Hopf algebras associated to transitive pseudogroups in codimension 2

89 Pages·2016·0.465 MB·English

by  José Rodrigo Cervantes Polanco

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
By
Jos´e Rodrigo Cervantes Polanco , MS in Mathematics
Graduate Program in Mathematics
The Ohio State University
2016
Dissertation Committee:
Henri Moscovici, Advisor
James Cogdell
Thomas Kerler
(cid:13)c Copyright by
Jos´e Rodrigo Cervantes Polanco
2016
ABSTRACT
We associate two different Hopf algebras to the same transitive but not primitive
pseudogrupoflocaldiffeomorphismsonR2 leavinginvariantthetrivialfoliationwhere
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
algebraassociatedtothepseudogroup, andthenitisrealizedasabicrossedproductof
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
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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´e Javier 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.
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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
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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 H . . . . . . . . . . . . . . . . . . . . . . . . . . 19
H
4 Bicrossed product realization. . . . . . . . . . . . . . . . . . . . . . . . 46
4.1 The bicrossed product Hopf algebra F (cid:73)(cid:67)U(g ). . . . . . . . . . . 46
H h
5 Hopf cyclic cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1 HP∗(H ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
H
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
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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 cohomol-
ogy 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 cer-
tain 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.
Ontheotherhand,cycliccohomologywasdiscoveredbyAlainConnesin1981; 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 H associated to the pseudogroup of local diffeomorphisms
n
of Rn 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 H , Henri Moscovici and Bahram Rangipour defined a Hopf algebra
n
H for each infinite primitive Lie-Cartan pseudogroup Π of local diffeomorphisms of
Π
Rn [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
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, 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 con-
struction 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
where G is the subgroup consisting of the affine transformations of Rn 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
ϕ(x ,x ) = (ϕ1(x ,x ),ϕ2(x ))
1 2 1 2 2
and one can consider two different splittings
G = G ·N , G ∩N = {Id}.
H H H H
where H is either the subgroup of upper triangular matrices or the subgroup of
diagonal matrices of GL (R). For each of this splittings, G is the subgroup of
2 H
2
“affine H-motions” transformations of the form L ◦y where L are translations by
x x
x ∈ R2 and y ∈ H, and N is the subgroup of elements of G preserving the origin
H
and with “H-tangent map” at zero the identity matrix. We will denoted by H the
H
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 H and the continuous cohomology
Π
of the Lie algebra of formal vector fields a related to the pseudogroup [5, 26]. This
Π
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
∂ ∂
v = p1(x ,x ) +p2(x )
1 2 2
∂x ∂x
1 2
where p1(x ,x ) and p2(x ) are formal polynomials in their respective variables.
1 2 2
In this dissertation, it is shown that the periodic Hopf cyclic cohomology for each
Hop algebra H is canonically isomorphic to the same continuous cohomology of the
H
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, coho-
mology and pseudogroups.
In chapter §3, the algebra H is defined via its natural action on the crossed
H
product algebra A = C∞(G )(cid:111)G. It is shown that every element h ∈ H satisfies
H c H H
a “Leibniz rule” of the form
h(ab) = h (a)h (b), for all a,b ∈ A .
(1) (2) H
This property together with an “invariant” trace τ gives us the coproduct structure
and the antipode map. Hence, we equipp H with a Hopf algebra structure.
H
3