The Algebra Of Multiple Zeta Values

Robert Henderson

Description

department of mathematics University of Durham The Algebra Of Multiple Zeta Values Robert Henderson Supervised by Dr Herbert Gangl

• braid lie algebra


• many relations known


• relation between


• multiple zeta


• hopf algebras


• shu?e theorem


• known riemann


• zeta values


• lie algebras

Sujets

• Henderson
• Lie algebra

Informations

• Publié par: pefav
• Nombre de lectures: 26
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

department of mathematics
University of Durham
The Algebra Of Multiple Zeta Values
Robert Henderson
Supervised by Dr Herbert GanglAbstract
The Multiple Zeta Values (or ‘MZVs’) have been investigated a great deal in recent years,
yielding a wealth of interesting results and exciting conjectures. The aim of this report is to
introduce the MZVs and their relations, and to display some of the techniques being used in
modern research to describe them. We being with an emphasis on the combinatorial methods of
describing the MZVs, and then develop this work into a more rigorous algebraic study. In
particular we investigate the Hopf algebras associated with the MZVs, which allow us to find
more general forms of known results. We then produce two Lie algebras which are shown to have
interesting connections to the MZVs, and indicate the need for further study.Contents
1 Introduction 3
1.1 Preliminary Definitions and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Relations amongs the MZVs 7
2.1 The Integral Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Duality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 The Shuffle Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 The Stuffle Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 The Double Shuffle Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 The Algebra of the MZVs 18
3.1 The Algebra of Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 The Shuffle and Stuffle Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 The Sum, Duality and Derivation Theorems . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 The Quasi-symmetric Functions 27
4.1 Symmetry and Quasi-symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 QSym and the multiple zeta values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Hopf Algebras 34
5.1 Bialgebras and Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2 The Hopf Algebras of QSym and h . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
6 A Hopf algebra action and Ohno’s Theorem 49
6.1 A Hopf algebra action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Ohno’s Theorem and special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.3 Other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7 A Hopf algebra investigation 57
7.1 Coproducts via products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.2 New Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
18 The new zeta space 63
8.1 The new zeta space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.2 The Braid Lie Algebra and the Stable Derivation Algebra . . . . . . . . . . . . . . . 65
8.3 The Drinfel’d Associator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.4 The double shuffle algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8.5 The formal zeta space and Lie coalgebras . . . . . . . . . . . . . . . . . . . . . . . . 75
9 Conclusion 80
Bibliography 82
A Tensors and Modules 85
B Long Proofs 87
B.1 The Kontsevich Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
B.2 The Derivation - Double Shuffle Theorem . . . . . . . . . . . . . . . . . . . . . . . . 88
B.3 The z-shuffle Hopf algebra is isomorphic to the stuffle Hopf algebra . . . . . . . . . . 89
C Long Calculations 92
C.1 The determination of Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
C.2 Relations between some elements of the Braid Lie algebra . . . . . . . . . . . . . . . 93
2Chapter 1
Introduction
The Multiple Zeta Values have received much attention in recent years, despite them being known
to Euler in the late 1700s. The modern general case was described and popularised by Hoffman and
Zagier [Ho1] [Za1] in the early nineties, and since then there has been a flurry of activity which
has continued up to the present day. The most exciting thing about this study is that despite
the recent boom of interest in the Multiple Zeta Values and the many relations known about
them, their precise structure still remains something of a mystery. For instance Euler proved that
2nζ(2n)∈Qπ ∀n∈N [Ca], but it is still not known whether or not all MZVs are transcendental
numbers. The main goal of the work being done on the MZVs is to classify their structure and
determine all possible relations between them. It is conjectured that all relations can be found
by formally extending just two relations. Another conjecture states that there exist no algebraic
relations between MZVs of different weights which cannot be reduced to relations between MZVs
of a single fixed weight. This second conjecture would imply that the space of multiple zeta values
is a graded algebra, which in turn would imply that all MZVs are transcendental [Fu1]. To prove
these claims is somewhat beyond the scope of this report. It is rather the author’s intention to
introduce the MZVs as combinatorial objects and thence describe how they can be studied in terms
of algebraic structures, in particular their relationship with Hopf and Lie algebras. In Section 1.1
we give the Multiple Zeta Values a proper introduction, along with a few other useful definitions.
Following this, in Section 1.2 we shall provide the reader with a brief overview of the rest of the
report.
1.1 Preliminary Definitions and Motivation
Multiple Zeta Values generalise the evaluation of the well-known Riemann Zeta Function evaluated
at positive integersa>1, which is defined as
∞￿ 1
ζ(a)= .
an
n=1
￿ 2∞ 1 πThe reader will almost certainly be aware of Euler’s famous declaration that = ,which2n=1 6n
2πwe now see can be concisely rewritten as ζ(2) = . Multiple Zeta Values, also known as MZVs or6
simply ‘zetas’, are a generalisation of this idea.
3Definition 1.1.1. For any ordered sequence of positive integers a=(a ,...,a)witha > 1, we1 r 1
define the Multiple Zeta Value (equivalently MZV or zeta) of a as
￿ 1
ζ(a)= (1.1.1)a a a1 2 rn n ...nr1 2n >n >...>n >01 2 r
For example the Riemann Zeta Function evaluated at a positive integer is simply the MZV with
r = 1. We now define two fundamental concepts.
Definition 1.1.2. For the Multiple Zeta Value ζ(a)witha an ordered sequence as above, we say￿ rthat ζ(a) has weight wt(a):= a , and depth dp(a):=r.ii=1
￿
1Example 1.1.3. ζ(2,1) = has depth 2 and weight 2+1=3.a bm>n>0 m n
There exist fascinating relationships between zetas of identical weights. For example it is true
that all MZVs of degree 4 are rational multiples of each other, and also that we have the following
surprising results
ζ(2,1) = ζ(3)
1
ζ(3,1) = ζ(4)
4
ζ(5) = ζ(3,1,1)+ζ(2,1,2)+ζ(2,2,1)
Weshallprovetheseresultslater. TheMZVscanbethoughtofasgeneratingavectorspaceoverQ,
￿
which we shall denote byZ as is done in [Fu1]. ForN ={0,1,2,...}, this is given byZ = Znn∈N
where Z is the vector space overQ generated by all MZVs of the same weight, in other wordsn
Z =￿ζ(a):wt(a)=n￿n Q
with Z = Q and Z = 0. These Z are graded by weight, i.e. for z ∈ Z , z ∈ Z we have0 1 n 1 n 2 m
z z ∈Z , and we sayZ is a graded vector space. Not all the MZVs of weightn are needed for a1 2 m+n
basis of Z as the examples above indicate. There are several conjectures concerning the structuren
of Z, for example
Conjecture 1.1.4. [Fu1] There exist no non-trivialQ-linear relations between MZVs of different
weights.
Here‘non-trivial’istakentomeanthatwecannotreducetherelationintosomeotherrelationsof
uniform weight. For example given the uniform-weight relations ζ(2,1) = ζ(3) and 4ζ(3,1) = ζ(4)
we have the relation ζ(2,1)+ζ(4) = ζ(3)+4ζ(3,1), which is ‘trivial’ as it is reducible into the two
uniform weight relations we started with. The evidence for this conjecture is overwhelming and
there are many theorems describing such relations. For instance
Theorem 1.1.5. (The Sum Theorem) [Ho2] For any positive integers r, n such thatr<n,
￿
ζ(a ,...,a)=ζ(n)1 r
a +...+a =n,a >1r1 1
4For which a particular case is ζ(5) = ζ(4,1) + ζ(3,2) + ζ(2,3) with n = 5, r = 2. There
are two particular relations which we shall discuss in detail, the stuffle and the stuffle. These are
particularly interesting as together they can be formally extended to a set of relations which appear
to generate all other relations between the zetas. This is conjectured by Hoffman [Ho1], Carr [Ca]
and others. We also have the following conjecture
Conjecture 1.1.6. (Zagier) [Za1] The dimensions of the graded levels of Z are given recursively
by dim(Z ) = 0, dim(Z )=dim(Z ) = 1 and1 0 2
dim(Z )=dim(Z )+dim(Z )n n−2 n−3
There is a lot of evidence for this conjecture too, but we are still very far from a proof. This
wealth of conjectures and relations pr