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- braid lie algebra
- many relations known
- relation between
- multiple zeta
- hopf algebras
- shu?e theorem
- known riemann
- zeta values
- lie algebras

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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)=nn 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

5zeta 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