9 The structure of the shuffle and harmonic al- gebras

Eighth lecture: April 25, 2011

9 The structure of the shuffle and harmonic al- gebras

We shall see that the shuffle and harmonic algebras are polynomial algebras in the Lyndon words.

Consider the lexicographic order on X ^∗ with x 0 < x 1 and denote by L the set of Lyndon words (see §3). We have seen that x 0 is the only Lyndon word which is not in H ¹ _x , while x 0 and x 1 are the only Lyndon words which are not in H ⁰ _x .

For instance, there are 1 + 2 + 2 ² + 2 ³ = 15 words of weight ≤ 3, and 5 among them are Lyndon words:

x 0 < x ² ₀ x 1 < x 0 x 1 < x 0 x ² ₁ < x 1 . We introduce 5 variables

T 10 , T 21 , T 11 , T 12 , T 01 , where T ij corresponds to x ⁱ ₀ x ^j ₁ .

9.1 Shuffle

The structure of the commutative algebra H x is given by Radford Theorem.

Theorem 33. The three shuffle algebras are (commutative) polynomial algebras

H x = K[ L ] x , H ¹ _x = K �

L \ { x 0 } �

x and H ⁰ _x = K �

L \ { x 0 , x 1 } �

x .

We write the 10 non-Lyndon words of weight ≤ 3 as polynomials in these

Lyndon words as follows:

e = e,

x ² ₀ = ¹ ₂ x 0 xx 0 = ¹ ₂ T ₁₀ ² , x ³ ₀ = ¹ ₃ x 0 xx ₀ xx ₀ = ¹ ₃ T ₁₀ ³ ,

x 0 x 1 x 0 = x 0 xx 0 x 1 − 2x ² ₀ x 1 = T 10 T 11 − 2T 21 , x ₁ x ₀ = x ₀ xx ₁ − x ₀ x ₁ = T ₁₀ T ₀₁ − T ₁₁ ,

x 1 x ² ₀ = ¹ ₂ x 0 xx ₀ xx ₁ − x 0 xx ₀ x 1 + x ² ₀ x 1 = ¹ ₂ T ₁₀ ² T 01 − T 10 T 11 + T 21 , x 1 x 0 x 1 = x 0 x 1 xx 1 − 2x 0 x ² ₁ = T 11 T 01 − 2T 12 ,

x ² ₁ = ¹ ₂ x 1 xx ₁ = ¹ ₂ T ₀₁ ² ,

x ² ₁ x 0 = ¹ ₂ x 0 xx 1 xx 1 − x 0 x 1 xx 1 + x 0 x ² ₁ = ¹ ₂ T 10 T ₀₁ ² − T 11 T 01 + T 12 , x ³ ₁ = ¹ ₃ x 1 xx ₁ xx ₁ = ¹ ₃ T ₀₁ ³ .

Corollary 34. We have

H _x = H ¹ _x [x 0 ] _x = H ⁰ _x [x 0 , x 1 ] _x and H ¹ _x = H ⁰ _x [x 1 ] _x .

9.2 Harmonic algebra

Hoffman’s Theorem gives the structure of the harmonic algebra H � :

Theorem 35. The harmonic algebras are polynomial algebras on Lyndon words:

H � = K[L] � , H ⁰ _� = K �

L \ {x 0 , x 1 } �

� and H ¹ _� = K �

L \ {x 0 , x 1 } �

� .

For instance, the 10 non-Lyndon words of weight ≤ 3 are polynomials in the

5 Lyndon words as follows:

e = e,

x ² ₀ = x 0 � x 0 = T ₁₀ ² , x ³ ₀ = x 0 � x 0 � x 0 = T ₁₀ ³ , x 0 x 1 x 0 = x 0 � x 0 x 1 = T 10 T 11 , x 1 x 0 = x 0 � x 1 = T 10 T 01 , x ₁ x ² ₀ = x ₀ � x ₀ � x ₁ = T ₁₀ ² T ₀₁ ,

x 1 x 0 x 1 = x 0 x 1 � x 1 − x ² ₀ x 1 − x 0 x ² ₁ = T 11 T 01 − T 21 − T 12 , x ² ₁ = ¹ ₂ x 1 � x 1 − ¹ ₂ x 0 x 1 = ¹ ₂ T ₀₁ ² − ¹ ₂ T 11 ,

x ² ₁ x 0 = ¹ ₂ x 0 � x 1 � x 1 − ¹ ₂ x 0 � x 0 x 1 = ¹ ₂ T 10 T ₀₁ ² − ¹ ₂ T 10 T 11 ,

x ³ ₁ = ¹ ₆ x 1 � x 1 � x 1 − ¹ ₂ x 0 x 1 � x 1 + ¹ ₃ x ² ₀ x 1 = ¹ ₆ T ₀₁ ³ − ¹ ₂ T 11 T 01 + ¹ ₃ T 21 . In the same way as Corollary 34 follows from Theorem 33, we deduce from Theorem 35:

Corollary 36. We have

H _� = H ¹ _� [x ₀ ] _� = H ⁰ _� [x ₀ , x ₁ ] _� and H ¹ _� = H ⁰ _� [x ₁ ] _� . Remark. Consider the diagram

H   _x −→ K[ L ] _x

 � ^f

 

 � ^g H _� −→ K[ L ] _�

The horizontal maps are just the identitication of H x with K[ L ] x and of H �

with K[L] � . The vertical map f is the identity map on H, since the algebras H _x and H _� have the same underlying set H (only the law differs). But the map g which makes the diagram commute is not a morphism of algebras: it maps each Lyndon word on itself, but consider, for instance, the image of the word x ² ₀ , as a polynomial in K[ L ] _x ,

x ² ₀ = (1/2)x 0 xx ₀ = (1/2)x ^x ₀ ² , but as a polynomial in K[ L ] � ,

x ² ₀ = x 0 � x 0 = x ^�2 ₀ ,

hence g(T ₁₀ ² ) = 2T ₁₀ ² .

10 Regularized double shuffle relations

10.1 Hoffman standard relations

The relation ζ(2, 1) = ζ(3) is the easiest example of a whole class of linear relations among MZV.

For any word w ∈ X ^∗ , each of x 1 � x 0 wx 1 and x 1 xx 0 wx 1 is the sum of x 1 x 0 wx 1 with other words in x 0 X ^∗ x 1 (i.e. convergent words):

x ₁ xw − x ₁ w ∈ H ⁰ , x ₁ � w − x ₁ w ∈ H ⁰ , Hence

x ₁ xw − x ₁ � w ∈ H ⁰ .

It turns out that these elements x ₁ xw − x ₁ � w are in the kernel of ζ � : H ⁰ → R.

Proposition 37. For w ∈ H ⁰ ,

ζ(x � ₁ � w − x ₁ xw) = 0.

Since x 1 xe = x 1 � e = e, these relations are can be written in an equivalent way ζ � (x ₁ � y _s − x ₁ xy _s ) = 0 whenever s ₁ ≥ 2.

They are called Hoffman standard relations between multiple zeta values.

Example. Writing

x ₁ � (x ^s ₀ ^{− 1} x ₁ ) = y ₁ � y _s = y ₁ y _s + y _s y ₁ + y _s+1 and

x 1 x(x ^s ₀ ^{− 1} x 1 ) = x 1 x ^s ₀ ^{− 1} x 1 + x 0 (x 1 xx ^s ₀ ^{− 2} x 1 ) =

� s

ν=1

y ν y s+1 − ν + y s y 1 , we deduce

x ₁ � (x ^s ₀ ^{− 1} x ₁ ) − x ₁ x(x ^s ₀ ^{− 1} x ₁ ) = x ^s ₀ x ₁ −

� s

ν=1

x ^ν ₀ ^{− 1} x ₁ x ^s ₀ ^{− ν} x ₁ , hence

ζ(p) = �

s+s�=p s≥2, s�≥1

ζ(s, s ^� ).

Remark. As we have seen, the word

x ₁ � x ₁ − x ₁ xx ₁ = x ₀ x ₁ is convergent, but

ζ � (x ₁ � x ₁ − x ₁ xx ₁ ) = ζ(2) � = 0.

On the other hand a word like

x ² ₁ � x ₀ x ₁ − x ² ₁ xx ₀ x ₁ = x ₁ x ² ₀ x ₁ + x ² ₀ x ² ₁ − x ₁ x ₀ x ² ₁ − 2x ₀ x ³ ₁

is not convergent; also the “convergent part” of this word in the concatenation algebra H, namely x ² ₀ x ² ₁ − 2x 0 x ³ ₁ = y 3 y 1 − 2y 2 y ² ₁ , does not belong to the kernel of ζ: �

ζ(3, 1) = 1

4 ζ(4), ζ(2, 1, 1) = ζ(4).

We shall see (theorem 40) that the convergent part in the shuffle algebra H x of a word of the form wxw ₀ − w � w 0 for w ∈ H ¹ and w 0 ∈ H ⁰ is always in the kernel of ζ � .

Hoffman’s operator d 1 : H → H is defined by δ(w) = x 1 � w − x 1 xw.

For w ∈ H ⁰ , we have d 1 (w) ∈ H ⁰ and the Hoffman standard relations (37) mean that it satisfies d 1 (H ⁰ ) ⊂ ker ζ. �

10.2 Ihara–Kaneko

There are further similar linear relations between MZV arising from regularized double shuffle relations

Recall Corollaries 34 and 36 of Radford and Hoffman concerning the struc- tures of the algebras H x and H � respectively (we take here for ground field K the field R of real numbers). From

H ¹ _x = H ⁰ _x [x 1 ] x and H ¹ _� = H ⁰ _� [x 1 ] �

we deduce that there are two uniquely determined algebra morphisms Z � _x : H ¹ _� −→ R[X] and Z � _� : H ¹ x −→ R[T ]

which extend ζ � and map x 1 to X and T respectively: for a i ∈ H ⁰ , Z � x

� �

i

a i xx ^x ₁ ⁱ

�

= �

i

ζ(a � i )X ⁱ and Z � �

� �

i

a i � x ^�i ₁

�

= �

i

ζ(a � i )T ⁱ .

Proposition 38. There is a R-linear isomorphism � : R[T ] → R[X] which makes commutative the following diagram:

R[X]

Z � x � H ¹ ↑ �

Z � �

�

R[T ]

The kernel of ζ � is a subset of H ⁰ which is an ideal of the algebra H ⁰ _x and also of the algebra H ⁰ _� . One can deduce from the existence of a bijective map

� as in proposition 38 that ker ζ � generates the same ideal in both algebras H ¹ _x and H ¹ _� .

Explicit formulae for � and its inverse � ⁻¹ are given by means of the gener- ating series

�

�≥0

�(T ^� ) t ^�

�! = exp

� Xt +

� ∞ n=2

(−1) ⁿ ζ(n) n t ⁿ

�

. (39)

and �

� ≥ 0

� ⁻¹ (X ^� ) t ^�

�! = exp

� T t −

� ∞ n=2

( − 1) ⁿ ζ (n) n t ⁿ

� . For instance

�(T ⁰ ) = 1, �(T ) = X, �(T ² ) = X ² + ζ(2), �(T ³ ) = X ³ + 3ζ(2)X − 2ζ(3),

�(T ⁴ ) = X ⁴ + 6ζ (2)X ² − 8ζ(3)X + 27 2 ζ(4).

It is instructive to compare the right hand side in (39) with the formula giving the expansion of the logarithm of Euler Gamma function:

Γ(1 + t) = exp

�

− γt +

� ∞ n=2

( − 1) ⁿ ζ(n) n t ⁿ

� . Also, � may be seen as the differential operator of infinite order

exp

� _∞

�

n=2

(−1) ⁿ ζ(n) n

� ∂

∂T

� n �

(consider the image of e ^tT ).

10.3 Shuffle regularization of the divergent multiple zeta values, following Ihara and Kaneko

Recall that H x = H ⁰ [x ₀ , x ₁ ] _x . Denote by reg _x the Q-linear map H → H ⁰ which maps w ∈ H onto its constant term when w is written as a polynomial in x 0 , x 1 in the shuffle algebra H ⁰ [x 0 , x 1 ] x . Then reg _x is a morphism of algebras H x → H ⁰ _x . Clearly for w ∈ H ⁰ we have

reg _x (w) = w.

Here are the regularized double shuffle relations of Ihara and Kaneko.

Theorem 40. For w ∈ H ¹ and w 0 ∈ H ⁰ ,

reg _x (wxw 0 − w � w 0 ) ∈ ker ζ. �

Define a shuffle regularized extension of ζ � : H ⁰ → R as the map ζ � x : H → R defined by

� ζ _x = � ζ ◦ reg _x .

Hence ζ � x is nothing else than the composite of Z � x with the specialization map R[X] → R which sends X to 0.

With this definition of ζ � _x Theorem 40 can be written ζ � x (wxw 0 − w � w 0 ) = 0 for any w ∈ H ¹ and w 0 ∈ H ⁰ .

Define a map D _x : H −→ H by D _x (e) = 0 and

D x (x �

x �

· · · x �

) =

� 0 if � 1 = 0, x �

· · · x �

if � 1 = 1.

For instance, for m ≥ 0 and for w 0 ∈ H ⁰ , D ⁱ _x (x ^m ₁ w 0 ) =

� x ^m−i ₁ w 0 for 0 ≤ i ≤ m, 0 for i > m.

One checks that D _x is a derivation on the algebra H x . Its kernel contains x ^m ₀ and H ⁰ _x . There is a Taylor expansion for the elements of H ¹ _x :

u = �

i ≥ 0

1

i! reg _x (D ⁱ _x u)xy ₁ ^{x i} , hence Z � _x (u) = �

i ≥ 0

1

i! reg _x (D _x ⁱ (u)X ⁱ , and

reg _x (u) = �

i ≥ 0

( − 1) ⁱ

i! y ^x ₁ ⁱ xD ⁱ _x u.

For w ₀ ∈ H ⁰ with w ₀ = x ₀ w (with w ∈ H ¹ ) and for m ≥ 0, we have reg _x (y ^m ₁ w 0 ) = ( − 1) ^m x 0 (wxy ^m ₁ ).

10.4 Harmonic regularization of the divergent multiple zeta values

There is also a harmonic regularized extension of ζ � by means of the star product.

Recall that, according to Hoffman’s Corollary 36, H � = H ⁰ _� [x 0 , x 1 ] � . Denote by reg _� the Q-linear map H → H ⁰ which maps w ∈ H onto its constant term when w is written as a polynomial in x ₀ , x ₁ in the harmonic algebra H ⁰ _� [x ₀ , x ₁ ] _� . Then reg _� is a morphism of algebras H � → H ⁰ _� . Clearly for w ∈ H ⁰ we have

reg _� (w) = w.

The map D � : H ¹ −→ H ¹ defined by D � (e) = 0 and

D � (y s

y s

· · · y s

) =

� 0 if s 1 = 1, y s

· · · y s

if s 1 ≥ 2,

is a derivation on the algebra H ¹ _� with kernel H ⁰ _� ; there is a Taylor expansion for the elements of H ¹ _� :

u = �

i ≥ 0

1

i! y ^�i ₁ � reg _� (D ⁱ _� u), hence Z � � (u) = �

i ≥ 0

1

i! reg _� (D _� ⁱ u)T ⁱ , and

reg _� (u) = �

i ≥ 0

(−1) ⁱ

i! y ^�i ₁ � (D ⁱ _� u).

For w 0 ∈ H ⁰ and for m ≥ 0, we have

reg _� (y ₁ ^m w 0 ) =

� m

i=0

(−1) ⁱ

i! (y ^m ₁ ^{− i} w 0 ) � y ₁ ⁱ .