ) = γjk(ψ)(0, Id) = αjk(ψ)

1 1 1 η2(ψ) = γ2 (ψ)(0, Id) = α2(ψ) where the indexes are taking depending on the subgroup H.

Next, one has

h i i i −1 H ρ γjkr ···r (ψ)(0, Id) = ∂xr ··· ∂xr (ψ )∗,ψ(x) ∂xj ∂xk ψ 1 s s 1 ρ x=0 h 1 i 1 −1 H 1 γ2r ···r (ψ)(0, Id) = ∂xr ··· ∂xr (ψ )∗,ψ(x) ∂x2 ψ 1 s s 1 1 x=0

34 2 note that γ2···2 is diﬀerent depending on H. The product rule implies

h i i −1 H ρ ∂xr ··· ∂xr (ψ )∗,ψ(x) ∂xj ∂xk ψ s 1 ρ x=0 i −1 H ρ = ∂xr ··· ∂xr (ψ )∗,ψ(x) ∂xj ∂xk ψ s 1 ρ x=0 x=0 + ...... i −1 H ρ + (ψ )∗,ψ(x) ∂xr ··· ∂xr ∂xj ∂xk ψ ρ x=0 s 1 x=0 h 1 i −1 H 1 ∂xr ··· ∂xr (ψ )∗,ψ(x) ∂x2 ψ s 1 1 x=0 1 −1 H 1 = ∂xr ··· ∂xr (ψ )∗,ψ(x) ∂x2 ψ s 1 1 x=0 x=0 + ...... 1 −1 H 1 + (ψ )∗,ψ(x) ∂xr ··· ∂xr ∂x2 ψ 1 x=0 s 1 x=0 i −1 H where the summands at the middle are product of higher derivatives of (ψ )∗,ψ(x) ρ ρ 1 and higher derivatives of ∂xj ∂xk ψ or ∂x2 ψ all of them evaluated at x = 0 and with the number of derivatives taken less or equal than s. −1 H H Furthermore, by diﬀerentiating Id = (ψ)∗,x (ψ)∗,x and using induction, the −1 −1 H H higher derivatives of (ψ )∗,ψ(x) = (ψ)∗,x can be expressed as product of higher

H derivatives of (ψ)∗,x . For example:

−1 h i h H H i 0 = ∂xr Id = ∂xr (ψ)∗,x (ψ)∗,x −1 −1 h H i H H h H i = ∂xr (ψ)∗,x (ψ)∗,x + (ψ)∗,x ∂xr (ψ)∗,x .

H After evaluating at x = 0 and using that (ψ)∗,0 = Id, we obtain

−1 h H i h H i ∂xr (ψ)∗,0 = −∂xr (ψ)∗,0 , which implies that

i i i ν ηjkr(ψ) = αjkr(ψ) − αrν(ψ)αjk(ψ)

35 In the general case, we have proved that

p p p λ λ ηq1···qs = αq1···qs + Pq1···qs (αµ, ··· , αµ1···µs−1 )

p where Pq1···qs is a polynomial. This implies that

X ηK = αK + cK0 αK0 K0

0 λ where K are made of αµ1···µs of length strictly less than the ones on K. λ Therefore, the transformations sending αK to ηK , with αµ1···µs of length less or equal than a fixed s, will be invertible linear maps sending basis to basis on these finite subspaces of FH . This proves that the monomials ηK are also a basis for FH because we are always using finite linear combination and we can pick a finite subspace with

λ a ﬁxed maximum length for αµ1···µs to check the independence of them.

Next, we prove Proposition 3.1.9:

Proof. We need to prove that if cI,K ∈ C are such that

X cI,K ξK ZI (a) = 0 , for all a ∈ AH , I,K then cI,K = 0 for any (I,K). To this end, we evaluate the previous expression on all elements of the form

∗ a = fUϕ ∈ AH at the point (x, y) ∈ GH . In particular, for any ﬁxed but arbitrary ϕ ∈ G, one obtains: ! X X ∞ cI,K γK (ϕ)(x, y) (ZI f)(x, y) = 0 , for all f ∈ Cc (GH ) , I K

Since the ZI ’s form a PBW basis for U(gh), which can be viewed as the algebra of left invariant diﬀerential operators on GH , the validity of the previous equality for

36 ∞ any f ∈ Cc (GH ) implies the vanishing for each I of the corresponding coeﬃcient. One therefore obtains, for any ﬁxed I,

X cI,K γK (ϕ)(x, y) = 0 , for all ϕ ∈ G . K

By using ψ ∈ NH and the identity element (0, Id) ∈ GH , we get

X cI,K ηK (ψ) = 0 , K which by Lemma 3.1.13 implies cI,K = 0.

1 p ⊗p We can associate to any element h ⊗ · · · ⊗ h ∈ HH a multi-diﬀerential operator, ⊗p acting on AH , by the following formula

T (h1 ⊗ · · · ⊗ hp)(a1 ⊗ · · · ⊗ ap) := h1(a1) ··· hp(ap),

1 p 1 p where h , . . . , h ∈ HH and a , . . . , a ∈ AH , the linear extension of this assignment will be denoted by the same letter.

Proposition 3.1.14. For each p ∈ N, the linear transformation

⊗p ⊗p T : HH → L(AH , AH ) is injective.

Proof. For p = 1, T gives the standard action of HH on AH , which was just shown to be faithful. To prove that Ker T = 0 for an arbitrary p ∈ N, assume that

X 1 p H = hρ ⊗ · · · ⊗ hρ ∈ Ker T ρ

j By using the basis as above, we may uniquely express each hρ in the form

j X hρ = Cρ,Ij ,Kj ξKj ZIj . Ij ,Kj 37 ∗ ∗ Evaluating T (H) on elementary tensors of the form f1Uϕ1 ⊗ · · · ⊗ fpUϕp , one obtains

X ∗ ∗ Cρ,I1,K1 ··· Cρ,Ip,Kp ξK1 (ZI1 (f1)Uϕ1 ) ··· ξKp (ZIp (fp)Uϕp ) = 0 . ρ,I,K

Evaluating further at a point u1 = (x, y) ∈ GH , and denoting

u2 := ϕe(u1) , . . . , up := ϕep−1(up−1) the above identity gives

X Cρ,I1,K1 ··· Cρ,Ip,Kp · γK1 (ϕ1)(u1) ··· γKp (ϕp)(up) ρ,I,K

· ZI1 (f1)(u1) ··· ZIp (fp)(up) = 0 .

Since ϕj are arbitrary then the points uj can be chosen arbitrarily, and using again the fact that ZI form a PBW basis for U(gh), we can infer that for each p-tuple of indices of the ﬁrst kind (I1,...,Ip) one has

X Cρ,I1,K1 ··· Cρ,Ip,Kp · γK1 (ϕ1)(u1) ··· γKp (ϕp)(up) = 0 . ρ,K

Finally, by taking all points xj = (0, Id) and ϕk = ψ ∈ NH we have

X Cρ,I1,K1 ··· Cρ,Ip,Kp · ηK1 (ψ) ··· ηKp (ψ) = 0 . ρ,K which implies that for any (K1,...,Kp)

X Cρ,I1,K1 ··· Cρ,Ip,Kp = 0 . ρ This proves H = 0.

∞ Deﬁnition 3.1.15. The algebra AH = Cc (GH ) o G carries a canonical trace, uniquely determined up to a scaling factor. It is deﬁned as the linear functional

τ : AH → C, Z   f ω, if ϕ = Id, ∗ G τ fUϕ = H  0, otherwise. 38 i k j Here ω is the volume form attached to the dual basis { ωj , θ } of { Yi ,Xk } on GH

^ i ^ k ω := ωj ∧ θ . (i,j) k=1,2 The tracial property

τ(ab) = τ(ba), ∀ a, b ∈ AH . is a consequence of the G- invariance of the volume form ω. In turn, the later follows from the fact that

∗ i i i k ∗ k k X p q ( ϕe ) (ωj) = ωj + γj,k(ϕ)θ and ( ϕe ) (θ ) = θ + γq (ϕ) θ q>k indeed, ! ! ∗ ^ i i k ^ k X p q ( ϕe ) (ω) = ωj + γj,k(ϕ)θ ∧ θ + γq (ϕ) θ (i,j) k=1,2 q>k

^ i ^ k = ωj ∧ θ = ω . (i,j) k=1,2 Therefore, we have

Z Z ∗ ∗ −1 ∗ τ f1Uϕ · f2Uϕ−1 = f1(f2 ◦ ϕg) ω = (f1 ◦ ϕe)f2 ( ϕe ) (ω) GH GH Z ∗ ∗ = f2(f1 ◦ ϕe) ω = τ f2Uϕ−1 · f1Uϕ , GH which proves the tracial property for the only non-trivial case.

Lemma 3.1.16. The pairing (a, b) → τ(ab) is non-degenerate.

Proof. Because of the deﬁnition and the tracial property of τ, it is enough to show that Z f1( f2 ◦ ϕe) ω = 0 for all f2 implies f1 = 0, GH which is the same that Z f1f2 ω = 0 for all f2, implies f1 = 0 . GH 39 Z If we assume f1f2 ω = 0 for all f2, but f1(g) 6= 0 at some g ∈ GH , without loss of GH generality suppose f1(g) > 0. Then there exits some small neighborhood U of g such that f1 is strictly positive, say f > c > 0 on U. We can choose a compact set V1 ⊂ U and a smaller compact set V2 $ V1. Take f2 to be a smooth cutoﬀ function which is the identity on V2 and vanish on V1, then Z Z Z Z f1f2 = f1f2 ω > f1f2 $ > c $ > 0 , GH V1 V2 V2 hence a contradiction.

This trace satisﬁes an invariant property relative to a character of HH which is deﬁned as follows:

Definition 3.1.17. The infinitesimal trace character δ(Z) = Tr(ad(Z)), Z ∈ gh, extends uniquely to a character δ : HH → C. It is defined on the algebra generators as follows:

j j i 1 δ(Yi ) := δi , δ(Xk) = 0 , δ(ξjk) := 0 , δ(ξ2 ) := 0 ,

This deﬁnition is compatible with all the relations between the generators of HH and therefore extends to a character of the algebra HH .

Proposition 3.1.18. The trace τ is Hn-invariant relative to the character δ, i.e.

τ( h(a) ) = δ( h )τ( a ), a ∈ AH , h ∈ HH .

Proof. It suﬃces to verify the stated identity when ϕ = Id on the generators of HH .

40 If h = Z ∈ gh, it is just the restatement of the invariance property of the left Haar measure on GH with respect to right translation at the level of the Lie algebra, i.e. Z ∗ ∗ τ Z(fUId) = τ Z(f)UId = Z(f) ω GH Z d = f((∗) exp(tZ)) ω dt GH t=0 Z d = f((∗) exp(tZ)) ω dt t=0 GH Z d = f det(Ad(exp(tZ))) ω dt t=0 GH Z d = det(Ad(exp(tZ))) f ω dt t=0 GH Z ∗ = Tr(ad(Z)) f ω = δ(Z)τ fUId . GH

i On the other hand, if h = ξjk we have

i ∗ i ∗ ∗ τ ξjk(fUId) = τ γjk(Id)fUId = τ 0 · fUId

∗ i ∗ = 0 = 0 · τ fUId = δ( ξjk) τ fUId .

1 Similarly if h = ξ2 when H = D.

Proposition 3.1.19. There exist a unique anti-automorphism Sδ : HH → HH such that

τ( h(a)b ) = τ( aSδ(h)(b)) , for any h ∈ HH and a, b ∈ AH . Moreover, Sδ is involutive:

2 Sδ = Id .

Proof. The idea behind the proof of this Proposition is to use the diﬀerent types of

“Leibniz rule” (Lemma 3.1.6) and the invariance property (Proposition 3.1.18).

i 1 First, let us consider the generators h = ξj,k or h = ξ2 which satisfy

h(ab) = h(a)b + ah(b) 41 by using the invariance property, one obtains

0 = δ( h )τ( ab ) = τ( h(ab) ) = τ( h(a)b ) + τ( ah(b)) , which implies

τ( h(a)b ) = −τ( ah(b)) .

1 Note that for H = D, the generator ξ12 satisﬁes

1 1 1 1 1 ξ12(ab) = ξ12(a)b + aξ12(b) + ξ11(a)ξ2 (b) , and for this case we get

1 1 0 = δ( ξ12 )τ( ab ) = τ( ξ12(ab))

1 1 1 1 = τ( ξ12(a)b ) + τ( aξ12(b) ) + τ(ξ11(a)ξ2 (b))

1 1 1 1 = τ( ξ12(a)b ) + τ( aξ12(b)) − τ(aξ11ξ2 (b))

1 1 where the last equality is obtain by using that τ( ξ11(a)b ) = −τ( aξ11(b) ), then

1 1 1 1 τ( ξ12(a)b ) = −τ( aξ12(b) ) + τ(aξ11ξ2 (b)) .

j Second, let us consider the generators Yi which satisfy

j j j Yi (ab) = Yi (a)b + aYi (b) , for this generators we have

j j j j j δi τ( ab ) = δ( Yi )τ( ab ) = τ( Yi (ab) ) = τ( Yi (a)b ) + τ( aYi (b)) , thus

j j j τ( Yi (a)b ) = −τ( aYi (b) ) + δi τ( ab ) .

Finally, for the operators Xk satisfying

i j Xk(ab) = Xk(a)b + aXkb + ξjk(a)Yi (b) 42 i the invariance property and the identities for ξjk give

0 = δ( Xk )τ( ab ) = τ( Xk(ab))

i j = τ( Xk(a)b ) + τ( aXk(b) ) + τ(ξjk(a)Yi (b))

i j = τ( Xk(a)b ) + τ( aXk(b)) − τ(aξjkYi (b)) , which can be written as

i j τ( Xk(a)b ) = −τ( aXk(b) ) + τ(aξj,kYi (b))

Note that for H = D, the generator X2 satisﬁes

1 2 2 1 1 X2(ab) = X2(a)b + aX2(b) + ξ2 (a)X1(b) + ξ22(a)Y2 (b) + ξ12(a)Y1 (b) , and by following the same procedure as before, for this case we have

0 = δ( X2 )τ( ab ) = τ( X2(ab))

1 = τ( X2(a)b ) + τ( aX2(b) ) + τ(ξ2 (a)X1(b))

2 2 1 1 + τ(ξ22(a)Y2 (b)) + τ(ξ12(a)Y1 (b))

1 = τ( X2(a)b ) + τ( aX2(b)) − τ(aξ2 X1(b))

2 2 1 1 1 1 1 − τ(aξ22Y2 (b)) − τ(aξ12Y1 (b)) + τ(aξ11ξ2 Y1 (b)) , therefore

1 τ( X2(a)b ) = − τ( aX2(b) ) + τ(aξ2 X1(b))

2 2 1 1 1 1 1 + τ(aξ22Y2 (b)) + τ(aξ12Y1 (b)) − τ(aξ11ξ2 Y1 (b)) .

Thus the generators of HH satisﬁes identities of the form τ( h(a)b ) = τ( aSδ(h)(b) ). By Lemma 3.1.16, this operators are uniquely determined.

This rule, being multiplicative, extends from generators to all elements h ∈ HH , and uniquely deﬁnes a map Sδ : HH → HH satisfying the identity form our Proposi- tion. In turn, this identity implies that Sδ is anti-automorphism as well as the fact that Sδ is involutive. 43 Finally, we can equip HH with a canonical Hopf structure.

Theorem 3.1.20. There exists a unique Hopf algebra structure on HH with respect to which AH is a left HH -module algebra.

Proof. The “Leibniz rules” given by Lemma 3.1.6 extend multiplicatively to any element h ∈ HH

h(ab) = h(1)(a)h(2)(b), h(1), h(2) ∈ HH , a, b ∈ AH , where we are using Sweedler notation.

By Proposition 3.1.14, this property uniquely determines the coproduct map

∆ : HH → HH ⊗ HH , ∆(h) := h(1) ⊗ h(2), that satisﬁes

T (∆h)(a ⊗ b) = h(ab).

Furthermore, the coassociativity of ∆ becomes a consequence of the associativity of

AH , because after applying T it amounts to the identity

h((ab)c) = h(a(bc)), h ∈ HH , a, b, c ∈ AH .

Similarly, the property that ∆ is an algebra homomorphism follows from the fact that

AH is a left HH -module. By the very definition of the coproduct, AH is actually a left HH -module algebra (see Definition §4.1). The counit is defined by

(h) = h(1) , when transported via T , its required properties amount to the obvious identities

h(a1) = h(1a) = h(a), h ∈ HH , a ∈ AH .

44 It remains to show the existence of the antipode. We ﬁrst check that the anti- automorphism Sδ is a twisted antipode, i.e., satisﬁes for any h ∈ HH ,

Sδ(h(1))h(2) = δ(h)1 = h(1)Sδ(h(2)) .

Indeed, with a, b ∈ AH arbitrary, one has,

τ aδ(h)b = δ(h)τ(ab) = τ h(ab) = τ h(1)(a)h(2)(b) = τ a Sδ(h(1))h(2) (b) which proves the left hand side equality. For the right hand side is checked case by case on generators, and hence for any h ∈ Hn. ˇ ∗ ∗ Now, let δ ∈ HH denote the convolution inverse of the character δ ∈ HH , which on generators is given by

ˇ j j ˇ ˇ i ˇ 1 δ(Yi ) := −δi , δ(Xk) = 0 , δ(ξjk) := 0 , δ(ξ2 ) := 0 ,

ˇ Then S(h) := δH (h(1))Sδ(h(2)) is an algebra anti-homomorphism which satisﬁes the antipode requirement

S(h(1))h(2) = (h)1 = h(1)S(h(2)) on the generators, and hence for any h ∈ Hn.

45 CHAPTER 4

BICROSSED PRODUCT REALIZATION.

The Hopf algebra HH can be reconstructed as a bicrossed product Hopf algebra [24]. This structure arises naturally from the canonical splitting of the group G as a

2 set-theoretical product G = GH · NH of the group GH of “aﬃne H-motions” of R and the group NH of elements of G with trivial “ﬁrst H-jet”.

4.1 The bicrossed product Hopf algebra FH IC U(gh).

We recall below the basic notions concerning the bicrossed product construction.

Let U be a Hopf algebra. Suppose an algebra A is a left U-module with action

. : U ⊗ A → A where . (u, a) = u . a, then A is a left H-module algebra if it satisﬁes the following conditions

u . (ab) = (u(1) . a)(u(2) . b),

u . (1A) = ε(u)1A.

Then we can form a left cross product algebra A>CU built on the vector space A⊗U with the product structure

(a>Cu)(b>Cv) = a(u(1) . b)>Cu(2)v, a, b ∈ A, u, v ∈ U

1A>C U = 1A ⊗ 1U .

46 Let F be a Hopf algebra. Suppose a coalgebra C is a right F-comodule with coaction

H : C → C ⊗ F where H(c) = c<0> ⊗ c<1>, then C is a right F−comodule coalgebra if it satisﬁes the following conditions

c<0>(1) ⊗ c<0>(2) ⊗ c<1> = c(1)<0> ⊗ c(2)<0> ⊗ c(1)<1>c(2)<1>

ε(c<0>)c<1> = ε(c)1F .

Then we can form a right cross coproduct coalgebra F I

∆(f I ⊗ f(2)c(1)<1> I

ε(f I

Deﬁnition 4.1.1. Let F, U be Hopf algebras, let F be a left U-module algebra and

U be a right F-comodule coalgebra such that the compatibility conditions

ε(u . f) = ε(u)ε(f) ,

∆(u . f) = u(1)<0> . f(1) ⊗ u(1)<1>(u(2) . f(2)) ,

H(1) = 1 ⊗ 1,

H(uv) = u(1)<0>v<0> ⊗ u(1)<1>(u(2) . v<1>) ,

u(2)<0> ⊗ (u(1) . f)u(2)<1> = u(1)<0> ⊗ u(1)<1>(u(2) . f) , are satisﬁed for all u, v ∈ U and f ∈ F.

Then F and U are said to form a matched pair, and the algebra F >C U and coalgebra F I

SIC(f ICu) = 1ICS(u<0>) S(fu<1>)IC1 .

47 The matched pair of Hopf Algebras constructed on this section arise form a matched pair of groups, via a splitting ´ala G.I. Kac [19].

Proposition 4.1.2. Given any diﬀeomorphism ϕ ∈ G, there is a unique φ ∈ GH ,

2 −1 the group of aﬃne “H-motions” of R , such that ψ = φ ◦ ϕ has the properties

H H 2 H 2 ψ(0) = 0 and (ψ)∗,0 = Id : T0 R → T0 R .

i.e. ψ ∈ NH (Deﬁnition 3.1.10) the group of elements of G with trivial “ﬁrst H-jet”.

Proof. Let Lx be the translation by x = ϕ(0) and let

−1 H H 2 H 2 y := (Lx ◦ ϕ)∗,0 : T0 R → T0 R .

Deﬁne

φ := Lx ◦ y ∈ GH .

Then ψ fulﬁlls the required properties. Furthermore, we have that GH ∩ NH = ∅ which implies the uniqueness of φ and ψ.

Proposition 4.1.2 gives a decomposition

G = GH · NH .

By reversing the order in the decomposition G = GH · NH one simultaneously obtains two well-deﬁned operations, of NH on GH and of GH on NH :

ψ ◦ φ = (ψ . φ) ◦ (ψ / φ), for φ ∈ GH and ψ ∈ NH .

Proposition 4.1.3. The operation . is a left action of NH to GH , and / is a right action of GH on NH . Both actions leave the neutral element ﬁxed.

Proof. This Proposition is true for any decomposition for a group and a complete proof is given in [25]. 48 2 ∼ Via the identiﬁcation FH R = GH , one can recognize the action . as the usual action of diﬀeomorphisms on the H-frame bundle.

Lemma 4.1.4. The left action . of NH on GH coincides with the left action of G on GH given by φe(x, y) for φ ∈ G and (x, y) ∈ GH (Section §2).

Proof. Let ψ ∈ NH and φ ∈ GH . Since ψ / φ ∈ NH , then

H ψ / φ(0) = 0 and (ψ / φ)∗,0 = Id .

Therefore, by the deﬁnition of the actions we obtain

ψ . φ(0) = (ψ . φ) ◦ (ψ / φ)(0) = ψ ◦ φ(0) and

H H H H (ψ . φ)∗,0 = (ψ . φ)∗,ψ/φ(0) ◦ (ψ / φ)∗,0 = (ψ ◦ φ)∗,0 .

These equalities prove that

H H ψ . φ(0), (ψ . φ)∗,0 = ψ(x), (ψ)∗,x · y .

Thus ψe(x, y) 'H ψ . φ where (x, y) 'H φ.

In this dissertation, the universal enveloping algebra U := U(gh) and the algebra of functions F := FH (Deﬁnition 3.1.10) are the candidates to form a matched pair of Hopf Algebras, where the module/comodule structures will be based on the previous decomposition.

The Hopf algebra structure of U(gh) is the standard structure for a universal enveloping algebra, given on the generators Z ∈ gh by

∆(Z) = Z ⊗ 1 + 1 ⊗ Z,S(Z) = −Z , (Z) = 0 ,

and the Hopf algebra structure of FH is given in the following proposition.

49 Proposition 4.1.5. With the coproduct ∆ : FH → FH ⊗ FH , the antipode S : FH →

FH , and the counit : FH → C determined by the requirements

∆(f)(ψ1, ψ2) = f(ψ1 ◦ ψ2), ψ1, ψ2 ∈ NH , f ∈ FH ,

−1 S(f)(ψ) = f(ψ ), ψ ∈ NH , f ∈ FH ,

ε(f) = f(Id), f ∈ FH ,

FH is a Hopf algebra.

Proof. We will explain the proof in two parallel ways: conceptual and explicit.

• Conceptual: we can consider the inverse limit group

N := { j∞(ψ) | ψ ∈ N } = lim Nk H 0 H ←− H

k with NH denoting the group of invertible k-jets at 0 of diﬀeomorphims in NH .

p The polynomial algebra FH generated by the jet components αq1···qs can be seen p as the algebra of regular functions on NH , i.e. , the jet components αq1···qs are viewed as aﬃne coordinates.

Thus FH inherits from the group NH a canonical Hopf algebra structure, in the standard fashion, with the coproduct, the antipode and counit given by the formulas of this Proposition.

• Explicit: the coassociativity and the counit diagrams are consequence of the following facts

f((ψ1 ◦ ψ2) ◦ ψ3) = f(ψ1 ◦ (ψ2 ◦ ψ3)) ,

f(ψ ◦ ψ−1) = f(Id) = f(ψ−1 ◦ ψ) .

Checking that

p p ∆(αq1···qs ) ∈ FH ⊗ FH and S(αq1···qs ) ∈ FH

50 only involves the generalized Fa´adi Bruno’s formula, the fact that ψ1(0) = ψ2(0) = 0

H H and (ψ1)∗,0 = (ψ2)∗,0 = Id. For example

p p p ∆(αqr)(ψ1, ψ2) = (αqr)(ψ1 ◦ ψ2) = ∂xr ∂xq (ψ1 ◦ ψ2) x=0 p ν µ = ∂xν ∂xµ (ψ1)(ψ2(0))∂xr (ψ1 )(0)∂xq (ψ1 )(0)

p λ + ∂xλ (ψ1)(ψ2(0))∂xr ∂xq (ψ1 )(0)

p p = ∂xr ∂xq (ψ1) + ∂xr ∂xq (ψ1) x=0 x=0 p p = (αqr ⊗ 1 + 1 ⊗ αqr)(ψ1 ⊗ ψ2) .

1 1 1 ∆(α2)(ψ1, ψ2) = (α2)(ψ1 ◦ ψ2) = ∂x2 (ψ1 ◦ ψ2) x=0 1 ν = ∂xν (ψ1)(ψ2(0))∂x2 (ψ2 )(0)

1 1 = ∂x2 (ψ1) + ∂x2 (ψ2) x=0 x=0 1 1 = (α2 ⊗ 1 + 1 ⊗ α2)(ψ1 ⊗ ψ2) . where the indexes depend on the subgroup H.

To deal with the antipode, one uses the same ideas together with the identity

ψ−1 ◦ ψ = Id. For example, by using the previous calculations one obtains

p p −1 p −1 p 0 = (αqr)(Id) = (αqr)(ψ ◦ ψ) = (αqr)(ψ ) + (αqr)(ψ) .

1 1 −1 1 −1 1 0 = (α2)(Id) = (α2)(ψ ◦ ψ) = (α2)(ψ ) + (α2)(ψ1) .

ab Furthermore, FH can be recognized as a Hopf subalgebra of HH . Let HH denote

i the (commutative) Hopf subalgebra of HH generated by the operators { ξjkr1···rs }.

Proposition 4.1.6. There is a unique isomorphism of Hopf algebras

ab cop i i ι :( HH ) → FH , ι(ξjkr1···rs ) := ηjkr1···rs ,

51 and extended as an algebra homomorphism, where the indexes depends of H and

i ηjkr1···rs as in Deﬁniton 3.1.12.

Proof. Suppose that we have a relation of the form

X i1 is c ξ ··· ξ s s (a) = 0 , for all a ∈ A , K j k r1···r1 jsksr ···r H 1 1 1 p1 1 ps K which is equivalent

X i1 is c γ (φ) ··· γ s s (φ) = 0 , for all φ ∈ G , K j k r1···r1 jsksr ···r 1 1 1 p1 1 ps K

Restricting the last equality to ψ ∈ NH and evaluating at (0, Id) ∈ GH , one obtains

X i1 is c η (ψ) ··· η s s (ψ) = 0 . K j k r1···r1 jsksr ···r 1 1 1 p1 1 ps

ab cop It follows that ι :( HH ) → FH give rise to a well deﬁned algebra homomorphism, which by Lemma 3.1.13 and Proposition 3.1.9 is bijective, since it sends basis to basis as vector spaces.

ab cop It remains to prove that ι :( HH ) → FH is a coalgebra map, which amounts to checking that

i op i ι ⊗ ι(∆(ξjkr1···rs )) = ∆ (ηjkr1···rs ) .

i Recall that ∆ : HH → HH ⊗ HH is determined by a “Leibniz rule”, which for ξjkr1···rs takes the form

i i i ξjkr1···rs (ab) = (ξjkr1···rs )(1)(a)(ξjkr1···rs )(2)(b) , for all a, b ∈ AH , which is the same as

i i i γjkr1···rs (φ2 ◦ φ1) = (γjkr1···rs )(1)(φ1)(γjkr1···rs )(2)(φ2) ◦ φe1 , for all φ1, φ2 ∈ G .

Restricting the last equality to ψ1, ψ2 ∈ NH and evaluating at (0, Id) ∈ GH , we get

op i i i i ∆ (ηjkr1···rs )(ψ1, ψ2) = (ηjkr1···rs )(ψ2 ◦ ψ1) = (ηjkr1···rs )(1)(ψ1)(ηjkr1···rs )(2)(ψ2)

52 Now, we proceed to equip FH with a left U(gh)-module algebra structure.

The right action / of GH on NH induces a operation . of U(gh) on FH , given by

d (1 . f) = f and (Z . f)(ψ) := f(ψ / φ ), dt t t=0 for f ∈ FH and Z ∈ gh, where φt is the one parameter subgroup of GH corresponding to Z ∈ gh, and extended to U(gh) as an algebra action.

ab On the other hand, there is a natural action of U(gh) on HH induced by the adjoint action of gh on the generators and extended as a derivation on polynomials

ab which make HH a U(gh)-module algebra.

ab cop Proposition 4.1.7. The algebra isomorphism ι :( HH ) → FH identiﬁes the

ab operation . of U(gh) on FH with the natural action of U(gh) on HH . In particular

FH is U(gh)-module algebra.

In order to prove this Proposition, we need a preparatory Lemma.

Lemma 4.1.8. Let φ ∈ GH and ϕ ∈ G. Then for valid indexes we have

i i γjkr1···rs (φ ◦ ϕ) = γjkr1···rs (ϕ) ,

i i γjkr1···rs (ϕ ◦ φ) = γjkr1···rs (ϕ) ◦ φe . j p i (i ) 1 1 p −1 Proof. Remember that by deﬁnition γjk = Γk and γ2 = Γ2,Γq(ϕ) = Γ(ϕ) , q and Γ satisﬁes the cocycle identity Γ(ϕ1 ◦ ϕ2) = Γ(ϕ1) ◦ ϕe2 · Γ(ϕ2). ∗ Note that Γ(φ) = Id for φ ∈ GH , because UφZUφ = Z which is equivalent to

−1 −1 (φg)∗(Z) = Z for any Z ∈ gh. The latter is true because φg can be realized as left

−1 multiplication by φ , and the one parameter subgroup φt corresponding to Z ∈ gh

−1 acts by right multiplication, therefore φg and φt commute. Using these facts one obtains

p p p −1 −1 Γq(φ ◦ ϕ) = Γ(φ ◦ ϕ) = Γ(φ) ◦ ϕ · Γ(ϕ) q e q p −1 p = Γ(ϕ) = Γq(ϕ) . q

53 which prove the ﬁrst equality. Similarly we have

p p p −1 −1 Γq(ϕ ◦ φ) = Γ(ϕ ◦ φ) = Γ(ϕ) ◦ φe · Γ(φ) q q p −1 p = Γ(ϕ) ◦ φe = Γq(ϕ) ◦ φe , q

−1 and, the same invariance property (φg)∗(Z) = Z for any Z ∈ gh gave us

p p Z(Γq(ϕ) ◦ φ) = Z(Γq(ϕ)) ◦ φ , which together implies the second equality.

Next, we prove Proposition 4.1.7:

Proof. Let φt the one parameter subgroup corresponding to Z ∈ gh. From the ﬁrst equality from Lemma 4.1.8 it follows that

i i i γjkr1···rs (ψ ◦ φ) = γjkr1···rs ((ψ . φt) ◦ (ψ / φt)) = γjkr1···rs (ψ / φt) , whence

d d (Z . ηi )(ψ) = ηi (ψ / φ ) = γi (ψ / φ )(0, Id) jkr1···rs dt jkr1···rs t dt jkr1···rs t t=0 t=0

d = γi (ψ ◦ φ )(0, Id) . dt jkr1···rs t t=0 Now using the second identity from Lemma 4.1.8 , one can continue as follows

d i d i γ (ψ ◦ φt)(0, Id) = γ (ψ)(φet(0, Id)) dt jkr1···rs dt jkr1···rs t=0 t=0

d = γi (ψ)((0, Id)φ ) = Z(γi (ψ))(0, Id) dt jkr1···rs t jkr1···rs t=0

By iterating this argument one obtains, for any u ∈ U(gh)

i i (u . ηjkr1···rs )(ψ) = u(γjkr1···rs (ψ))(0, Id) , ψ ∈ NH .

The right-hand side of the previous equality, before evaluation at (0, Id), describes

i ab the eﬀect of the action of u ∈ U(gh) on ξjkr1···rs ∈ HH . In view of the deﬁnition of ab cop the isomorphism ι :( HH ) → FH (Proposition 4.1.6) this achieves the proof. 54 Finally, we proceed to equip U(gh) with a right FH -comodule coalgebra structure.

The left action . of NH on GH by Proposition 4.1.3 satisﬁes (ψ .)(0, Id) = (0, Id), then we have ∼ ∼ (ψ .)∗,(0,Id) : gh = T(0,Id)GH → gh = T(0,Id)GH .

Let { Zr }r∈R be the basis gh as in Deﬁnition 3.1.4, one has

∗ (ψ .)∗,(0,Id) Zq = ( ψe)∗,(0,Id) Zq = ( ψe)∗ Zq = Uψ Zq Uψ (0,Id) (0,Id) (0,Id) (0,Id)

p −1 p −1 = Γq(ψ )Zp = Γq(ψ )(0, Id)Zp (0,Id) (0,Id) p p −1 −1 = Γq(ψ ) ◦ ψe(0, Id)Zp = Γ(ψ) (0, Id)Zp (0,Id) q (0,Id)

p = Γq(ψ)(0, Id)Zp (0,Id)

Therefore, we obtain a map

p Hgh : gh → gh ⊗ FH , HZq := Zp ⊗ ι(Ξq) , where ι is deﬁned in Proposition 4.1.6.

Proposition 4.1.9. The map Hgh : gh → gh⊗FH endows gh with a right FH comodule structure.

Proof. By using the Proposition 4.1.6 and the second “Leibniz rule” from Lemma

3.1.6, one has

p r p r p (Hgh ⊗ Id)(Hgh Zq) = Hgh Zp ⊗ ι(Ξq ) = Zr ⊗ ι(Ξp ) ⊗ ι(Ξq ) = Zr ⊗ (ι ⊗ ι)( Ξp ⊗ Ξq )

cop r r = Zr ⊗ (ι ⊗ ι)(∆ (Ξq )) = Zr ⊗ ∆(ι(Ξq )) = (Id ⊗∆)(Hgh Zq) .

We also have

p ∼ p (Id ⊗ ε)(Hgh Zq) = Zp ⊗ ε(ι(Ξq )) = Zp Γq(Id)(0, Id) = Zq

∗ because UId Zq UId = Zq. 55 We extend the coaction Hgh to a function H : U(gh) → U(gh) ⊗ FH via

H(1) = 1 ⊗ 1 and H(uv) := u(1)<0>v<0> ⊗ u(1)<1>(u(2) . v<1>) ,

for u, v ∈ U(gh), which are two of the compatibility condition from Deﬁnition 4.1.1.

To prove that H is a coaction we will prove that the other compatibility conditions for a bicrossed product Hopf algebra from Deﬁnition 4.1.1 are satisfy and that H satisﬁes the comodule coalgebra conditions

Lemma 4.1.10. For any u ∈ U(gh) and f ∈ FH one has

ε(u . f) = ε(u)ε(f) .

Proof. By using the deﬁnition of the action and Proposition 4.1.3, we have

ε(1 . f) = f(Id) = ε(1)ε(f) ,

d d ε(Z . f) = (Z . f)(Id) = f(Id / φ ) = f(Id) = 0 = ε(Z)ε(f) , dt t dt t=0 t=0 for Z ∈ gh and f ∈ FH .

Finally, let assume that the equality is held for u, v ∈ U(gh) and any f ∈ F, then

ε(uv . f) = ε(u . (v . f)) = ε(u)ε(v . f) = ε(u)ε(v)ε(f) = ε(uv)ε(f) .

Lemma 4.1.11. For any u ∈ U(gh) and f ∈ FH one has

∆(u . f) = u(1)<0> . f(1) ⊗ u(1)<1>(u(2) . f(2)) .

Proof. First, we prove this equality for Zq ∈ gh and ι(f) ∈ F as follows

cop cop ∆ Zq . ι(f) = ∆ ι [Zq, f] = ι ⊗ ι ∆ [Zq, f] = ι ⊗ ι ∆ (Zqf − fZq)

cop cop cop cop = ι ⊗ ι ∆ (Zq)∆ (f) − ∆ (f)∆ (Zq) .

56 Using the ﬁrst “Leibniz rule” from Lemma 3.1.6, ones has

cop cop p ∆ (Zq)∆ (f) = (1 ⊗ Zq + Zp ⊗ Ξq)(f(2) ⊗ f(1))

p = f(2) ⊗ Zqf(1) + Zpf(2) ⊗ Ξqf(1) ,

cop cop p ∆ (f)∆ (Zq) = (f(2) ⊗ f(1))(1 ⊗ Zq + Zp ⊗ Ξq)

p = f(2) ⊗ Zqf(1) + f(2)Zp ⊗ f(1)Ξq .

ab By using that the elements on HH commute and subtracting these equalities, we obtain

p ∆ Zq . ι(f) = ι ⊗ ι f(2) ⊗ [Zq, f(1)] + [Zp, f(2)] ⊗ Ξqf(1)

p = ι(f(2)) ⊗ ι [Zq, f(1)] + ι [Zp, f(2)] ⊗ ι(Ξq )ι(f(1))

p = ι(f(2)) ⊗ Zq . ι(f(1)) + Zp . ι(f(2)) ⊗ ι(Ξq )ι(f(1))

= (Zq)(1)<0> . ι(f(2)) ⊗ (Zq)(1)<1>((Zq)(2) . ι(f(1)))

= (Zq)(1)<0> . ι(f)(1) ⊗ (Zq)(1)<1>((Zq)(2) . ι(f)(2)) .

To ﬁnish the proof, suppose that the equality is true for u, v ∈ U(gh) and any f ∈ F, by using the fact that F is U(gh)-module algebra and the deﬁnition of H we obtain ∆(uv . f) = u(1)<0> . v . f ⊗ u(1)<1> u(2) . v . f (1) (2) = u(1)<0> . v(1)<0> . f(1) ⊗ u(1)<1> u(2) . v(1)<1> v(2) . f(2) = u(1)<0>v(1)<0> . f(1) ⊗ u(1)<1> u(2) . v(1)<1> u(3)v(2) . f(2) = (uv)(1)<0> . f(1) ⊗ (uv)(1)<1> (uv)(2) . f(2) .

Lemma 4.1.12. For any u ∈ U(gh) and f ∈ FH one has

u(2)<0> ⊗ (u(1) . f)u(2)<1> = u(1)<0> ⊗ u(1)<1>(u(2) . f)

Proof. Since U(gh) is cocommutative and FH is commutative, the equality is automatically satisﬁed. 57 Lemma 4.1.13. The function H : U(gh) → U(gh) ⊗ FH satisﬁes the comodule coalgebra conditions.

Proof. From the deﬁnition of ε, one has that for any u ∈ U(gh)

ε(u<0>)u<1> = 0 = ε(u)1 .

Now, the equation

u<0>(1) ⊗ u<0>(2) ⊗ u<1> = u(1)<0> ⊗ u(2)<0> ⊗ u(1)<1>u(2)<1> ,

is true for 1 ∈ U(gh)

1<0>(1) ⊗ 1<0>(2) ⊗ 1<1> = 1 ⊗ 1 ⊗ 1 = 1(1)<0> ⊗ 1(2)<0> ⊗ 1(1)<1>1(2)<1> .

For Zq ∈ gh, one has

p (Zq)<0>(1)⊗(Zq)<0>(2) ⊗ (Zq)<1> = (Zp)(1) ⊗ (Zp)(2) ⊗ ι(Ξq)

p p = Zp ⊗ 1 ⊗ ι(Ξq) + 1 ⊗ Zp ⊗ ι(Ξq)

= (Zq)<0> ⊗ 1<0> ⊗ (Zq)<1>1<1> + 1<0> ⊗ (Zq)<0> ⊗ 1<1>(Zq)<1>)

= (Zq)(1)<0> ⊗ (Zq)(2)<0> ⊗ (Zq)(1)<1>(Zq)(2)<1>.

Finally, suppose that the equation is true for u, v ∈ U(gh), by using the deﬁnition of

H, the fact that FH is U(gh)-module algebra, that U(gh) is cocommutative and FH is commutative, we observe that

58 (uv)<0>(1) ⊗ (uv)<0>(2) ⊗ (uv)<1>

= (u(1)<0>v<0>)(1) ⊗ (u(1)<0>v<0>)(2) ⊗ u(1)<1>(u(2) . v<1>)

= u(1)<0>(1)v<0>(1) ⊗ u(1)<0>(2)v<0>(2) ⊗ u(1)<1>(u(2) . v<1>)

= u(1)<0>v(1)<0> ⊗ u(2)<0>v(2)<0> ⊗ u(1)<1>u(2)<1>(u(3) . (v(1)<1>v(2)<1>))

= u(1)<0>v(1)<0> ⊗ u(2)<0>v(2)<0> ⊗ u(1)<1>u(2)<1>(u(3) . v(1)<1>)(u(4) . v(2)<1>)

= u(1)<0>v(1)<0> ⊗ u(3)<0>v(2)<0> ⊗ u(1)<1>(u(2) . v(1)<1>)u(3)<1>(u(4) . v(2)<1>)

= (u(1)v(1))<0> ⊗ (u(2)v(2))<0> ⊗ (u(1)v(1))<1>(u(2)v(2))<1>

= (uv)(1)<0> ⊗ (uv)(2)<0> ⊗ (uv)(1)<1>(uv)(2)<1> .

Proposition 4.1.14. The map H : U(gh) → U(gh) ⊗ FH endows U(gh) with a right

FH -comodule structure. Furthermore, U(gh) is a right FH -comodule coalgebra under this coaction.

Proof. Let us begin with the ﬁrst commutative diagram, i.e. ,

(H ⊗ Id)(H) = (Id ⊗∆)(H) .

By using the deﬁnition of the function H, we have

(H ⊗ Id)(H1) = (H ⊗ Id)(1 ⊗ 1) = 1 ⊗ 1 ⊗ 1 = (Id ⊗∆)(1 ⊗ 1) = (Id ⊗∆)(H1)

and by Lemma 4.1.9 it is true for Z ∈ gh. Suppose this equality is true for u, v ∈ U(gh), by using Lemma 4.1.13, the deﬁnition of H, and Lemma 4.1.11 then one has

59 H ⊗ Id Huv = H ⊗ Id u(1)<0>v<0> ⊗ u(1)<1>(u(2) . v<1>)

= u(1)<0>(1)<0>v<0><0> ⊗ u(1)<0>(1)<1>(u(1)<0>(2) . v<0><1>)

⊗ u(1)<1>(u(2) . v<1>)

= u(1)<0><0>v<0><0> ⊗ u(1)<0><1>(u(2)<0>) . v<0><1>)

⊗ u(1)<1>u(2)<1>(u(3) . v<1>)

= u(1)<0>v<0> ⊗ u(1)<1>(1)(u(2)<0> . v<1>(1))

⊗ u(1)<1>(2)u(2)<1>(u(3) . v<1>(2))

= u(1)<0>v<0> ⊗ u(1)<1>(1)(u(2) . v<1>)(1) ⊗ u(1)<1>(2)(u(2) . v(1))(2) = u(1)<0>v<0> ⊗ u(1)<1>(u(2) . v<1>) ⊗ u(1)<1>(u(2) . v(1)) (1) (2) = Id ⊗∆ u(1)<0>v<0> ⊗ u(1)<1>(u(2) . v<1>) = Id ⊗∆) Huv .

For the second commutative diagram, one has

∼ (Id ⊗ ε)(H1) = (Id ⊗ε)(1 ⊗ 1) = 1 ⊗ ε(1) = 1 ,

by Lemma 4.1.9 it is true for Z ∈ gh and if it is true for u, v ∈ U(gh) then we obtain

(Id ⊗ ε)(Huv) = (Id ⊗ ε)(u(1)<0>v<0> ⊗ u(1)<1>(u(2) . v<1>)

= u(1)<0>v<0> ⊗ ε(u(1)<1>(u(2) . v<1>) ∼ = u(1)<0>v<0>ε(u(1)<1>)ε(u(2))ε(v<1>)

= u(1)<0>ε(u(1)<1>)ε(u(2))v<0>ε(v<1>)

= u1ε(u(2))v = uv .

Therefore H is a coaction and together with Proposition 4.1.13, it provides U(gh) with a right FH -comodule structure. 60 Proposition 4.1.15. The Hopf algebras U(gh) and FH form a matched pair of Hopf algebras. Therefore, we can form the bicrossed product Hopf algebra FH IC U(gh).

Proof. Proposition 4.1.7 and Proposition 4.1.14 show that FH is a U(gh)-module algebra and U(gh) is a FH -comodule coalgebra, respectively. In addition, the compatibility conditions for a bicrossed product Hopf algebra (Definition 4.1.1) are sat- isfied because of the very definition of the coaction and Lemmas 4.1.10, 4.1.11 and

4.1.12.

Finally, we are ready to write the principal result of this section.

cop Theorem 4.1.16. The Hopf algebras HH and FH IC U(gh) are isomorphic.

Proof. Proposition 3.1.9 provides us with ξK ZI as a basis for the Hopf algebra HH . Let us deﬁne

cop I : HH →FH IC U(gh)

I(Id) = 1 IC 1 , I(ξK ZI ) = ι(ξK ) IC ZI ,

cop where ι is deﬁned in Proposition 4.1.6, and linearly extended it on HH . The map I is an isomorphism between vector spaces since it sends basis to basis.

Let us check that the map I is an algebra homomorphism. It is enough to prove this fact for elements of the form Zf ∈ HH , with Z ∈ gh and f ∈ FH , since this is the relation used to write any element in HH as a linear combination of the basis

ξK ZI and need it to rewrite the image of the element in FH IC U(gh). By using proposition 4.1.7 and the deﬁnition of the product on a bicrossed product

Hopf algebra, we have

61 I(Zf) = I([Z, f] + fZ)

= ι([Z, f]) IC 1 + ι(f) IC Z

= Z . ι(f) IC 1 + ι(f) IC Z

= (Z IC 1)(1 IC ι(f)) = I(Z)I(f) .

So HH and F are isomorphic as algebras. To show that I is a coalgebra map it is necessary and suﬃcient to check that I commutes with coproduct on the generators.

Indeed,

∆FH IC U(gh)(I(Zq)) = ∆FH IC U(gh)(1 IC Zq)

= 1 IC (Zq)(1)<0> ⊗ (Zq)(1)<1> IC (Zq)(2)

= 1 IC 1<0> ⊗ 1<0> IC Zq + 1 IC (Zq)<0> ⊗ (Zq)<1> IC 1

p = 1 IC 1 ⊗ 1 IC Zq + 1 IC Zp ⊗ ι(Ξq ) IC 1

p = I(Id) ⊗ I(Zq) + I(Zp) ⊗ I(Ξq )

p = ( I ⊗ I )(Id ⊗Zq + Zq ⊗ Ξq)

= ( I ⊗ I )∆cop (Z ) , HH q and

p p ∆FH IC U(gh)(I(Ξq )) = ∆FH IC U(gh)(ι(Ξq ) IC 1)

p p = ι(Ξq )(1) IC 1 ⊗ ι(Ξq )(2) IC 1

p µ = ι(Ξµ ) IC 1 ⊗ ι(Ξq ) IC 1

p µ = I(Ξµ ) ⊗ I(Ξq )

p µ = I ⊗ I(Ξµ ⊗ Ξq )

= ( I ⊗ I )∆cop (Ξp )) . HH q

62 Remark 4.1.17. The antipode of a Hopf algebra is unique. Therefore, the antipode

S on HH can be expressed by using the antipode SIC on FH IC U(gh) as follows:

−1 S = I ◦ SIC ◦ I .

63 CHAPTER 5

HOPF CYCLIC COHOMOLOGY

The periodic Hopf cyclic cohomology of each Hopf algebra HH is shown to be quasi-isomorphic to the continuous cohomology of the Lie algebra a given by formal vector ﬁelds of the form

1 ∂ 2 ∂ v = p (x1, x2) + p (x2) , ∂x1 ∂x2

1 2 where p (x1, x2) and p (x2) are formal series in their respective variables.

∗ 5.1 HP (HH )

Remember that the crossed product algebra AH comes equipped with a trace

τ : AH → C (Deﬁnition 3.1.15) which is HH -invariant relative to the character

δ : HH → C (Deﬁnition 3.1.17). Using this trace τ the standard Hopf cyclic model for HH is “imported” from the standard cyclic model of the algebra AH , via the characteristic map

1 n ⊗n 1 n n h ⊗ · · · ⊗ h ∈ HH 7→ χτ (h ⊗ · · · ⊗ h ) ∈ C (AH )

1 n 0 n 0 1 1 n n i χτ (h ⊗ · · · ⊗ h )(a ⊗ · · · ⊗ a ) = τ(a h (a ) ··· h (a )) , a ∈ AH .

64 n ⊗n This map is faithful and give rise to a family of F-modules { C (HH ) := HH }n≥0, with faces, degenerates and cyclic operators

n−1 n δi : C (HH ) → C (HH ) , 0 ≤ i ≤ n ,

n+1 n σi : C (HH ) → C (HH ) , 0 ≤ i ≤ n .

n n τn : C (HH ) → C (HH ) . given by

1 n−1 1 n−1 δ0(h ⊗ · · · ⊗ h ) = 1 ⊗ h ⊗ · · · ⊗ h ,

1 n−1 1 i n−1 δi(h ⊗ · · · ⊗ h ) = h ⊗ · · · ⊗ ∆(h ) ⊗ · · · ⊗ h , 1 ≤ i ≤ n − 1,

1 n−1 1 n−1 δn(h ⊗ · · · ⊗ h ) = h ⊗ · · · ⊗ h ⊗ 1,

1 n+1 1 i+1 n+1 σi(h ⊗ · · · ⊗ h ) = h ⊗ · · · ⊗ ε(h ) ⊗ · · · ⊗ h , 1 ≤ i ≤ n,

1 n 1 2 1 n 1 τn(h ⊗ · · · ⊗ h ) = S(h(n))h ⊗ · · · ⊗ S(h(2))h ⊗ Sδ(h(1))1 . where in the last formula we are using Sweedler notation.

∗ The periodic Hopf cyclic cohomology HP (HH ) of HH is, by deﬁnition [5,6], the

•,• Z2-graded cohomology of the total complex C (HH ) associated to the bicomplex p,q { C (HH ), b, B }, where   q−p C (HH ), if q ≥ p, p,q  C (HH ) =  0, if q < p, where

n n+1 n+1 n b : C (HH ) → C (HH ),B : C (HH ) → C (HH ) ,

n n X i X ni i n b = (−1) δi,B = (−1) τn σnτn+1(1 − (−1) τn+1). i=0 i=0 Note that the pair (δ, 1) is an example of a modular pair in involution [6,7] and

1 Cδ is a particular case of stable-anti-Yetter-Drinfeld module-comodule [15]. Also, 65 ∗ the Hopf cyclic cohomology HP (HH ) is a particular case of Hopf cyclic cohomology with stable-anti-Yetter-Drinfeld module-comodule coeﬃcients [16].

In general the computation of this cohomology poses quite a challenge. Exploiting

•,• the bicrossed product structure from the previous section, the total complex C (HH ) can be replaced [25, 26] by quasi-isomorphic bicocyclic bicomplexes. A bicocyclic bicomplex is a bigraded module whose rows and columns are cocyclic modules and, in addition, the vertical operators and the horizontal operators commute. One of these bicocyclic bicomplexes amalgamate two classical types of cohomological constructs,

Lie algebra cohomology with coefficients and coalgebra cohomology with coefficients, with the essential distinction though that the coefficients are not only acted upon but also “act back”.

The ﬁrst such bigraded module is given by

p,q ⊗q ⊗p C (U(gh), FH , C) := C ⊗ U(gh) ⊗ FH .

To simplify the next formulas, we introduce the abbreviated notation

f˜ = f 1 ⊗ · · · ⊗ f p, u˜ = u1 ⊗ · · · ⊗ up,

1 p 1 p u˜h0i ⊗ u˜h1i = uh0i ⊗ · · · ⊗ uh0i ⊗ uh1i ··· uh1i,

and the Hopf algebra U(gh) admits the following action on FH deﬁned by

˜ 1 2 n u • f := u(1)h0i . f ⊗ u(1)h1iu(2)h0i . f ⊗ · · · ⊗ u(1)hn−1i ··· u(n−1)h1iu(n) . f

With these conventions, the horizontal operators

→ p−1,q p,q ∂ i : C (U(gh), FH , C) → C (U(gh), FH , C), 0 ≤ i ≤ p

→ p+1,q p,q σ j : C (U(gh), FH , C) → C (U(gh), FH , C), 0 ≤ j ≤ p

→ p,q p,q τ : C (U(gh), FH , C) → C (U(gh), FH , C),

66 are deﬁned by

→ ˜ ˜ ∂ 0(1 ⊗ u˜ ⊗ f) := 1 ⊗ u˜ ⊗ 1 ⊗ f, → ˜ 1 i p−1 ∂ i(1 ⊗ u˜ ⊗ f) := 1 ⊗ u˜ ⊗ f ⊗ · · · ⊗ ∆(f ) ⊗ · · · ⊗ f , → ˜ ˜ ∂ p(1 ⊗ u˜ ⊗ f) := 1 ⊗ u˜h0i ⊗ f ⊗ S(˜uh1i),

→ ˜ 1 j+1 p+1 σ j(1 ⊗ u˜ ⊗ f) := 1 ⊗ u˜ ⊗ f ⊗ · · · ⊗ (f ) ⊗ · · · ⊗ f ,

→ ˜ 1 2 p τ (1 ⊗ u˜ ⊗ f) := 1 ⊗ u˜h0i ⊗ S(f ) · (f ⊗ · · · ⊗ f ⊗ S(˜uh1i)), while the vertical operators

p,q−1 p,q ↑∂i : C (U(gh), FH , C) → C (U(gh), FH , C), 0 ≤ i ≤ q

p,q+1 p,q ↑σj : C (U(gh), FH , C) → C (U(gh), FH , C), 0 ≤ j ≤ q

p,q p,q ↑τ : C (U(gh), FH , C) → C (U(gh), FH , C), are deﬁned by

˜ ˜ ↑∂0(1 ⊗ u˜ ⊗ f) := 1 ⊗ 1 ⊗ u˜ ⊗ f,

˜ 1 i q−1 ˜ ↑∂i(1 ⊗ u˜ ⊗ f) := 1 ⊗ u ⊗ · · · ⊗ ∆(u ) ⊗ · · · ⊗ u ⊗ f, ˜ ˜ ↑∂q(1 ⊗ u˜ ⊗ f) := 1 ⊗ u˜ ⊗ 1 ⊗ f,

˜ 1 j+1 q+1 ˜ ↑σj(1 ⊗ u˜ ⊗ f) := 1 ⊗ u ⊗ · · · ⊗ (u ) ⊗ · · · ⊗ u ⊗ f,

˜ 1 1 2 q 1 ˜ ↑τ(1 ⊗ u˜ ⊗ f) := δ(u(1)) ⊗ S(u(3)) · (u ⊗ · · · ⊗ u ⊗ 1) ⊗ S(u(2)) • f,

Deﬁne the vertical and horizontal operators

p p ! → → → X i X pi →i → → p → b = (−1) ∂ i, B = (−1) τ σ p τ 1 − (−1) τ , i=0 i=0 q q ! X i X qi i q ↑b = (−1) ↑∂i, ↑B = (−1) ↑τ ↑σq ↑τ 1 − (−1) ↑τ , i=0 i=0

67 The bicocycle module C is given by the following diagram ......

↑b ↑B ↑b ↑B ↑b ↑B → → → ⊗2 b ⊗2 b ⊗2 ⊗2 b ⊗ U(gh) ⊗ U(gh) ⊗ FH ⊗ U(gh) ⊗ F ··· C → C → C H → B B B ↑b ↑B ↑b ↑B ↑b ↑B → → → b b ⊗2 b ⊗ U(gh) ⊗ U(gh) ⊗ FH ⊗ U(gh) ⊗ F ··· C → C → C H → B B B ↑b ↑B ↑b ↑B ↑b ↑B → → → b b ⊗2 b ⊗ FH ⊗ F ··· C → C → C H → B B B

(p,q) Let Tot(C) denote the total complex of C deﬁned by Tot(C)n = ⊕p+q=nC for → → n ≥ 0, with bC = b + ↑ b and BC = B + ↑ B. Since the total complex of a bicocycle module is a mixed complex, its cyclic cohomology is well deﬁned.

The next bigraded module is given by

q p,q ∗ ^ ∗ ⊗p C ( gh , FH ) := gh ⊗ FH . related with the following bicocyclic bicomplex

......

∂g∗ ∂g∗ ∂g∗

bF bF bF 2 ∗ 2 ∗ 2 ∗ ⊗2 ∧ gh ∧ gh ⊗ FH ∧ gh ⊗ FH ··· BF BF BF

∂g∗ ∂g∗ ∂g∗

bF bF bF ∗ ∗ ∗ ⊗2 gh gh ⊗ FH gh ⊗ FH ··· BF BF BF

∂g∗ ∂g∗ ∂g∗

bF bF bF ⊗2 FH FH ··· C BF BF BF

68 p,q p,q+1 The vertical operator ∂g∗ : C → C is the Lie algebra cohomology coboundary

⊗p of the Lie algebra gh with coeﬃcients in FH , where the right action of gh is given by

(f 1 ⊗ · · · ⊗ f p) /Z =

1 2 p − Z(1)<0> . f ⊗ Z(1)<1> (Z(2)<0> . f ) ⊗ · · · ⊗ Z(1) ··· Z(p−1)<1> (Z(p) . f ).

The horizontal operator bF is the coboundary operator and the horizontal operator

• ∗ BF is the Connes boundary operator for coalgebra FH with coeﬃcients in ∧ gh, where the left coaction is given by

r s s H(ζ ) = ι(Ξr) ⊗ ζ ,

r • ∗ (where ζ are a dual basis to Zr), and then extend to ∧ gh. More precisely,

1 p−1 1 p−1 bF (α ⊗ f ⊗ · · · ⊗ f ) = α ⊗ 1 ⊗ f ⊗ · · · ⊗ f X + (−1)iα ⊗ f 1 ⊗ · · · ⊗ ∆(f i) ⊗ · · · ⊗ f p−1 1≤i≤p−1

p 1 p−1 + (−1) α<0> ⊗ f ⊗ · · · ⊗ f ⊗ S(α<−1>) .

p X B = ( (−1)piτ i )σ τ (1 − (−1)pτ ), with F p,F p,F p+1,F p+1,F i=0

1 p 1 2 p τp,F (α ⊗ f ⊗ · · · ⊗ f ) = α<0> ⊗ S(f ) · (f ⊗ · · · ⊗ f ⊗ S(α<−1>))

1 p p 1 p−1 σp,F (α ⊗ f ⊗ · · · ⊗ f ) = ε(f )α ⊗ f ⊗ · · · ⊗ f .

The next bigraded module is a homogeneous version of the previous one

q FH p,q ∗ ^ ∗ ⊗p+1 Ccoinv( gh , FH ) := gh ⊗ FH ,

X p,q ∗ deﬁned as follows: an element α ⊗ fe ∈ Ccoinv( gh , FH ) if it satisﬁes the FH - coinvariant condition

X X α<0> ⊗ fe⊗ S(α<1>) = α ⊗ fe<0> ⊗ fe<1> 69 here for fe = f 0 ⊗ · · · ⊗ f p, we have denoted

0 p 0 p fe<0> ⊗ fe<1> = f(1) ⊗ · · · ⊗ f(1) ⊗ f(2) ··· f(2).

The identiﬁcation between the two module is made by the isomorphism

q q FH ^ ∗ ⊗p ^ ∗ ⊗p+1 I : gh ⊗ FH → gh ⊗ FH ,

1 1 2 p−1 p p I(α ⊗ fe) := α<0> ⊗ f(1) ⊗ S(f(2))f(1) ⊗ · · · ⊗ S(f(2) )f(1) ⊗ S(α<1>f(2)) .

p,q ∗ By transfer of structure, Ccoinv( gh , FH ) can be equipped with vertical and horizontal operators

coinv −1 ∂g := I ◦ ∂g∗ ◦ I ,

coinv −1 bF := I ◦ bF ◦ I ,

coinv −1 BF := I ◦ BF ◦ I , and one obtains the following bicocyclic bicomplex

......

coinv coinv coinv ∂g ∂g ∂g

coinv coinv coinv bF bF bF 2 ∗ FH 2 ∗ ⊗2 FH 2 ∗ ⊗3 FH (∧ gh ⊗ FH ) coinv (∧ gh ⊗ FH ) coinv (∧ gh ⊗ FH ) coinv ··· BF BF BF

coinv coinv coinv ∂g ∂g ∂g

coinv coinv coinv bF bF bF ∗ FH ∗ ⊗2 FH ∗ ⊗3 FH (gh ⊗ FH ) coinv (gh ⊗ FH ) coinv (gh ⊗ FH ) coinv ··· BF BF BF

coinv coinv coinv ∂g ∂g ∂g

coinv coinv coinv bF bF bF FH ⊗2 FH ⊗3 FH (C ⊗ FH ) coinv (C ⊗ FH ) coinv (C ⊗ FH ) coinv ··· BF BF BF

To deﬁne the next bigraded module we are going to use the inverse limit group

N = lim Nk H ←− H 70 k with NH denoting the group of invertible k-jets at zero of diﬀeomorphims in NH .

Recall that the algebra FH is precisely the polynomial algebra generated by the components of such jets. When there is no danger of confusion, we shall abuse the

∞ notation by simply writing ψ ∈ NH instead of j0 (ψ) ∈ NH , where we are denoting

∞ by j0 (ψ) the jet at zero of ψ ∈ NH . We deﬁne

p q ∗ p+1 q ∗ Cpol(NH , ∧ gh) := { c : NH → ∧ gh |

−1 c(ψ0ψ, . . . , ψpψ) = ψ . c(ψ0, . . . , ψp) ∀ψ ∈ NH } where . stands for

−1 ∗ q ∗ ψ . ω = (ψ ) ω (0,Id) , for ω ∈ ∧ gh .

The vertical operators are given by

X r ( ∂polc )(ψ0, . . . , ψp) = ∂(c(ψ0, . . . , ψp)) − ζ ∧ (Zr . c)(ψ0, . . . , ψp) . r

Here ∂ stands for the Lie algebra cohomology coboundary of gh with trivial coeﬃ-

r cients, { Zr }r∈R is a basis of gh with dual basis { ζ }r∈R, and the action of gh on a

p,q ∗ cochain c ∈ Cpol(NH , gh) is given by

X d (Z . c)(ψ , . . . , ψ ) = (ψ , . . . , ψ / exp(tZ), . . . , ψ ) 0 p dt 0 r p r t=0 The horizontal operators are given by

p X r ˆ ( bpol c )(ψ0, . . . , ψp) = (−1) c(ψ0,..., ψr, . . . , ψp) r=0 p X pi i p Bpol = ( (−1) τp,pol)σp,polτp+1,pol(1 − (−1) τp+1,pol), with i=0

( τp,pol, c )(ψ0, . . . , ψp) = c(ψ1, . . . , ψp, ψ0)

( σp,pol c )(ψ0, . . . , ψp−1) = c(ψ0, . . . , ψp−1, ψp−1) .

71 The bicocyclic bicomplex in this case is given by

......

∂pol ∂pol ∂pol

bpol bpol bpol 0 2 ∗ 1 2 ∗ 2 2 ∗ Cpol(NH , ∧ gh) Cpol(NH , ∧ gh) Cpol(NH , ∧ gh) ··· Bpol Bpol Bpol

∂pol ∂pol ∂pol

bpol bpol bpol 0 ∗ 1 ∗ 2 ∗ Cpol(NH , gh) Cpol(NH , gh) Cpol(NH , gh) ··· Bpol Bpol Bpol

∂pol ∂pol ∂pol

bpol bpol bpol 0 1 2 Cpol(NH , ) Cpol(NH , ) Cpol(NH , ) ··· C Bpol C Bpol C Bpol

This bicomplex is isomorphic to the previous one via the chain map

q FH ^ ∗ ⊗p+1 p q ∗ J : gh ⊗ FH → Cpol(NH , ∧ gh) ,

0 p J (α ⊗ fe)(ψ0, . . . , ψp) := f (ψ0) ··· f (ψp)α .

0 p ⊗p+1 where fe = f ⊗ · · · ⊗ f ∈ FH .

We record the principal results from section §1 [26] in the following proposition.

Proposition 5.1.1. The total cohomology of all the above bicomplexes are canonically isomorphic to the standard total Hopf cyclic cohomology.

Proof. It is a restatement of Propositions 1.5 (were the bicrossed realization is used),

1.11, 1.15, and 1.16 from [26].

Finally, we will show that the proof for the quasi-isomorphim given by Moscovici and Rangipour [26] between the Hopf cyclic cohomology of the Hopf algebra related to a primitive pseudogroup and the continuous cohomology of the inﬁnite Lie algebra related to the same pseudogroup is still valid in our case for both Hopf algebras HH . 72 ∞ Letting J0 (G) stand for the manifold of invertible jets of inﬁnite order at zero

∞ of local diffeomorphism in G, and denoting by j0 (ϕ) the jet at zero of ϕ ∈ G, any formal vector field v ∈ a can be represented via the jet at zero of a 1-parameter group of local diffeomorphisms { ϕt }t∈R, ! ∂ v = j∞ ϕ , 0 ∂t t t=0

∞ ∞ and give rise to a left invariant vector ﬁeld ve on J0 (G), whose value at j0 (ϕ) ∈ J ∞(G) is given by 0 ! ∂ ∞ v j∞(ϕ) = j0 (ϕ ◦ ϕt) e 0 ∂t t=0 m ∞ We can thus associate to each ω ∈ Ccont(a) the left invariant form ωe on J0 (G) deﬁned by ωe ve1,..., vem = ω(v1, . . . , vm) .

∞ ∞ We denote by πGH : J0 (G) → GH and πNH : J0 (G) → NH the projections

∞ relative to the decomposition J0 (G) = GH · NH , for each H.

r To construct the desired homomorphism, we associate to any ω ∈ Ccont(a) and any pair of integers (p, q), p + m = q + r, 0 ≤ q ≤ m = dim GH , a p-cochain

Cp,q(ω)(ψ0, . . . , ψp) on the group NH with values in q-currents on GH as follows. For

q each q-form with compact support ζ ∈ Ωc(GH ), one sets Z h C (ω)(ψ , . . . , ψ ) , ζ i = π∗ (ζ) ∧ ω , p,q 0 p GH e Σ(ψ0,...,ψp)

∞ the integration is taken over the ﬁnite dimensional cycle in J0 (G)

∞ ∞ −1 Σ(ψ0, . . . , ψp) = { j0 (ρ) ∈ J0 (G) | πNH (ρ ) ∈ ∆(ψ0, . . . , ψp) } where ( p p )

X ∞ X ∆(ψ0, . . . , ψp) = tij0 (ψi) ti ≥ 0 , ti = 1 . i=0 i=0

73 We note that ∆(ψ0, . . . , ψp) ⊂ NH , because at the “ﬁrst H-jet” level p X ∞ tij0 (ψi) = (0, Id) H-version i=0

Also, the integration makes sense because the form ωe lives on a ﬁnite order jet bundle and has an algebraic expression, with polynomial coeﬃcients, in the canonical coordinates.

Proposition 5.1.2. The map

• • C : Ccont(a) → Ccont(NH , Ω•(GH )) , is a map of complexes.

Proof. The proof is the same as for the primitive pseudogroups, Lemma 2.2 [26].

Remember our convention that we often abbreviate the notation by not making

∞ the distinction between ψ and its jet j0 (ψ). Manifestly the simplex deﬁnition is covariant with respect to right translations by elements of NH ,

∆(ψ0 ◦ ψ, . . . , ψp ◦ ψ) = ∆(ψ0, . . . , ψp) ◦ ψ , ψ ∈ NH .

A similar relation holds with respect to the right action of GH on NH . For φ ∈ GH this action amounts to passing from ψ ∈ NH to ψ / φ ∈ NH . Recall from Lemma 4.1.8 that one has

i i i γjkr1···rs (ψ / φ) = γjkr1···rs (ψ ◦ φ) = γjkr1···rs (ψ) ◦ φe , in particular

i i jkr1···rs (ψ / φ) = γjkr1···rs (ψ)( φe(0, Id) )

Thus, tautologically, one has

∆(ψ0 / φ, . . . , ψp / φ) = ∆(ψ0, . . . , ψp) / φ , φ ∈ GH . 74 Therefore, we have proven the covariance property of the simplex with respect to the right action of G on NH

∆(ψ0 / ϕ, . . . , ψp / ϕ) = ∆(ψ0, . . . , ψp) / ϕ , ϕ ∈ G .

Lemma 5.1.3. The group cochain Cp,•(ω) satisﬁes the covariance property

t ∗ Cp,•(ω)(ψ0 / ϕ, . . . , ψp / ϕ) = Lϕ−1.Cp,•(ω)(ψ0, . . . , ψp) , ϕ ∈ G , where the left translation operator in the right-hand side refers to the left action of G on GH .

Proof. The proof is analogous to Lemma 2.1 [26], it is just necessary to check the coinvariance on the simplex for our cases which was proved above.

The current Cp,•(ω)(ψ0, . . . , ψp) is defined as the integration along the fiber, in the fibration π : J ∞(G) → G , of the current ω ∩ ∆. In order to better understand GH 0 H e it, we shall revisit its definition.

Let ρ = φ · ψ ∈ G, with φ ∈ GH and ψ ∈ NH . Then

ρ−1 = ψ−1 · φ−1 = (ψ−1 . φ−1)(ψ−1 / φ−1) , and so

∞ X −1 −1 j0 (ρ) ∈ (ψ0, . . . , ψp) ⇐⇒ ψ / φ ∈ ∆(ψ0, . . . , ψp) −1 ⇐⇒ ψ ∈ ∆(ψ0, . . . , ψp) / φ

X In view of the covariance property for the simplex, this shows that the cycle (ψ0, . . . , ψp) can be described as the image of the map

∞ σ(ψ0,...,ψp) : GH × ∆p → J0 (G) −1 σ(ψ0,...,ψp)( φ , t ) = φ · s(ψ0,...,ψp)(t) / φ ,

75 where s(ψ0,...,ψp) : ∆p → NH is the map

p X ∞ s(ψ0,...,ψp)(t) = tij0 (ψi) i=0 n p o X with ∆p = t = (t0, . . . , tp) ti ≥ 0 , ti = 1 . i=0

Therefore the deﬁnition of the current Cp,•(ω)(ψ0, . . . , ψp) can be restated as

Z h C (ω)(ψ , . . . , ψ ) , ζ i = σ∗ π∗ (ζ) ∧ ω . p,q 0 p (ψ0,...,ψp) GH e GH ×∆p

Since πGH ◦ σ(ψ0,...,ψp) : GH × ∆p → GH coincides with the projection on the first factor π1 : GH × ∆p → GH , it follows that Z h C (ω)(ψ , . . . , ψ ) , ζ i = π∗(ζ) ∧ σ∗ ( ω ) p,q 0 p 1 (ψ0,...,ψp) e GH ×∆p Z Z = ζ ∧ σ∗ ( ω ) , (ψ0,...,ψp) e GH π∆ Z where stands for the integration along the fiber of π∆ : GH × ∆p → GH . π∆ Thus, the current Cp,•(ω)(ψ0, . . . , ψp) is actually the Poncarédual of the form Z D (ω)(ψ , . . . , ψ ) = σ∗ ( ω ) ∈ Ω dim GH −•(G ) . p,dim GH −• 0 p (ψ0,...,ψp) e H π∆ Transposing the covariance property in Lemma 5.1.3, by Poincaréduality yields the identity

∗ Dp,•(ω)(ψ0 / ϕ, . . . , ψp / ϕ) = Lϕ.Dp,•(ω)(ψ0, . . . , ψp) , ϕ ∈ G .

In particular, for any φ ∈ GH and any q-tuple of left invariant tangent vector ﬁelds

1 q (X ,...,X ) on GH one has

1 q Dp,•(ω)(ψ0, . . . , ψp)(Xφ,...,Xφ)

1 q = Dp,•(ω)(ψ0 / φ, . . . , ψp / φ) (Lφ−1 )∗Xφ,..., (Lφ−1 )∗Xφ

1 q = Dp,•(ω)(ψ0 / φ, . . . , ψp / φ) (X(0,Id),...,X(0,Id)

76 • This equality shows that, as a group cochain with values in Ω (GH ), Dp,•(ω) is com- pletely determined by the values taken at the unit e ∈ GH , i.e. by the “core” cochain Z ∗ • ∗ Ep,•(ω)(ψ0, . . . , ψp) = σ(ψ ,...,ψ )(ω) ∈ ∧ gh . 0 p e φ=e π∆

To give a precise meaning to the localization at the unit e ∈ GH , let us ﬁx a basis of left invariant forms on GH

{ αI = αi1 ∧ · · · ∧ αiq | I = (i1 < ··· < iq) , q ≥ 0 } , then there is a uniquely expression as a sum for the form

Z X D (ω)(ψ , . . . , ψ ) = σ∗ ( ω ) = hI α p,dim GH −• 0 p (ψ0,...,ψp) e (ψ0,...,ψp) I π∆ I

I ∞ with h(ψ0,...,ψp) ∈ C (GH ). Thus, one has Z ∗ X I • ∗ Ep,•(ω)(ψ0, . . . , ψp) = σ(ψ ,...,ψ )(ω) = h(ψ ,...,ψ )αI ∈ ∧ gh . 0 p e φ=e 0 p φ=e π∆ I

I Denoting by Z the basis of left invariant vector fields on GH dual to { αI }, one can express the above coefficients as genuine integrals of forms over simplices Z I ∗ h = ι I σ (ω) , (ψ0,...,ψp) Xe (ψ0,...,ψp) e ∆p which leads to the following unambiguous definition Z ! X ∗ q ∗ Ep,q(ω)(ψ0, . . . , ψp) = ιXI σ(ψ ,...,ψ )(ω) αI ∈ ∧ gh e 0 p e φ=e |I|=q ∆p

Observe now that one can factor the map σ(ψ0,...,ψp) as follows

∗ σ(ψ0,...,ψp) = ν ◦ (IdGH × s(ψ0,...,ψp)) , where

∞ ν : GH × NH → J0 (G) , ν(φ, ψ) = φ i(ψ / φ) ,

−1 i : GH → GH , i(ϕ) = ϕ .

77 Substituting this in the unambiguous deﬁnition of Ep,q(ω) yields the equivalent deﬁ- nition Z ! X ∗ ∗ q ∗ Ep,q(ω)(ψ0, . . . , ψp) = i ιXI ν (ω) αI ∈ ∧ gh e e φ=e |I|=q ∆(ψ0,...,ψp) Lemma 5.1.4. The form

X ∗ ∗ p q ∗ µ(ω) = i ιXI ν (ω) ⊗ αI ∈ Ω NH , ∧ gh e e φ=e |I|=q is invariant under the action of NH , operating on NH by right translation and on the coeﬃcients via the diﬀerential of the action .. In addition, Z q ∗ Ep,q(ω)(ψ0, . . . , ψp) = µ(ω) ∈ ∧ gh ∆(ψ0,...,ψp)

Proof. The new expression for Ep,q(ω)(ψ0, . . . , ψp) is merely a reformulation under the new notation using µ(ω). The invariance is a consequence of the covariance property.

For details of the proof see Lemma 2.4 [26].

Lemma 5.1.5. The map

• • • ∗ E : Ccont(a) → Cpol(NH , ∧ gh) , is a map of complexes.

Proof. The proof is the same as for the primitive pseudogroups, Lemma 2.5 [26].

We have constructed the maps of complexes

s X p ω ∈ Ccont(a) 7→ Cp,r(ω) ∈ Cpol(NH , Ωr(GH )) , p+dim GH −r=s so that after the passage from Cp,r(ω) to Ep,q(ω), r + q = dim GH , by Poincar´eduality it takes the form

s X p q ∗ ω ∈ Ccont(a) 7→ Ep,q(ω) ∈ Cpol(NH , ∧ gh) . p+q=s 78 Furthermore, the last assignment can be promoted to a map of bicomplexes, if one identify the Lie algebra a with the double crossed sum Lie algebra a = gh BC nh and by using the following Lemma.

• Lemma 5.1.6. The complex { Ccont(a), ∂a } is isomorphic to the total complex of

• • the bicomplex { C (gh) ⊗ Ccont(nh), ∂gh , ∂nh }, where ∂gh and ∂nh denote the respective diﬀerentials for Lie algebra cohomology with coeﬃcients.

Proof. The isomorphism is implemented by the map

s M p p \ : C (gh BCnh) → C (gh) ⊗ C (nh) p+q=s

\(ω)(Z1,...,Zp|Z1,..., Zq) = ω(Z1 ⊕ 0,...,Zp ⊕ 0, 0 ⊕ Z1,..., 0 ⊕ Zq).

For details of the proof see Lemma 2.7 [26].

Then, we have the following Proposition.

• • ∗ • • ∗ Proposition 5.1.7. The map E•,• : Ccont(nh, ∧ gh) → Cpol(NH , ∧ gh) induces a quasi-isomorphism of total complexes.

Proof. The proof is analogous to Proposition 2.10 [26], it is suﬃcient to note that nh is a projective limit of nilpotent Lie algebras in our cases for each h.

Therefore, we can conclude with the following Theorem which implies that we have two diﬀerent Hopf algebras with the same peridic Hopf cyclic cohomology.

Theorem 5.1.8. There is a quasi-isomorphism between the periodic Hopf cyclic cohomology of HH and the continuous cohomology of the Lie algebra a.

p q ∗ Proof. It follows from the fact that Cpol(NH , ∧ gh) is one of the bicomplexes that calculates the periodic Hopf cyclic cohomology of HH (Proposition 5.1.1) and it also calculates the continuous cohomology of the Lie algebra a (Proposition 5.1.7), both calculations up to a quasi-isomorphism.

79 BIBLIOGRAPHY

[1] Abe, E. Hopf algebras, vol. 74 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge-New York, 1980. Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka.

[2] Cartan, E. Les groupes de transformations continus, infinis, simples. Annales scientifiques de l’Ecole´ Normale Supérieure 26 (1909), 93–161.

[3] Cartan, H., and Eilenberg, S. Homological algebra. Princeton University Press, Princeton, N. J., 1956.

[4] Connes, A. Spectral sequence and homology of currents for operator algebras. Math. Forschungsinst. Oberwolfach Tagungsber, 41/81, Funktionalanalysis und C∗-Algebren, 27-9/3-10 (1981).

[5] Connes, A., and Moscovici, H. Hopf algebras, cyclic cohomology and the transverse index theorem. Comm. Math. Phys. 198, 1 (1998), 199–246.

[6] Connes, A., and Moscovici, H. Cyclic cohomology and Hopf algebras. Lett. Math. Phys. 48, 1 (1999), 97–108. Mosh´eFlato (1937–1998).

[7] Connes, A., and Moscovici, H. Cyclic cohomology and Hopf algebra sym- metry. Lett. Math. Phys. 52, 1 (2000), 1–28. Conference Mosh´eFlato 1999 (Dijon).

[8] Dasc˘ alescu,˘ S., Nast˘ asescu,˘ C., and Raianu, S¸. Hopf algebras, vol. 235 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, 2001. An introduction.

[9] Di Bruno, F. Note sur une nouvelle formule de calcul diﬀerentiel. Quart. Jour. of Pure and App. Math 1 (1857), 359–360.

[10] Drinfel’d, V. G. Quantum groups. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) (1987), Amer. Math. Soc., Providence, RI, pp. 798–820.

[11] Gel’fand, I. M., and Fuks, D. B. Cohomologies of the Lie algebra of tangent vector ﬁelds of a smooth manifold. Funkcional. Anal. i Priloˇzen. 3, 3 (1969), 32–52. 80 [12] Gel’fand, I. M., and Fuks, D. B. Cohomologies of the Lie algebra of formal vector ﬁelds. Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 322–337.

[13] Guillemin, V. Inﬁnite dimensional primitive Lie algebras. J. Diﬀerential Geometry 4 (1970), 257–282.

[14] Guillemin, V., Quillen, D., and Sternberg, S. The classiﬁcation of the complex primitive inﬁnite pseudogroups. Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 687–690.

[15] Hajac, P. M., Khalkhali, M., Rangipour, B., and Sommerhauser,¨ Y. Hopf-cyclic homology and cohomology with coeﬃcients. C. R. Math. Acad. Sci. Paris 338, 9 (2004), 667–672.

[16] Hajac, P. M., Khalkhali, M., Rangipour, B., and Sommerhauser,¨ Y. Stable anti-Yetter-Drinfeld modules. C. R. Math. Acad. Sci. Paris 338, 8 (2004), 587–590.

[17] Hernandez´ Encinas, L., and Munoz˜ Masque,´ J. A short proof of the generalized Fa`adi Bruno’s formula. Applied Mathematics Letters 16, 6 (2003), 975–979.

[18] Hopf, H. Uber¨ die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verall- gemeinerungen. Ann. of Math. (2) 42 (1941), 22–52.

[19] Kac, G. I. Extensions of groups to ring groups. Mathematics of the USSR- Sbornik 5, 3 (1968), 451.

[20] Kobayashi, S., and Nomizu, K. Foundations of diﬀerential geometry. Vol. I. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1996. Reprint of the 1963 original, A Wiley-Interscience Publication.

[21] Loday, J.-L., and Quillen, D. Cyclic homology and the Lie algebra homology of matrices. Comment. Math. Helv. 59, 4 (1984), 569–591.

[22] Ma, T.-W. Higher chain formula proved by combinatorics. The Electronic Journal of Combinatorics [electronic only] 16, 1 (2009), Research Paper N21, 7 p.–Research Paper N21, 7 p.

[23] Mac Lane, S. Homology. Die Grundlehren der mathematischen Wissenschaften, Bd. 114. Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin- G¨ottingen-Heidelberg, 1963.

[24] Majid, S. Foundations of Quantum Group Theory. Cambridge University Press, 2000.

[25] Moscovici, H., and Rangipour, B. Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology. Adv. Math. 220, 3 (2009), 706–790. 81 [26] Moscovici, H., and Rangipour, B. Hopf cyclic cohomology and transverse characteristic classes. Adv. Math. 227, 1 (2011), 654–729.

[27] Rangipour, B., and Sutl¨ u,¨ S. A van Est isomorphism for bicrossed product Hopf algebras. Comm. Math. Phys. 311, 2 (2012), 491–511.

[28] Shnider, S. Classification of the real infinite simple and real infinite primitive Lie algebras. Proc. Nat. Acad. Sci. U.S.A. 64 (1969), 466–467.

[29] Singer, I. M., and Sternberg, S. The inﬁnite groups of Lie and Cartan. I. The transitive groups. J. Analyse Math. 15 (1965), 1–114.

[30] Sweedler, M. E. Hopf algebras. W. A. Benjamin, 1969.

[31] Tsygan, B. L. Homology of matrix Lie algebras over rings and the Hochschild homology. Uspekhi Mat. Nauk 38, 2(230) (1983), 217–218.

[32] Weibel, C. A. An Introduction to Homological Algebra. Cambridge University Press, 1995.