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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 provide ample motivation for our study. We shall now describe
the structure of the report in more detail.
1.2 Contents
The main body of this report is divided as follows. In Chapter 2 we introduce the integral formu-
lation of the MZV as discovered by Kontsevich [Wa], and we explain how it implies the duality
theorem. We go on to provide definition and illustration of the shuffle and stuffle products of Hoff-
man [Ho1], Zagier et al. via exercises in combinatorics. We then develop these notions to describe
the double shuffle and the extended double shuffle relations. We begin Chapter 3 by defining what
is meant by an algebra, and proceed to assess the operations defined in Chapter 2 with a more
algebraic approach, considering the shuffle and stuffle products in terms of the algebra and com-
binatorics of words as in [Ho2], [HO]. We then illustrate the correspondence between the algebra
of words and the MZVs, and describe the sum, duality and derivation theorems in this algebraic
context.
InChapter4weintroducethequasi-symmetricfunctionsQSym. Thisseeminglyunrelatedarea
of mathematics is demonstrated to have a clear connection with the stuffle algebra, and thus an
algebraic relationship with the MZVs. This is done via work of Hazewinkel [Ha2], Hoffman [Ho1]
and others. In Chapter 5 we define bialgebras and Hopf algebras, and provide an explicit example
of a Hopf algebra through QSym, from which we can induce Hopf algebra structure on the MZVs.
In Chapter 6 we are then able to present a more general relation, Ohno’s Theorem [Oh], via an
action of the stuffle Hopf algebra on the algebra of words [HO]. This Hopf algebra is so useful
that we wonder if other Hopf algebras can be obtained from aforementioned products of the MZVs,
and if they are equally useful. In Chapter 7 we prove that in fact the only Hopf algebras to do
with the shuffle, stuffle and concatenation are the shuffle and stuffle Hopf algebras. We do this
by reformulating work in [Cr] for a different space. This observation will prove useful in our final
chapter.
We begin Chapter 8 by defining a quotient space of our original algebraZ which forms the ‘new
zeta space’ NZ. This space seems to be deeply related with two Lie algebras, the double shuffle
Lie algebra ds and the stable derivation Lie algebra D. The stable derivation algebra comes from
certain derivations acting on the pure braid Lie algebra B , whereas the double shuffle algebra is5
an attempt to unify the dual shuffle and stuffle Hopf algebras of Chapter 7 into a single space. We
explicitly construct these algebras and describe in some detail their relationship withNZ.Inthis
description we shall introduce the Drinfel’d associatorΦ and certain formal versions of the newKZ
5￿zeta space, FNZ and FNZ. We discuss the implications of these relationships, and indicate the
need for further study. The work in this chapter comes principally from the theses of Carr [Ca]
and Racinet [Rac], and the work on the stable derivation algebra developed by Ihara [Ih1] and
Furusho [Fu1].
It is recommended that the reader has a working knowledge of algebra and number theory,
with an understanding of of rings, fields, vector spaces and dual vector spaces. Knowledge of
modules, tensor products and direct sums is desirable, but a brief introduction to these is provided
in Appendix A.
1.3 Acknowledgements
Thanks to Herbert Gangl for his advice and originally introducing me to the area, and to Michael
Hoffman and Sarah Carr for their personal correspondence.
6Chapter 2
Relations amongs the MZVs
Having previously defined the MZV as a sum, we shall now formulate it as an integral as
in [Za2], [Wa] and others. This will allow us to introduce two important relations between ze-
tas, the shuffle product [BB] [Ree] and the duality theorem conjectured by Hoffman [Ho1] and
proved by Zagier [Za1]. We will then use the definition of the MZV to define a new product, the
stuffle [Ho1], and by combining the two as in we shall calculate even more interesting relationships
between the MZVs. We then give a main conjecture described in [IKZ].
2.1 The Integral Formulation
As we have seen, any MZV can be written as an infinite sum of rationals. Consider the iterated
integral ￿ ￿ ￿
1 t2dt dt2 1ω (t )ω (t )= (2.1.1)0 2 1 1
t t2 10<t <t <1 0 01 2
where ω (t)=dt/t and ω (t)=dt/(1−t), and also the identity0 1
∞￿
nt =1/(1−t) (2.1.2)
n=0
for |t|< 1. Substituting ( 2.1.2) into ( 2.1.1) gives us the following:
￿ ￿t￿ ￿ ￿ ￿ 2∞ ∞1 t 1 n2￿ ￿dt dt dt dt t2 1 2 2n 1= t dt =11t 1−t t t n0<t <t <1 2 1 0 2 0 0 21 2 n=0 n=1 0
￿ ￿∞ ∞1 n 1 n−1￿ ￿dt t t2 22= = dt2
t n n20 0n=1 n=1
￿ ￿∞ 1n￿ t2= = ζ(2)
2n 0n=1
In fact any multiple zeta value can be written as an iterated integral. To show this explicitly, we
first define alternative notation for an MZV as in [Za2], [Wa]. For a given positive integer a wei
a −1 niassociate the binary string 000...01 = 0 1, where 0 means a string of 0’s of length n.Thenwe
7concatenate (stick together) the strings associated with each a for i=1,2,...,r, and therefore cani
a −1 a −1 a −11 2 rassociate the r-tuplea=(a ,a,...,a)withthestringa˜=0 10 1...0 1. We refer toa˜ as1 2 r
a −1 a −1 a −11 2 r ˜thebinaryformof a,anddenotetheassociationbya =a ,a,...,a ↔ 0 10 1...0 1=a.1 2 r
The original form a is called the integer form.Wethenwrite
a −1 a −1 a −11 2 rζ(a ,a,...,a)=ζ(0 10 1...0 1)1 2 r
Example 2.1.1.
ζ(2) = ζ(01)
ζ(4,1) = ζ(00011)
ζ(2,4,3,1) = ζ(0100010011)
1We shall use the convention that ζ(a+b)=ζ(a)+ζ(b), and similarly for binary forms . Also
any binary form a˜ has weight given by the number of 0’s and 1’s in a˜ and depth given by the
number of 1’s in a˜, which naturally are equal to the weight and depth of a. Note that although
every sequence of positive integers can be associated with a binary string, not every binary string
can be associated with a sequence of integers. We require that a binary string ends with a 1 in
order to be associated with an admissible integer form, and to also start with a 0 in order to be
associated with a convergent zeta, i.e. ensuring that the first integer in the sequence is greater than
1. We can now provide the iterated integral formula for an MZV by the following result due to
Kontsevich [BB], [Za1].
Theorem 2.1.2. (The Kontsevich Formula) Leta=(a ,...,a) be a string of positive integers1 r
of weight n and depth r with binary form a˜=(￿ ···￿ ), where each ￿ is either 1 or 0 for alln 1 i
i=1,..,n. Then for the iterated integral given by
￿ ￿ ￿ ￿1 t tn 2
It(￿ ···￿ ):= ω (t )···ω (t )= ω (t ) ω (t )··· ω (t )n 1 ￿ n ￿ 1 ￿ n ￿ n−1 ￿ 1n n n n−1 1
0<t <t <...<t <1 0 0 01 2 n
(2.1.3)
we have
ζ(a ,a,...,a)=It(￿ ···￿ ) (2.1.4)1 2 r n 1
Example 2.1.3. ζ(2,3,1) =It(010011), ζ(5,2) =It(0000101).
For a detailed proof see App. B, Sec. B.1. Our alternative notation ζ(a)=ζ(a˜) now makes a
great deal of sense. The integral formulation of an MZV has some interesting consequences. We
can use certain properties of iterated integrals to determine relations between the multiple zeta
values.
2.2 The Duality Theorem
We have seen that any string of positive integers (a ,...,a) can be associated with a binary string1 r
of 0’s and 1’s, namely
(a ,...,a)↔ 00...0100...01...00...011 r ￿￿￿￿ ￿￿￿￿ ￿￿￿￿
a −1 a −1 a −11 2 r
1This is valid as we can think of ζ as a homomorphism to the real numbers, as we shall see later.
8

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