Hopf Algebra Structures on Quasi-shuffle Algebras and ... · Hopf Algebra Structures ME Ho man Outline Introduction The algebra QSym and multiple zeta values Hopf algebra structure

Hopf AlgebraStructures

ME Hoffman

Outline

Introduction

The algebraQSym andmultiple zetavalues

Hopf algebrastructure

Action of theHopf algebraQSym on H

Generalquasi-shuffleHopf algebras

Hopf Algebra Structures onQuasi-shuffle Algebrasand their Applications

Michael E. Hoffman

U. S. Naval Academy

Algebra and Number TheoryUniversity of Rochester

29 November 2017

ME Hoffman Hopf Algebra Structures


ME Hoffman

Outline

Introduction





Outline

1 Introduction

2 The algebra QSym and multiple zeta values

3 Hopf algebra structure

4 Action of the Hopf algebra QSym on H

5 General quasi-shuffle Hopf algebras



ME Hoffman

Outline

Introduction





Introduction

I have been thinking about the multiple zeta values

ζ(i1, i2, . . . , ik) =∑

n1>n2>···>nk≥1

1

ni11 ni2

2 · · · nikk

since 1988. One of the early results I noticed was the“derivation theorem”

k∑j=1

ζ(i1, . . . , ij + 1, . . . , ik) =

∑j such that ij > 1

ij−2∑p=0

ζ(i1, . . . , ij−1, ij − p, p + 1, ij+1, . . . , ik)

for any sequence (i1, . . . , ik) of integers.



ME Hoffman

Outline

Introduction





Introduction cont’d

Of course this statement is awkward, and early on I startedthinking about an algebraic model for the multiple zeta values.Suppose we form monomials in the noncommuting variables xand y , and associate with the integer sequence (i1, . . . , ik) themonomial x i1−1y · · · x ik−1y . Let H0 be the rational vectorspace spanned by monomials that begin with x and end with y(and the empty monomial 1). We can think of ζ as ahomomorphism from H0 to R sending x i1−1y · · · x ik−1y toζ(i1, . . . , ik). Then there are derivations D and D̄ of H0, bothof degree 1, with D(x) = 0, D(y) = xy and D̄(x) = xy ,D̄(y) = 0. The derivation theorem says thatζ(D(w)) = ζ(D̄(w)) for all monomials w .



ME Hoffman

Outline

Introduction






For example, the derivation theorem applied to (2, 1, 3) is

ζ(3, 1, 3) + ζ(2, 2, 3) + ζ(2, 1, 4) =

ζ(2, 1, 1, 3) + ζ(2, 1, 3, 1) + ζ(2, 1, 2, 2). (1)

But in algebraic guise, the sequence (2, 1, 3) is xy2x2y . Then

D(xy2x2y) = x2y2x2y + xyxyx2y + xy2x3y

andD̄(xy2x2y) = xy3x2y + xy2xyxy + xy2x2y2

so that Eq. (1) is ζ(D(xy2x2y)) = ζ(D̄(xy2x2y)).



ME Hoffman

Outline

Introduction






In fact, we can do a bit better notationally. There is anantiautomorphism τ : H0 → H0 that exchanges x and y , e.g.,τ(x2yxy) = xyxy2. The signficance of τ is that the “dualitytheorem” for multiple zeta values is ζ(τ(w)) = ζ(w) for wordsw of H0, e.g., ζ(2, 1, 1) = ζ(xy3) = ζ(x3y) = ζ(4). I actuallydiscovered this by playing around with examples back in 1988;it has an almost immediate proof from the representation ofmultiple zeta values by iterated integrals. But now thederivation theorem is ζ(D(τ(w)) = ζ(D(w)). (We’ll see ageneralization later.)



ME Hoffman

Outline

Introduction






So now I was thinking of derivations on the algebra H0, and it’snatural to try to extend this to an algebra of operators actingon H0. But how should such an algebra behave? If w = uv inH0 and we want c to act on it, we need a coproduct

∆(c) =∑c

c ′ ⊗ c ′′

so thatc · w =

∑c

(c ′u)(c ′′v),

i.e., the operators should form a bialgebra. Since everything’sgraded here, the bialgebra is automatically a Hopf algebra.This was the germ of my interest in applying Hopf algebras inthis situation.



ME Hoffman

Outline

Introduction





The harmonic product

Let’s back up a bit. Let Q〈x , y〉 be the noncommutativepolynomial algebra on x and y , H its underlying rational vectorspace. Let H1 be the subspace of H generated by 1 andmonomials that end with y . In my 1997 paper I constructedthe “harmonic algebra” as the additive group H1 together withthe product ∗ defined inductively by v ∗ 1 = 1 ∗ v = v for allmonomials v and

x iyv1 ∗ x jyv2 = x iy(v1 ∗ x jv2) + x jy(x iyv1 ∗ v2)

+ x i+j+1y(v1 ∗ v2)

for all monomials v1, v2 and nonnegative integers i , j .



ME Hoffman

Outline

Introduction





The harmonic algera

Then (H1, ∗) is a commutative, associative graded algebra. Infact, it isomorphic to the algebra QSym of quasi-symmetricfunctions as defined by Gessel in 1984. Malvenuto andReutenauer showed in 1995 that QSym is a polynomial algebraon certain words we will define in a moment. Now (H1, ∗) hasthe subalgebra (H0, ∗) generated by 1 and all monomialsstarting in x and ending in y . There is a homomorphismζ : (H0, ∗)→ R given by ζ(1) = 1 and

ζ(x i1−1y · · · x ik−1y) = ζ(i1, . . . , ik)

for all compositions (i1, . . . , ik) with i1 > 1.



ME Hoffman

Outline

Introduction





Quasi-symmetric functions

The quasi-symmetric functions are usually described as follows:let t1, t2, . . . be commuting indeterminates, all thought of ashaving degree 1, and let P be the set of formal power series inthe ti having bounded degree. A formal power series u ∈ P iscalled quasi-symmetric if the coefficient of tp1i1 tp2i2 · · · t

pkik

in u

equals the coefficient of tp11 tp22 · · · tpkk in u whenever

i1 < i2 < · · · < ik . The quasi-symmetric functions form asubalgebra QSym of P, and a vector space basis for QSym isgiven by the monomial quasi-symmetric functions

MI =∑

j1<j2<···<jk

t i1j1 t i2j2 · · · tikjk

for compositions I = (i1, . . . , ik).



ME Hoffman

Outline

Introduction





Quasi-symmetric functions cont’d

The algebra QSym contains the well-known algebra Sym ofsymmetric functions. Indeed M(i) is the power-sum symmetricfunction pi , and the elementary symmetric function ei isM(1,...,1), where (1, . . . , 1) is the composition with i repetitionsof 1. The complete symmetric function hi is the sum of MI

over all compositions I of i . Recognizing that MZVs arehomomorphic images of quasi-symmetric functions was a keyinsight of my 1997 Journal of Algebra paper: sinceSym ⊂ QSym, identities of MZVs can be obtained fromidentities of symmetric functions.



ME Hoffman

Outline

Introduction






There is an isomorphism QSym→ H1 sending M(i1,...,ik ) to

x ik−1y · · · x i1−1y ; we can think of it as sending ti to 1i . (The

latter description doesn’t really make sense on all of QSym, butit sends all MI with I = (i1, . . . , ik) ending in an integer ik > 1to the convergent series ζ(ik , . . . , i1).) Here is anotherformulation of the algebra (H1, ∗). Let z1, z2, . . . be formalnoncommuting indeterminates, and define a multiplication onwords in the zi by w ∗ 1 = 1 ∗ w = w for all words w , and

ziv ∗ zjw = zi (v ∗ zjw) + zj(ziv ∗ w) + zi+j(v ∗ w)

for all positive integers i , j and words v ,w .



ME Hoffman

Outline

Introduction






Then zi1 · · · zik corresponds to x i1−1y · · · x ik−1y . We can nowdescribe the Malvenuto-Reutenauer result giving QSym as apolynomial algebra: first, order the letters zi as z1 > z2 > · · ·and then put words in the zi in lexicographic order, i.e.,

z2 < z2z3 < z22 < z1 < z1z3 < z1z2 < z2

1 < z21 z2 < · · ·

Call such a word Lyndon if it is less than its proper rightfactors. For example, thinking of zi as having weight i , theLyndon words of weight 4 are z4, z3z1, and z2z2

1 . Then QSymis the polynomial algebra on the Lyndon words. Note that theonly Lyndon word starting with z1 is z1 itself.



ME Hoffman

Outline

Introduction





Hopf algebra structure

It is now easy to describe the Hopf algebra structure on QSym.The algebra structure is the ∗-product of words in the zi . Thecoalgebra structure is deconcatenation of such words, i.e., f

∆(zi1zi2 · · · zik ) =k∑

j=0

zi1 · · · zij ⊗ zij+1· · · zik ,

where the first and last terms are 1⊗ zi1 · · · zik and zi1 · · · zik ⊗ 1respectively. This gives QSym the structure of a graded Hopfalgebra, where the weight of zi1 · · · zik is i1 + · · ·+ ik .



ME Hoffman

Outline

Introduction





Generalizations

The nice thing is that this generalizes easily to a much largerclass of examples. We just need a collection A of letters withan algebra structure � : kA⊗ kA→ kA, where k is a field andkA is the set of k-linear combinations of letters. Then if k〈A〉is the noncommutative polynomial algebra in the letters, thereis a quasi-shuffle algebra (k〈A〉, ∗) given inductively byw ∗ 1 = 1 ∗ w = w for all words w , and

au ∗ bv = a(u ∗ bv) + b(au ∗ v) + a � b(u ∗ v)

for all letters a, b and words u, v . Deconcatenation provides acoalgebra structure; this gives a Hopf algebra structure onk〈A〉. It is not necessary to have a grading; word length gives afiltration that make the Hopf algebra connected and filtered.



ME Hoffman

Outline

Introduction





Multiple polylogarithms

For a positive integer r , we can define multiple polylogarithmsat r th roots of unity by

Li(i1,...,ik )(ωj1 , . . . , ωjk ) =

∑n1>···>nk≥1

ωn1j1···+nk jk

ni11 · · · n

ikk

,

where ω = e2πir . These are homormorphic images of the algebra

(k〈A〉, ∗) with A = {zi ,j : i ≥ 1, 0 ≤ j ≤ r − 1} and product

zi ,j � zp,q = zi+p,j+q,

with the second subscript is understood mod r . The case r = 1is MZVs; for r ≥ 2 one has “colored” MZVs.



ME Hoffman

Outline

Introduction





q-MZVs

Another example is the q-version of MZVs. If one defines

ζq(i1, . . . , ik) =∑

n1>n2>···>nk≥1

qn1(i1−1)+···+nk (ik−1)

[n1]i1 [n2]i2 · · · [nk ]ik,

where [n] = (1− qn)/(1− q), then the algebra of thesequantities the homomorphic image of the quasi-shuffle algebra(k〈A〉, ∗) with A = {z1, z2, . . . } and

zi � zj = zi+j + (1− q)zi+j−1.

In this case the ∗ product isn’t compatible with the naturalgrading, but we still have a Hopf algebra structure as notedabove.



ME Hoffman

Outline

Introduction





Action of Hopf algebra QSym on H

Define a Q-linear map . : QSym⊗H→ H by 1 · w = w for anyword w , and for any u ∈ QSym of positive degree let u · x = 0and

u · y =

{xky , u = zk ,

0, otherwise.

Now extend the action to any element of H by

u · w1w2 =∑u

(u′ · w1)(u′′ · w2)

for words w1,w2 of H, where we use the Sweedler notation

∆(u) =∑u

u′ ⊗ u′′.

Then the map · makes Q〈x , y〉 a QSym-module algebra. Infact, the derivation D discussed above is give by D(w) = z1 ·w .



ME Hoffman

Outline

Introduction





Applications

This action makes it easy to state some results about multiplezeta values. For example, Ohno’s theorem (1999), which givesa generalization of both the derivation and duality theorems,has the following succinct statement.

Theorem

For all integers n ≥ 0 and words w of H0,ζ(hn · τ(w)) = ζ(hn · w).

The case n = 0 is the duality theorem, and n = 1 is thederivation theorem. There are further results; for example,from the cyclic derivation theorem of myself and Ohno one hasthe following.

Theorem

For all n,m ≥ 1, ζ(pn · xym) = ζ(pm · xyn).



ME Hoffman

Outline

Introduction





General quasi-shuffle Hopf algebras

Recall that k〈A〉 has a Hopf algebra structure with thequasi-shuffle product ∗ and deconcatenation coproduct ∆.Given a word w = a1a2 . . . an of length n and a compositionI = (i1, . . . , ik) of n, there is an action of I on w given by

I [w ] = (a1 � · · · � ai1)(ai1+1 � · · · � ai1+i2) · · · (an−in+1 � · · · � an).

Then Σ : k〈A〉 → k〈A〉 is defined by

Σ(w) =∑

compositions I of `(w)

I [w ]

There is also a function T : k〈A〉 → k〈A〉 given byT (w) = (−1)`(w)w .



ME Hoffman

Outline

Introduction





General quasi-shuffle Hopf algebras cont’d

Let R : k〈A〉 → k〈A〉 reverse words, i.e.,

R(a1a2 · · · an) = anan−1 · · · a1.

Then we have the following result.

Theorem

The antipode of (k〈A〉, ∗,∆) is S∗ = ΣTR.

The function Σ is of interest since it takes multiple zeta valuesto multiple zeta-star values

ζ?(i1, . . . , ik) =∑

n1≥n2≥···≥nk≥1

1

ni11 ni2

2 · · · nikk

.



ME Hoffman

Outline

Introduction





A noncommuative algebra

There is a noncommutative product � on k〈A〉 defined byw � 1 = w = 1 � w and

a1 · · · an � b1 · · · bm = a1 · · · an−1(an � b1)b2 · · · bm

for m, n ≥ 1. If we let ∆̃ be the reduced coproduct, i.e.,∆̃(1) = 0 and ∆̃(w) = ∆(w)− w ⊗ 1− 1⊗ w , then (k〈A〉, �)has a canonical derivation D = �∆̃, i.e., D(1) = D(a) = 0 forall a ∈ A, and

D(a1a2 · · · an) =n−1∑i=1

a1 · · · ai � ai+1 · · · an

for n ≥ 2. Note that Dn(w) = 0 when n is greater than orequal to the length of the word w .



ME Hoffman

Outline

Introduction





An infinitesimal Hopf algebra

In fact we have the following result.

Theorem

(k〈A〉, �, ∆̃) is an infinitesimal Hopf algebra, with antipodeS� = −Σ−1.

This means that

∆̃(w � v) =∑v

(w � v(1))⊗ v(2) +∑w

w(1) ⊗ (w(2) � v),

where

∆̃(w) =∑w

w(1) ⊗ w(2), ∆̃(v) =∑v

v(1) ⊗ v(2),

and also∑w

S�(w(1))�w(2)+S�(w)+w = 0 =∑w

w(1)�S�(w(2))+S�(w)+w .d =



ME Hoffman

Outline

Introduction





An infinitesimal Hopf algera cont’d

Since D is nilpotent on any given word,

eD =∞∑n=0

Dn

n!

makes sense as an element of Homk(k〈A〉, k〈A〉). From thegeneral theory of infinitesimal Hopf algebras it follows thatΣ−1 = −S� = e−D , and this can easily be improved toΣr = erD for any r ∈ k , where

Σr (w) =∑

compositions I = (i1, . . . , ik ) of `(w)

rk I [w ].

It follows (since the exponential of a derivation is anautomorphism) that Σr is an automorphism of (k〈A〉, �) for allr ∈ k.


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Hopf Algebra Structures on Quasi-shuffle Algebras and ... · Hopf Algebra Structures ME Ho man Outline Introduction The algebra QSym and multiple zeta values Hopf algebra structure