Page 1
Michiel Hazewinkel
Direct line: +31-20-5924204
Secretary: +31-20-5924233
Fax: +31-20-5924166
E-mail: mich@cwi.nl
1
CWI
POBox 94079
1090GB Amsterdam
original version: 20 October, 2001
revised version: 26 December, 2001
The primitives of the Hopf algebra of noncommutative symmetric functions
by
Michiel Hazewinkel
CWI
POBox 94079
1090GB Amsterdam
The Netherlands
Abstract. Let NSymm be the Hopf algebra of noncommutative
symmetric functions over the integers. In this paper a description is given
of its Lie algebra of primitives over the integers, Prim(NSymm), in terms
of recursion formulas. For each of the primitives of a basis of
Prim(NSymm), indexed by Lyndon words, there is a recursively given
divided power series over it. This gives another proof of the theorem that
the algebra of quasi-symmetric functions is free over the integers.
MSCS: 16W30, 05E05, 17A50
Key words and key phrases: noncommutative symmetric functions,
quasisymmetric functions, Lyndon word, Hopf algebra, primitive in a
Hopf algebra, curve in a Hopf algebra, divided power series, Ditters-Shay
bi-isobaric decomposition, Newton primitive, symmetric functions,
Leibniz Hopf algebra, Lie Hopf algebra, free Lie algebra, graded Hopf
algebra, coalgebra, free coalgebra, graded coalgebra, free associative
algebra, Newton primitive, Verschiebung morphism, Frobenius morphism.
1. Introduction.
Let NSymm be the Hopf algebra of noncommutative symmetric functions, also known as the
Leibniz Hopf algebra. As an algebra (over the integers) NSymm is simply the free algebra in
countably many indeterminates:
NSymm = Z Z1,Z2,L
and the comultiplication is given by
(1.1)
(Zn) =
Zi⊗ Zj
i+j=n
∑
, Z0=1, i,j ∈N∪{0} (1.2)
NSymm is the noncommutative analogue of the Hopf algebra of symmetric functions
Symm = Z[c1,c2L], (cn)=
ci⊗cj
i+j=n
∑
(1.3)
and more or less recently it has been discovered that very many of the remarkable structures and
properties of the symmetric functions have natural noncommutative analogues in NSymm (or
noncocommutative analogues in the graded dual QSymm of NSymm, the Hopf algebra of
quasisymmetric functions); for instance, Schur functions, Newton primitives, representation
theoretic interpretations, Frobenius reciprocity, ... ; see [3, 7, 8, 9, 10, 14],[6], and other papers.
As often happens a number of things even become nicer or more transparent in the natural
noncommutative generalization.
Page 2
Michiel Hazewinkel2Primitives of NSymm
A primitive element in a Hopf algebra is an element P such that
(P) =1⊗ P +P ⊗1(1.4)
The set of primitives is a Lie algebra under the commutator difference product. For any Hopf
algebra, and in particular for NSymm, there is interest in having a good description of its Lie
algebra of primitives. One reason is that over a field of characteristic zero a cocommutative Hopf
algebra is (isomorphic to) the universal enveloping algebra of its Lie algebra of primitives.
An outstanding question about NSymm is a good description of its Lie algebra of
primitives over the integers, in particular writing down an explict basis of it as a free Abelian
group. This is easy in the case of Symm, where the Lie algebra of primitives is commutative and
a (rather canonical) basis is given by the Newton primitives (given by the same formula (1.7)
below with the Zi replaced by the ci ). This is an instance of where the noncommutative
version is more transparent and easier to prove than its commutative version. Formula (1.7) is
clear and fits well with the recursion fomula
Pn(Z) =nZn− Zn−1P1(Z)−L− Z1Pn−1(Z)
In the commutative case the same recursion formula holds but it takes more than casual
inspiration to guess that the coeffient of a monomial in the c’s in Pn(c) is in fact the sum of the
last indices of all noncommutative monomials that give rise to the same commutative monomial;
and even so the explanation runs via noncommutative monomials.
This matter of primitives in the case of NSymm is vastly more complicated (and more
interesting). To give some indications, lets first consider the matter over the rational numbers
(which simplifies things quite a good deal). To do this consider yet another Hopf algebra
U =Z U1,U2,L , (Un) =1⊗Un+Un⊗1(1.5)
In this case, the Un are primitives and the Lie algebra of primitives, Prim(U), is the free Lie
algebra (over the integers) generated by the Un . This is an object of sufficient interest and
complexity that a book and more can be and has been devoted to it, [11].
Now, over the rationals NSymm and U are isomorphic as Hopf algebras. The
isomorphism is given by setting
1+Z1t + Z2t2+ Z3t3+L = exp(U1t +U2t2+ U3t3+L)
which gives formulas for the Zn as polynomials in U1,U2,L,Un, which are then used to define
an algebra morphism of NSymm into U, which turns out to be a Hopf algebra isomorphism; see
[4] for two proofs of the latter fact.
Thus, up to isomorphism, there is a good description of the primitives of NSymm over the
rationals. Actually, one can do even better (from certain points of view). Define the
(noncommutative) Newton primitives by
∑
(1.6)
Pn(Z) =
(−1)k +1rkZr1Zr2LZrk , ri∈N={1,2,L }
r1+Lrk=n
(1.7)
It is easily proved by induction that the Pn(Z) are primitives of NSymm, and it is also easy to
see that over the rationals NSymm is the free associative algebra generated by the Pn(Z). Thus
over the rationals the Lie algebra of primitives of NSymm is simply the free Lie algebra
generated by the Pn(Z). Over the integers things are vastly different. For one thing
Prim(NSymm) is most definitely not a free Lie algebra; rather it tries to be something like a
divided power Lie algebra (though I do not know what such a thing would be). As it turns out
Page 3
Michiel Hazewinkel3Primitives of NSymm
there is a basis of Prim(NSymm) that includes the Pn(Z) and these generate a free sub Lie
algebra of Prim(NSymm). This free sub Lie algebra is a rather small part of Prim(NSymm). Both
are infinite dimensional free Abelian groups, and the one is a full rank subgroup of the other
(meaning that over the rationals they span the same vectorspace). So, to indicate how small one
is within the other we have to consider some of the extra structure that is present, viz that we are
dealing with graded Abelian groups. The grading is given by weight. In NSymm give Zn
weight n, and a monomial Za1Za2LZam weight a1+ a2+L+ am. This makes NSymm a graded
Abelian group. If an element P is primitive so are all its homogeneous components. Thus
Prim(Symm) is also a graded Abelain group. Further Pn(Z), as defined by formula (1.7) above, is
homogeneous of weight n. Thus the free Lie algebra FL(P) generated by them is also graded.
The homogenous components of both Prim(NSymm) and FL(P) are free of finite rank,
FL(P)n⊂ Prim(NSymm)n and we can study the value of the index of one in the other. The
results for the first few values of n are as follows
n
Index
123456
1126 57669120
,
a sequence of numbers that grows far faster than any exponential. There is in fact a formula. For
any word = [a1,a2,L,an] over the natural numbers let g( ) be the gcd of its entries and let
k( ) be the product of its entries, then
Index of FL(P)n in Prim(NSymm)n = k( )
g( )
∈LYN, wt()=n
∏
.
Here LYN is the set of Lyndon words over N (defined below).
To describe the basis of Prim(NSymm) alluded to (of Prim(NSymm) as a free Abelian
group) a number of definitions are needed.
A word over the natural numbers N = { 1,2,L} is simply a sequence of natural numbers
= [a1,a2,L,am]. The length, lg( ), of the word is m, and its weight is
wt( ) = a1+ a2+L+am. A word over N of weight n is also a called a composition of n.
The proper tails of are the words [ai,ai+1,L,am], i= 2,L,m. A word = [a1,L,am] is
lexicographically larger than a word = [b1,L,bn] iff there is an index i (1≤i ≤min{m,n})
such that a1= b1, L,ai−1= bi−1, ai> bi, or m > n and aj= bj, j = 1,L,n. A word is Lyndon if it
is lexicographicaly smaller than each of its proper tails.
Let H be a Hopf algebra. A curve in H , also called a DPS (divided power series), is a
sequence of elements from H , d =(d(0) =1,d(1),d(2),L) such that for all n
d(i)⊗d(j)
i+ j=n
H(d(n)) =
∑
(1.8)
Here
curve is d(t) =1+ d(1)t+ d(2)t2+L∈H[[t]]; i.e. it is written as a power series in t over H
(where t is a (counting) variable commuting with all elements of H).
With this notation, if d(t) is a curve than so is d(tr), which corresponds to the sequence
H is the comultiplication of the Hopf algebra H. An often convenient notation for a
Page 4
Michiel Hazewinkel4 Primitives of NSymm
1,0,L,0
r−1
1 2 3 ,d(1),0,L,0
r−1
1 2 3 ,d(2),L .
And if d(t),d1(t),d2(t) are curves than so are the power series product d1(t)d2(t) and the power
series inverse d(t)−1. The latter corresponds to the sequence 1, (d(0)), (d(2)),L where is
the antipode of H.
Now consider the free product 2NSymm of NSymm with itself
2NSymm = Z〈X1,Y1,X2,Y2,L〉, (Xn) =
Xi⊗ Xj, (Yn)=
i+ j=n
∑
Yi⊗Yj
i+ j=n
∑
(1.9)
In 2NSymm there are two obvious (natural) curves, viz
X(s) =1+ X1s+ X2s2+L, Y(t)= 1+Y1t +Y2t2+L
(where s is a second counting variable). Now, consider the commutator product
X(s)−1Y(t)−1X(s)Y(t). The Shay-Ditters bi-isobaric decomposition theorem says that this
commutator product can be written uniquely as an ordered product
(1.10)
X(s)−1Y(t)−1X(s)Y(t)=
(1+ La,b(X,Y)satb+ L2a,2b(X,Y)s2at2b+L)
gcd(a,b)=1
→
∏
(1.11)
Here the ordered product is over all pairs of natural numbers (a,b)∈N × N with greatest
common divisor, gcd, equal to 1, and the ordering is
(a,b)>( ′
a , ′
b ) ⇔ a +b > ′
a + ′
b or (a+ b = ′
a + ′
b and a > ′
a )(1.12)
Actually it does not matter much which ordering is used. This decomposition theorem is pretty
obvious once one observes that the monomials of the form sa and tb in the commutator
product have coefficient zero and that for each (k,l)∈N× N there is precisely one pair
(a,b), gcd(a,b)=1, such that the monomial skt
l occurs in
da, b(s,t) = 1+ La,bsatb+ L2a,2bs2at2b+L
viz (a,b)= (k / gcd(k,l),l / gcd(k,l)). The Lk,l(X,Y) have weight k in X and weight l in Y.
Writing (1.11) in the form
(1.13)
X(s)Y(t) =Y(t)X(s)(1+ La,b(X,Y)satb+ L2a,2 b(X,Y)s2at2b+L)
gcd(a,b)=1
→
∏
there immediately follows a recursion formula for the Lra,rb(X,Y); see below in section 3 for a
description of the explicit formula.
It follows readily from (1.11) that for all (a,b)∈N × N , gcd(a,b) =1, the sequences
1,La,b(X,Y),L2a,2 b(X,Y),L
are curves in 2NSymm. It follows also, because of the nature of the comultiplication (1.9) of
2NSymm, that the curves (1.14) can be used to define a new curve from two old ones: if d1,d2
are two curves in a Hopf algebra H, then for each (a,b)∈N × N, gcd(a,b) = 1,
(1.14)
Page 5
Michiel Hazewinkel5Primitives of NSymm
da, b(d1,d2) = (1,La,b(d1,d2),L2a,2b(d1,d2),L)
is also a curve in H.
(1.15)
There is now sufficient notation to give an explicit recursive description and construction
of a basis for Prim(NSymm). The basis consists of primitives P , one for each Lyndon word .
To each Lyndon word there are associated three things, viz a number g( ), the gcd of the
entries of = [a1,L,am], a curve d , and a primitive P that is homogeneous of weight
wt( ). The curves associated to and r
description. For of length 1 , i.e.
= [n], we have
=[ra1,L,ram] are the same. Here is the recursive
g([n]) =n, d[n]= (1,Z1,Z2,L), P[n]= Pn(Z)
For a Lyndon word of length >1 let ′ ′ =[ai,L,am] be its lexicographically smallest
proper tail (suffix), and let ′ = [a1,L,ai−1] be the corresponding prefix of
′ , ′ ′ are Lyndon words. Of course g( ) = gcd(g( ′ ),g( ′ ′ )). The entities associated to
are now
(1.16)
. Then both
g( ) = gcd(g( ′ ),g( ′ ′ ))= gcd(a1,Lam),
d = (1,Lg( ′ )/ g( ),g( ′ ′ )/ g( )(d′ ,d′ ′ ),L2g( ′ )/ g(
P = Pg(
),2g( ′ ′ )/ g()(d′ ,d′ ′ ),L),
)(d )
(1.17)
The main theorem is
Theorem. The P , where runs over all Lyndon words, form a basis (over the integers)
of Prim(NSymm).
From the construction above it is immediate that for Lyndon words with g( ) =1, the
corresponding primitive P is the first term of a curve (DPS). If g( ) >1, there is also such a
curve. To see that there is a second bi-isobaric decomposition theorem. This time we work in
NSymm itself and consider
Z(s)−1Z(t)−1Z(s +t)∈NSymm[[s,t]], Z(s)= 1+Z1s+ Z2s2+L
Again it is clear that the monomials sk and tl have coefficient zero in Z(s)−1Z(t)−1Z(s +t) ,
and again there is a natural bi-isobaric decomposition
(1.18)
Z(s)−1Z(t)−1Z(s +t) =
(1+ Na,b(Z)satb+ N2a,2b(Z)s2at2b+L)
gcd(a,b)=1
→
∏
(1.19)
And again it follows that for each pair (a,b)∈N × N, gcd(a,b) = 1 the sequences
1,Na,b(Z),N2a,2b(Z),L
are curves, and again these curves can be used to construct a new one from a known one. In
particular, if d is a curve in NSymm, then for each n >1
(1.20)
1,N1,n−1(d),N2,2n−2(d),L
is again a curve. A quick check shows that
(1.21)
Page 6
Michiel Hazewinkel6Primitives of NSymm
N1,n−1(Z)= Pn(Z) (1.22)
as defined by (1.7) above. Thus for all Lyndon words the associated primitive P is the first
term of a curve (DPS). It follows that the graded dual Hopf algebra of NSymm, the algebra
QSymm of quasisymmetric functions, is free as an algebra over the integers. A proof of the
implication (all primitives in NSymm extend to curves (DPS’s)) ⇒ QSymm is free) is given in
an appendix (for the case at hand. It is adapted from the one in [12] and the one in [13] (for the
case of Hopf algebras over a field of characteristic zero) and uses ingredients from both. So
Theorem. The Hopf algebra of quasisymmetric functions, the graded dual of NSymm, as an
algebra is (commutative) free.
For another proof of this theorem and more about the algebra QSymm of quasisymmetric
functions as the dual of NSymm, see [5, 6].
Acknowledgements. The description of the primitives of NSymm given in this paper is
essentially the same as the one in the preprint [12]. There are sign differences and the set of
words used by Brian Shay is quite different from the set of Lyndon words used here. So the
actual explicit formulas, when written out completely, are quite a bit different. These differences
probably do not really matter. Any Hall-like or Lazard-like set of words should work. Also
different orderings of the set {(r,s):r,s ∈N∪{0},gcd(r,s)=1} can no doubt be used.
One of the most essential ingredients of the construction, the Ditters-Shay bi-isobaric
decomposition theorem, cf above and below, is due, independently, to both Shay and Ditters;
see, [1, 2, 12] where it occurs in somewhat different forms than here. Another nice notion, not
crucial but very nice and useful, that of a V-curve, see below, is also due to Ditters.
The preprint [12] is very difficult to decipher; first because of the horrendous notations
used and second because of a dozen or more typos and/or inaccuracies on practically all the more
important pages. Still, currently, I have the impression that the preprint is basically correct. In
that case the first proof that the algebra of quasisymmetric functions is free as an algebra over the
integers is due to Shay. The second proof is in [5] and the present paper provides a third one
based on rather similar ideas as those in [12], but with quite different proofs and some new
constructions such as the second bi-isobaric decomposition theorem.
2. Curves, 2-curves, and V-curves.
2.1. Curves.
A curve in a Hopf algebra H is a sequence of elements
d =(d(0) =1,d(1),d(2),L)(2.1.1)
such that
H(d(n)) =
d(i)⊗d(j)
i+ j=n
∑
(2.1.2)
A curve in a Hopf algebra is also called a divided power series. When a curve or DPS is written
as a sequence like (2.1.1), the term d(0)=1 is often omitted. Note that d(1) is a primitive.
A convenient way, in many situations, to write a curve is as a power series in a counting
variable t (which commutes with all elements of H)
d(t) =1+ d(1)t+ d(2)t2+L
Noncommutative multiplication and inversion of power series with coefficients in H turns the
set of curves in a Hopf algebra into a (usually noncommutative) group. In the case of
(2.1.3)
Page 7
Michiel Hazewinkel7 Primitives of NSymm
commutative formal groups these groups, enriched with a number of functorial operations on
them, are classifying. It is a matter of absorbing interest to investigate to what extent this may
still be true in more general situations. The inverse of d(t) in (2.1.3) is the power series
d
−1(t)= 1+H(d(1))t +
H(d(2))t2+L
(2.1.4)
where H is the antipode of the Hopf algebra H.
There are two more useful ways of looking at curves. Let C be the coalgebra
C = Z⊕⊕
i=1
∞
ZZi, (Zn)=
Zi⊗ Zj
i+j=
∑
, Z0=1 (2.1.5)
With the co-unit given by projection on the zero-th factor. Then a curve in a Hopf algebra H is
exactly the same as a coalgebra morphism
C → H, Zia d(i)(2.1.6)
Further the multipication of two curves d, ′
coresponds to the convolution
d with corresponding coalgebra morphisms , ′
C → C ⊗C
⊗ ′
H⊗ H
→
m
→ H
(2.1.7)
Finally, the Hopf algebra of noncommutative symmetric functions NSymm = Z〈Z1,Z2,L〉 is the
free associative algebra on the Zi. Thus a curve in H is also the same as a Hopf algebra
morphism
NSymm → H, (Zi) =d(i)(2.1.8)
and this is the point of view that shall be frequently used below. There is, however, some danger
in this. It is very tempting to write down a similar diagram as (2.1.7)
NSymm → NSymm ⊗ NSymm
⊗ ′
H ⊗ H
→
m
→ H
(2.1.9)
and to think that this is the morphism of Hopf algebras corresponding to the product of d and
′
d . As a rule it is not if H is noncommutative; in particular this is not the case for
H = NSymm. The problem is that if H is noncommutative then H⊗ H
algebra morphism.
m
→ H is not an
The terminology ‘curve’ comes from a special case. Let F be a formal group over a ring A,
R(F)= A[[X1,X2,L,Xn]] its contravariant bialgebra, and U(F) its covariant Lie algebra. A
curve in U(F) is a coalgeba morphism C →
R(F) → A[[t]], i.e. a curve in the sense of formal geometry.
U(F) ; duality gives an algebra morphism
2.2. 2-Curves.
A 2-curve in a Hopf algebra H is a collection of elements c(n,m)∈H , indexed by pairs of
nonnegative integers (n,m) ∈N∪{0}× N∪{0}, c(0,0) =1, such that
∑
(c(n,m)) =
c(n1,m1)⊗c(n2,m2)
n1+n2=n
m1+m2=m
(2.2.1)
Page 8
Michiel Hazewinkel8 Primitives of NSymm
A convenient way of writing a 2-curve is as a powerseries in two variables
∑
c(s,t) =1+
c(n,m)sntm
n+ m>0
(2.2.2)
Power series products and power series inverses of 2-curves are again 2-curves. As in the case of
curves (i.e. 1-curves) there are interpretations in terms of coalgebra morphisms and Hopf algebra
morphisms, but those will not be needed here.
A curve
d(s) = 1+ d(1)s + d(2)s2+L
can be seen as a (degenerate) 2-curve. An example of a most important 2-curve is
(2.2.3)
X(s)−1Y(t)−1X(s)Y(t)∈2NSymm[[s,t]]
where
X(s) =1+ X1s+ X2s2+L and Y(t) =1+ Y1t+ Y2t2+L
are the two (canonical) natural curves in 2NSymm, see (1.9) above.
2.2.4. Lemma. Let a,b ∈N ∪{0} , not both zero, and let d(s,t) be a two curve of the form
d(s,t) =1+ c1sat
b+c2s2at
2b+L
(2.2.5)
Then c(t)= 1+c1t +c2t2+L is a curve.
Proof. Because d s,t
() is a 2-curve we have
∑
(d(n,m)) =
d(n1,m1)⊗ d(n2,m2)
n1+ n2=n
m1+m2=m
(2.2.6)
Now most of the terms on the right hand side of (2.2.6) are zero. The only ones that are possibly
nonzero are of the form (n1,m1) = r1(a,b), (n2,m2)= r2(a,b) and then
(n,m) = (r1+r2)(a,b) = r(a,b), r1+ r2= r and d(n,m) = cr, d(n1,m1)= cr1, d(n2,m2) = cr2 so
that
∑
(cr)=
cr1⊗cr2
r1+r2=r
proving that c(t) is a curve.
2.3. V-curves.
On NSymm and 2NSymm there are some some remarkable Hopf algebra endomorphisms,
called Verschiebung. There is one for every r ∈N , and they are defined as follows (on NSymm
and 2NSymm respectively):
Vr(Zn) =
Zn /r if r divides n
0 otherwise
(2.3.1)
Page 9
Michiel Hazewinkel9Primitives of NSymm
Vr(Xn) =
Xn/r if r divides n
0 otherwise
, Vr(Yn) =
Yn/r if r divides n
0 otherwise
(2.3.2)
A curve
d =(d(0),d(1),d(2),L)
in NSymm or 2NSymm is a V-curve if
Vr(d(n))=
d(n / r) if r divides n
0 otherwise
Let :NSymm →
algebras corresponding to the curve d in NSymm , respectively, 2NSymm. Then d is a V-curve
if and only if commutes with the endomorphisms Vr.
NSymm, respectively, :NSymm → 2NSymm be the morphism of Hopf
2.4. Substituting curves in curves.
Let d be a curve in NSymm, and ′
sequence of elements of H obtained by replacing the Z1,Z2,L in d with ′
More precisely, d(i) is a polynomial in the Zj , d(i)(Z1,Z2,L), and then
d( ′
2.4.1. Proposition. As above let d be a curve in NSymm and ′
algebra H. Then
(i) d( ′
d ) is a curve in H.
(ii) When H = NSymm and both d, ′
d a curve in a Hopf algebra H. Define d( ′
d ) as the
d (2),L.
d (1), ′
d )(i) = d(i)( ′
d (1), ′
d (2),L)
d a curve in a Hopf
d are V-curves, then d( ′
d ) is a V-curve.
Proof. Let
algebras corresponding to d, ′
the composed morphism
d:NSymm → NSymm and
d . Then the morphism of Hopf algebras coresponding to d( ′
′
d :NSymm → H be the morphisms of Hopf
d ) is
NSymm
d
→ NSymm
′
d
→ H
(2.4.2)
Composing Hopf algebra morphisms gives a Hopf algebra morphism, so d( ′
H = NSymm and d, ′
d are V-curves, then
does their composed morphism (2.4.2), proving that d( ′
d ) is a curve. If
d,
′
d both commute with the Vr , and hence so
d ) is a V-curve.
Now let c be a curve in 2NSymm , and d, ′
c(d, ′
d ) is obtained from c by replacing the Xi in c by d(i) and the Yj by ′
d curves in a Hopf algebra H. The sequence
d (j).
2.4.3. Proposition. As above let c be a curve in 2NSymm , and d, ′
algebra H. Then
(i) c(d, ′
d ) is a curve in H
(ii) When H = NSymm and all three curves c,d, ′
V-curve.
d curves in a Hopf
d are V-curves, then c(d, ′
d ) is a
Proof. The pair of curves d, ′
d defines a Hopf algebra morphism
d, ′
d :2NSymm → H, Xia di,Yja ′
d j
(2.4.4)
Page 10
Michiel Hazewinkel10Primitives of NSymm
Then c(d, ′
d ) corresponds to the composed morphism
NSymm
c
→ 2NSymm
d , ′
H
d
→
(2.4.5)
and as a composition of Hopf algebra morphisms this is a Hopf algebra morphism, so that
c(d, ′
d ) is a curve. If H = NSymm and d, ′
d are V-curves, then
(on 2NSymm and NSymm respectively). Hence the composed morphism (2.4.5) also commutes
with the Vr making c(d, ′
d ) a V-curve.
d, ′
d commutes with the Vr
2.4.6. Comments. Write Curve(H) for the group of curves in a Hopf algebra H where
the group multiplication is multiplication of power series. Write E = EndHopf(NSymm), then the
first construction above amounts to defining a right action of the semigroup E on Curve(H).
For d ∈Curve(H) and ∈E, the curve d
is the one corresponding to the composition of
Hopf algebra morphisms
NSymm →
NSymm
d
→ H
(2.4.7)
If H is commutative this action respects the group structure on Curve(H). If H is not
commutative, in particular when H = NSymm, this is not the case. This here is an instance of
the possible pitfalls in looking at curves as Hopf algebra morphisms. If, erroneously, one took
(2.1.9) as corresponding to the power series product of curves, it would follow that this right
action does respect the group structure.
In case H = NSymm there is also a left action. For d ∈Curve(NSymm),
the curve coresponding to the composition of Hopf algebra morphisms
∈E, d is
NSymm
d
→ NSymm → NSymm
(2.4.8)
This one does respect the group structure. In this case of course
EndHopf(NSymm) =E = Curve(NSymm)
and Curve(NSymm)is a set with a noncommutative addition on it (power series multiplication of
curves), a noncommutative multiplication on it (composition of endomorphisms) and the
multiplication is distributive over the addition on the left but not on the right. There is also a unit
(the identity endomorphism, or, as a curve, the natural curve 1+Z1t + Z2t2+L).
3. Isobaric decomposition.
Consider again
2NSymm = Z〈X1,Y1,X2,Y2,L〉, (Xn) =
Xi⊗ Xj, (Yn)=
i+ j=n
∑
Yi⊗Yj
i+ j=n
∑
(3.1)
and the two natural curves
X(s) =1+ X1s+ X2s2+L, Y(t)= 1+Y1t +Y2t2+L
and consider the commutator product
(3.2)
X(s)−1Y(t)−1X(s)Y(t)(3.3)
Page 11
Michiel Hazewinkel11Primitives of NSymm
On the set of pairs of nonnegative integers consider the ordering
(u,v)<wl( ′
u , ′
v ) ⇔ u +v < ′
u + ′
v or (u+ v = ′
u + ′
v and u < ′
u )(3.4)
(Here the index wl on <wl is supposed to be a mnemonic for weight first, then lexicographic.)
3.5. Theorem (Ditters-Shay bi-isobaric decomposition theorem). There are ‘higher
commutators’ (or perhaps better ‘corrected commutators’)
Lu,v(X,Y)∈Z〈X,Y〉, (u,v)∈N×N
(3.6)
such that
X(s)−1Y(t)−1X(s)Y(t)=
(1+ La,b(X,Y)satb+ L2a,2b(X,Y)s2at2b+L)
gcd(a,b)=1
→
∏
(3.7)
where the product is an ordered product for the ordering <wl just introduced, (3.4). Moreover
(i) Lu,v(X,Y) =[Xu,Yv]+(terms of length ≥3) (3.8)
(ii) Lu,v(X,Y) is homogeneous of weight u in X and of weight v in Y. (3.9)
(iii) For gcd(a,b) =1, 1+ La,b(X,Y)satb+ L2a,2b(X,Y)s2at2b+L is a curve. It is also
a V-curve.
Proof. All this basically follows from two simple observations. First that putting s or t zero in
the left hand side of (3.7) gives 1, so that there are no pure powers of s or t in (3.7); second
that for each (u,v)∈N× N there is precisely one (a,b)∈N × N, gcd(a,b) = 1, such that sutv
occurs in
1+ La,b(X,Y)satb+ L2a,2b(X,Y)s2at2b+L
viz (a,b)= (u,v) / gcd(u,v). In more detail, rewrite (3.7), as
(3.10)
X(s)Y(t) =Y(t)X(s)(1+ La,b(X,Y)satb+ L2a,2 b(X,Y)s2at2b+L)
gcd(a,b)=1
→
∏
(3.11)
Comparing coefficients of sutv left and right one finds
XuYv= YvXu+
Yv0Xu0
Lui,vi(X,Y)
i
→
∏
u0,v0≥0;k≥1
ui,vi≥1,i=1,L,k
u0+u1+L+uk=u
v0+v1+L+vk=v
∑
(3.12)
where the product is an ordered one for the ordering
(u1,v1) /gcd(u1,v1) <wl (u2,v2) / gcd(u2,v2) <wlL<wl(uk,vk) / gcd(uk,vk)
This is really a recursion formula for Lu,v(X,Y) (this term being the case u0= 0,v0= 0,k =1).
Explicitely
(3.13)
Page 12
Michiel Hazewinkel12Primitives of NSymm
Lu,v(X,Y) =[Xu,Yv]−
Yv0Xu0
Lui,vi(X,Y)
i
→
∏
u0,v0≥0;k≥1
ui,vi≥1,i=1,L,k
u0+u1+L+uk=u
v0+v1+L+vk=v
u0+v0≥1 or k≥2
∑
(3.14)
The last restriction in the sum in (3.14) is simply a way of saying that every term under the sum
sign has at least two factors (and not of the form Yv0Xu0 by the first condition in the sum of
(3.14)) so that all the L′
u , ′
′
u + ′
v < u +v.
Now define the Lu,v(X,Y) by formula (3.14), then (3.12) holds, and hence (3.11) and
(3.7). This takes care of existence (and uniqueness for that matter). Statements (i) and (ii) follow
immediately from the recursion formula (3.14).
To show that the series (3.10) are curves use induction with respect to bidegree for the
wl-ordering. Suppose that that all the factors on the right of (3.7) have been shown to be 2-curves
up to (but not including) bidegree (u,v). Let (a,b)= (u,v) / gcd(u,v). For all the power series
v (X,Y) on the right hand side of (3.14) have lower weight,
1+ L′
b ) ≠ (a,b), the coefficient of sutv is zero. Also a term from
∑
a , ′
b (X,Y)s
′
a t
′
b + L2 ′
a ,2 ′
b (X,Y)s2 ′
a t2 ′
b +L
(3.15)
with ( ′
a , ′
coeff(s
u =u
v + ′ ′
v =v
′
u t
′
v )
′
′
u + ′ ′
⊗ coeff(s
′ ′
u t
′ ′
v ) (3.16)
in (3.15) can be nonzero only if ( ′
would imply that (u,v) is a multiple of ( ′
right of (3.7), except possibly (3.10), are 2-curves up to and including didegree (u,v). Because
the left hand side of (3.7) is a 2-curve, it follows that the last remaining term, (3.10), is also a
2-curve up to and including bidegree (u,v).
The proof that the series (3.10) are V-curves goes exactly the same way. Again suppose
that this has been proved up to bidegree (u,v). The coefficient of sutv is zero and so is the
coefficient of all su , ′
v ) = r
(3.7) are V-curves up to and including bidegree (u,v) except possibly (3.10) itself. But the left
hand side of (3.7) is a V-curve. Hence (3.10) is also a V-curve up to and including bidegree
(u,v).
u , ′
v ) and ( ′ ′
a , ′
u , ′ ′
b ), which is not the case. Thus all the factors on the
v ) are both multiples of ( ′
a , ′
b ). But that
′
u t
′
v for ( ′
−1(u,v), r |gcd(u,v). Thus all the factors on the right of
Before stating the second isobaric decomposition theorem some preparation is needed. Consider
the natural curve
Z(t) =1+ Z1t + Z2t2+L
in NSymm.
3.17. Lemma. Z(s +t) is a 2-curve.
Proof. The coefficient of satb in Z(s +t) is
a +b
a
Za+b. Let u = a+ b. Applying the
comultiplication gives
u
a
u1+ u2=u
∑
Zu1⊗Zu2
(3.17)
Page 13
Michiel Hazewinkel13Primitives of NSymm
On the other hand
coeff(s
a1tb1)⊗
a1+a2=a
b1+b2=b
∑
coeff(sa2tb2)=
a1+ b1
a1
Za1+ b1⊗
a1+a2= a
b1+b2=b
∑
a2+b2
a2
Za2+b2
(3.18)
Take any u1, u2= u−u1. Then the coefficient of Zu1⊗ Zu2 in (3.18) is equal to
u1
a1
a1∑
u2
a −a1
=
u1+ u2
a
=
u
a
(3.19)
Where the binomial coefficient identity (3.19) follows from looking at the coefficient of ta in
(1+t)u1(1+ t)u2= (1+t)u1+u2
which proves the lemma. From the dual point of view things (when applicable, which is certainly
the case here), things are much easier. If d(t) is a curve in H and H∗ →
corresponding morphisms of algebras, then d(t + s) is the 2-curve correponding to the
composed morphism
Z[[t]] is the
H∗ → Z[[t]]
tas+t
Z[[s,t]].
→
The next bit of preparation concerns the noncommutative Newton primitives
∑
Pn(Z) =
(−1)k +1rkZr1Zr2LZrk , ri∈N={1,2,L }
r1+Lrk=n
(3.20)
(Note that these differ by a sign factor (−1)n+1 from the slightly more often used Newton
primitives, Z1, Z1
recursion relation
2− 2Z2, Z1
3−2Z1Z2−Z2Z1+3Z3,L.). The Newton primitives (3.20) satisfy the
Pn(Z) =nZn− Zn−1P1(Z)− Zn− 2P2(Z) −L− Z1Pn−1(Z)
Now, over NSymm, consider the 2-curve Z(s)−1Z(t)−1Z(s +t).
(3.21)
3.22. Theorem (Second isobaric decomposition theorem). There are unique homogeneous
noncommutative polynomials Nu,v(Z)∈NSymm such that
Z(s)−1Z(t)−1Z(s +t) =
(1+ Na,b(Z)satb+ N2a,2b(Z)s2at2b+L)
a,b∈N
gcd(a,b)=1
→
∏
.(3.23)
Moreover
(i)
Nu,v(Z)=
u+ v
u
Zu+v+ (terms of length ≥2)(3.23)
(ii)
(iii) For each a,b ∈N2, gcd(a,b) =1,
Nu,v(Z) is homogeneous of weight u +v
(3.24)
1+ Na,b(Z)sat
b+ N2a,2b(Z)s2 at2b+L (3.25)
Page 14
Michiel Hazewinkel14Primitives of NSymm
is a 2-curve.
(iv) For each n ≥2 , N1,n−1(Z)= Pn(Z) (3.26)
Proof. The situation is very like the one of the first decomposition theorem above. Again putting
s or t equal to zero in the left hand side of (3.23) gives 1, so that there are no pure powers of s
and t in (3.23), and, again, for every pair (u,v)∈N2 there is precisely one of the factors on the
right hand side of (3.23) in which sutv occurs. Proceeding as, before, i.e. bring Z(s) and Z(t)
over to the right hand side, and compare coefficients, one finds a recursion formula for the
Nu,v(Z)
Nu,v(Z)=
u+ v
u
Zu+v−
Zv0Zu0Nu1,v1(Z)LNuk,vk(Z)
u0+L+uk=u
v0+L+vk=v
ui,vi≥1, for i≥1
u0+v0>0 or k≥2
∑
(3.27)
where the product is again an ordered one; i.e. (3.13) must hold. Using (3.27) as a definition it
follows that (3.23) holds. This takes care of existence and uniquesness. Properties (3.23) and
(3.24) follow immediately from the recursion formula (3.27). Note that there are at least two
factors in each of the terms under the sum sign in (3.27) so that all terms there have weight less
than u +v.
The proof that the (3.25) are curves is exactly the same as in the case of the first isobaric
decomposition theorem, using that the left hand side of (3.23) is a 2-curve; see Lemma 3.17.
Finally, take u =1= v in (3.27). The only possible term in this case under the sum sign on
the right has k = 0, u0=1, v0=1 and thus
N1,1(Z) =2Z2− Z1
2
Now let v ≥ 2. Then the recursion formula (3.27) gives
N1,v=
v +1
1
Zv+1− Z1N1,v−1− Z2N1,v−2−L−Zv−1N1,1− ZvZ1
(3.28)
Noting that P1(Z) = Z1 it follows with induction that this is the same recursion formula as for the
Pn(Z) (with n = v −1, see (3.21) above), proving the last statement of the theorem.
3.29. Remarks. Using a different ordering in the ordered products occurring in (3.7) and
(3.23), and using slightly different ‘commutation formulae’ one obtains different versions of the
Lu,v(X,Y) and Nu,v(Z) with nice symmetry properties. This does not matter for the purposes of
the present paper but probably deserves further exploration for future applications and
calculations.
The first thing one needs is an ordering on the set J ={(a,b)∈N× N: gcd(a,b) = 1} with
the property that (a,b)>swl( ′
a , ′
b ) ⇔ ( ′
b , ′
a ) >swl(b,a). There are many such. Here is one that
fits the present circumstances rather well. Divide J into three parts as follows:
J = J−∪{(1,1)}∪ J+
(3.30)
where J−={(a,b)∈J: a > b}, J+={(a,b)∈J: a < b}Now on J− take the ordering ‘weight
first and lexicographic afterwards, and on J+ take the reverse ordering; i.e (a,b)>swl(c,d) for
(a,b),(c,d)∈J+ if and only if (d,c) >swl(b,a) in J−. Further set J−<swl(1,1)<swlJ+ Thus
restricting to weight ≤ 7, the resulting ordering is
Page 15
Michiel Hazewinkel15 Primitives of NSymm
(2,1) <swl(3,1) <swl(3,2) <swl(4,1)<swl(5,1)<swl(4,3) <swl(5,2) <swl(6,1) <swlL<swl(1,1)
L<swl(1,6)<swl(2,5) <swl(3,4) <swl(1,5)<swl(1,4) <swl(2,3) <swl(1,3) <swl(1,2)
The suffix ‘wsl’ is supposed to be an acronym for ‘symmetric weight first lexicographic after’.
There is a first element, viz (2,1) and a last element, viz. (1,2), and there are inifitely many
elements between the first one and the ‘middle one’, (1.1), and between the middle one and the
last one. In this it is a somewhat unusual order . In each equal weight segment the order is
lexicographic.
Now there are a bi-isobaric decompositions
X(s)−1Y(t)X(s)Y(t)−1=
(1+ ′
L a,b(X,Y)satb+ ′
L 2a,2b(X,Y)s2at2b+L)
gcd(a,b)=1
→
∏
(3.31)
Z(s)−1Z(s+t)Z(t)−1=
(1+ ′
N a,b(Z)satb+ ′
N 2a,2b(Z)s2at2b+L)
a,b∈N
gcd(a,b)=1
→
∏
(3.32)
where now the ordering is the one just defined, i.e. the <swl-ordering, with corresponding
recursion forrmulas
′
L u,v(X,Y) =−[Xu,Yv]−
Xu0
′
L ui,vi(X,Y)
i
→
∏
u0,v0≥0;k≥1
ui,vi≥1,i=1,L,k
u0+u1+L+uk=u
v0+v1+L+vk=v
u0+v0≥1 or k≥2
∑
Yv0
(3.33)
′
N u,v(Z)=
u+ v
u
Zu+v−
Zu0
′
N u1,v1(Z)L
′
N uk,vk(Z)
u0+L+uk=u
v0+L+vk=v
ui,vi≥1, for i≥1
u0+v0>0 or k≥2
∑
Zv0
(3.34)
Let on Z〈X;Y〉 or Z〈Z〉 be the anti-isomorphism of algebras that reverses the order of
multiplication. Thus, e.g. (Z1Z3Z2) = Z2Z3Z1, ([Xi,Yj])= [Yj,Xi]= −[Xi,Yj]. Then there are
the symmetry properties
′
L v,u(Y,X) = ( ′
L u,v(X,Y))(3.35)
′
N v,u(Z)= ( ′
N u,v(Z)) (3.36)
In this case
′
N k,1= Pk+1(Z),
′
N 1,k(Z)= Qk+1(Z)
where the Qn(Z) are the second family of ‘power sum primitives’ defined by the recursion
formula
Qn(Z) = nZn− Q1(Z)Zn −1−Q2(Z)Zn−2−L−Qn−1(Z)Z1
One reason that the ordering <swl is esthetically nice is that one can insert the element (1,0) at
the beginning, before (2,1), and (0,1) at the end, after (1,2). Now write
(3.37)
Lu,0(X,Y) = Xu, L0,v(X,Y) = Yv, Nu,0(Z) = Zu= N0,u(Z)
Page 16
Michiel Hazewinkel16Primitives of NSymm
and then the products in the sums in (3.33) and (3.34) are all according to the ordering
(1,0)<swl(2,1) <swl(3,1)<swl(3,2) <swl(4,1) <swl(5,1) <swl(4,3) <swlL<swl(1,1)
L<swl(3,4) <swl(1,5) <swl(1,4) <swl(2,3) <swl(1,3)<swl(1,2) <swl(0,1)
There is also a second symmetry property, obtained by taking inverses of (3.31) and (3.32)
involving the antipode. Here also an order like <wsl is important because taking inverses
reverses the product order and then switching s and t restores it again.
4. The primitives of NSymm.
The construction of the basis P ,
here for convenience of reference and with more detail, proofs, and explanations.
∈LYN was already given in the introduction. It is repeated
At this stage we have the naturally given (supreme) curve
z; z(i) = Zi; z(t) = 1+ Z1t + Z2t2+L
in NSymm (using all three notations so far used); and, using theorem 3.5 (The Ditters-Shay
bi-isobaric decomposition theorem) and Lemma 2.2.4, a large number of curves
(4.1)
ca,b, ca,b(i) = Lia,ib(X,Y), ca,b(t) = 1+ La,b(X,Y)t+ L2a,ab(X,Y)t2+L
in 2NSymm, one for each pair of positive integers (a,b),a,b ∈N2,N ={1,2,3,L} with
gcd(a,b) =1. Substiting curves in curves in a suitable way gives (potentially) new curves. Below
there is a systematic procedure of doing this that leads to a basis of Prim(NSymm).
(4.2)
Let be a composition (word) over the positive integers
= [a1,a2,L,am], ai∈N
Its weight and length are respectively
(4.3)
wt( ) = a1+L+ am, lg( ) = m
The corresponding noncommutative monomials
(4.4)
Z = Za1Za2LZam
(4.5)
have the same weight and length. The empty composition and the monomial 1 have length and
weight zero.
For use in a minute or so, here is the wll-ordering on compositions and monomials:
<wll
⇔
wt( ) < wt( )
or (wt(a)= wt( ) and lg( )< lg( ))
or (wt( ) = wt( ) and lg( )= lg( ) and
<lexico
)
(4.6)
The index ‘wll’ is supposed to be a mnemonic for ‘weight first, than length, and thereafter
lexocographic’.
Page 17
Michiel Hazewinkel17 Primitives of NSymm
A (nonempty) composition is Lyndon it it is lexicographically smaller than all its
proper tails, [ai,L,am], i =2,Lm. The set of Lyndon words is denoted LYN . To each Lyndon
word there are associated three things
• a positive integer g( ) = gcd(a1,a2,Lam)
• a curve d
• a primitive P = Pg( )(d )
(4.7)
(4.8)
where the Pn(Z) are the Newton primitives as defined by (3.20). That P is indeed a primitive
follows from the fact that the Pn(Z) are primitives (see e.g. [3]) and that, for any curve d,
Pn(d) is the image of Pn(Z) under the Hopf algebra morphism corresponding to the curve d,
and hence a primitive.
It remains to give the recursive construction of the d . This goes by induction on the
length of .
• For lg( ) =1, i.e.
= [n] for some n, d[n]= z
• For lg( ) >1, let ′ ′ =[ai,L,am] be the lexicographically smallest proper tail of
, and let ′ = [a1,L,ai−1]. Then ′ , ′ ′ are both Lyndon. (Indeed, ′ ′ is Lyndon because it
is the lexicographically smallest proper tail of ; further, if [aj,L,ai−1] were lexicographicaly
larger than ′ , then [aj,L,am] would be lexicographically larger than [a1,L,am]=
is not the case; so ′ is also Lyndon). The concatenation factorization
canonical factorization of the Lyndon word . The curve d is now defined by
, which
=′ ∗ ′ ′ is called the
d = cg( ′ )/ g(),g( ′ ′ )/g( )(d′ ,d′ ′ )(4.9)
Note that dr = d for any r ∈N, where r
4.10. Proposition. The wll-smallest term in d (i) is Zig( )−1 and it occurs with coefficient
1.
=[ra1,ra2,L,ram].
Proof. This is proved by induction on length. The case of length 1 is obvious as d[n]= z for all
n. Now let
=′ ∗ ′ ′ be the canonical factorization of . First observe that by theorem 3.5
(ii) (formula (3.9)), there are no monomials in the Lu,v(X,Y) that consist purely of X’s or purely
of Y’s. Using theorem (3.5) (i) and calculating modula lg( )+ 1 is follows that
d (i) = Lig( )−1g( ′ ),ig( )−1g( ′ ′ )(d
′ ,d′ ′ )≡[d′ (ig( )−1g( ′ )),d′ ′ (ig( )−1g( ′ ′ ))](4.11)
By induction the wll-smallest terms of the two expressions in the commutator on the right of
(4.11) are, respectively
Zig( )−1
′ , Zig( )−1
′ ′
and thus the wll-smallest term in d (i) is either
Zig( )−1
′ Zig( )−1
′ ′ = Zig( )−1
with coefficient 1, or Zig( )−1
lexicographically larger than ig( )−1
concatenation.
′ ′ Zig()−1
′ with coefficient -1. However, ig( )−1
′ ∗ig( )−1
′ ′ because
′ ′ ∗ig( )−1′ is
is Lyndon. Here ‘∗‘ denotes
Page 18
Michiel Hazewinkel18Primitives of NSymm
4.12. Corollary. The wll-smallest term in the primitive P is g( )Z for all
∈LYN.
Proof. this follows from the defining formula (4.8) and the observation that Pn(Z) ≡nZn modulo
(length ≥ 2).
4.13. Corollary. The P ,
∈LYN , form a basis of Prim(NSymm) over the rationals.
Proof. Let
weight n are homogeneous of weight n and they are linearly independent because when tested
against the monomials Z ,
∈LYNn={ ∈LYN: wt( ) = n} they form a triangular matrix
(with g( )‘s on the diagonal (ordering LYNn by the wll-ordering)). Now over the rationals
NSymm is isomorphic to
n be the number of Lyndon words of weight n. The P with Lyndon of
U =Z〈U1,U2,L〉, (Un) =1⊗Un+Un⊗1
(see (1.6)); and the isomorphism is degree preserving. The Lie algebra of primitives of U,
Prim(U) is the free Lie algebra generated by the U1,U2,L and has a homogeneous basis
indexed by Lyndon words, Q ,
∈LYN, wt(Q )= wt( ). In particular dim(Prim(U)n) =
follows that the isomorphism maps the space spanned by the P ,
proving the corollary.
n. It
∈LYN onto Prim(U)n ,
The collection of independent primitives P ,
Verschiebung Hopf algebra morphisms Vr.
∈LYN is very well behaved with respect to the
4.14. Theorem. For all ∈LYN
Vr(P ) =
rPr−1 if r divides g( )
0 if r does not divide g( )
(4.15)
Proof. First consider the case of length 1, P[n]= Pn(Z), for which there is an explicit formula, viz
Pn(Z) =
(−1)k +1rkZr1Zr2LZrk , ri∈N={1,2,L }
r1+Lrk=n
∑
(4.16)
Now Vr applied to one of the monomials making up (4.16) gives zero unless each of the
ri, i =1,L,k is divisible by r. This can happen only if r divides n = r1+L+rk, proving that
VrPn(Z) = 0 if r does not divide n
(4.17)
If r does divide all the ri then the result of applying Vr to (−1)k+1rkZr1Zr2LZrk is
r(−1)k+1ukZu1Zu2LZuk, ruj= rj, u1+L+uk=r
and summing this over all possibilities precisely gives rPr−1n(Z). Thus
−1n
VrPn(Z) = rPr−1n(Z) if r divides n
(4.18)
Writing (4.17) and (4.18) out a bit more gives also
Page 19
Michiel Hazewinkel19 Primitives of NSymm
Pn(0,L,0
1 2 3 ,Z1,0,L,0
r−1
r−1
1 2 3 ,Z2,0,L,0
r−1
1 2 3 ,Z3,L) =
0 if r does not divide n
rPr−1n(Z1,Z2,Z3,L) if r divides n
(4.19)
Now let be any Lyndon word. Then P = Pg( )(d (1),L,d (g( ))) . Now by theorem 3.5
(iii), the ca,b are V-curves, and the recursive construction of d starts with the V-curve z so
d is a V-curve by proposition 2.4.3. Thus , using (4.19),
VrPg( )(d (1),d (2),L) = Pg()(Vrd (1),Vrd (2),L) =
0 if r does not divide n
rPr−1g(
= Pg( )(0,L,0
r−1
1 2 3 ,d (1),0,L,0
r−1
1 2 3 ,d (2),L) =
)(d (1),d (2),L)= rPr−1 if r divides n
proving the theorem.
In the proof below of the main theorem, the theorem that the P ,
the integers for Prim(NSymm), the following lemma is used.
∈LYN , form a basis over
4.17. Lemma. Let p be a prime number, then
VpPrim(NSymm) ⊂ pPrim(NSymm)
The proof of this involves the graded dual Hopf algebra of NSymm, the Hopf algebra QSymm
of quasisymmetric functions. This Hopf algebra will again turn up later in this paper. For a good
deal of information on the Hopf algebra of symmetric functions see [6] and the references
therein. Here is a brief description of only some of its structure. Being the graded dual of
NSymm, which has the monomials Z , running over all words over the positive integers, as
an Abelian group basis one can take as a basis for QSymm those words themselves with the
duality pairing
〈 , 〉:NSymm× QSymm → Z, 〈Z , 〉 =
(4.18)
(Kronecker delta). The multiplication on QSymm induced by the pairing (4.18) is the
overlapping shuffle multiplication. It can be described as follows. Let = [a1,L,am] and
= [b1,L.bn] be two compositions. For each k, 0 ≤ k ≤ min(m,n) take a word of length
m + n− k with sofar empty slots. Divide the slots in three disjoint sets A, B, O of sizes m − k,
m+ n −k
m− k,n − k,k
the a’s in their original order; in the slots of B∪O put the b’s in their original order. In the slots
of O there is both an a and a b entry; add those. The product
n −k, k respectively. There are
ways of doing this. In the slots of A∪O put
=×osh
is the sum of all
the words thus obtained. Thus the product has
m+ n−k
m − k,n− k,k
k∑
terms. For instance
[a][b1,b2] =[a,b1,b2]+[b1,a,b2]+[b1,b2,a]+[a +b1,b2]+ [b1,a +b2]
The comultiplication on QSymm is ‘cut’
([a1,L,am]) =[]⊗[a1,L,am]+
[a1,L,ai]⊗[ai+1,L,am]
i=1
m−1
∑
+[a1,L,am]⊗[]
Page 20
Michiel Hazewinkel20Primitives of NSymm
where [] is the empty word which is the unit element of QSymm.
It is useful to have a description in one go of the overlapping shuffle product of possibly
more than two factors. For a matrix M with entries from N∪{0} let
compostion of the column sums of the entries of M. Let 1,
consider all matrices M of k rows that are as follows:
• the i-th row of M consists of zeros and the entries of
• there are no columns consisting entirely of zeros.
Then the product 1
M∑
c(M) denote the
k be k compositions. Now,
2,L,
i in their original order,
2L
k is equal to
12L
k=
ac(M)(4.19)
where M runs over all different matrices satisfying the two conditions just described.
The endomorphisms of Hopf algebras of QSymm dual to the Verschiebung
endomorphisms of NSymm are the Frobenius morphisms
fn([a1,a2,L,am])= [na1,na2,L,nam]
This is immediate from the duality pairing (4.18).
(4.20)
4.21. Lemma. For each prime number p
fp
≡
p mod p
(4.22)
Proof. Let = [a1,L,am] . Consider a matrix M such as in the description of the multiple
overlapping shuffle product of compositions given above and consider a (nonidentity) cyclic
permutation of its rows. Unless all the rows are identical the result is a different matrix (because
p is a prime number and the cyclic group of of p elements has no proper subgroups and the
stabilizer of an element of a set acted on by a group must be a subgroup of that group). The
column sums of a matrix and a row permuted version of that matrix are the same. There is only
one matrix with all rows identical and this one gives the term fp
matrices of the required properties the p cyclic permutations are all different and thus give rise
to terms of the form p(something). This proves the lemma.
=[pa1,L, pam]. For all other
4.23. Proof of Lemma 4.17. Let P be a primitive element of NSymm and let
composition. Then (modulo p)
be any
〈VpP, 〉 = 〈P,fp 〉≡ 〈P,
p〉 = 〈
p(P),
⊗L⊗
p
1 2 4 3 4 〉 =
= 〈1⊗L⊗1
1 2 4 3
p−1
4 ⊗ P+1⊗L⊗1
p−2
1 2 4 3 4 ⊗ P⊗1+L+ P ⊗1⊗L⊗1
p−1
1 2 4 3 4 ,
⊗L⊗
p
1 2 4 3 4 〉 = 0
where
p is the p-fold coproduct ( 2=
Things are now ready for the main theorem and its proof.
,
3=(id ⊗
) ,
4= (id⊗id ⊗
)
3L ).
4.24. Theorem. The P ,
∈LYN form a basis (over the integers) for Prim(NSymm).
Proof. Prim(NSymm) is a graded subgroup of NSymm; let Prim(NSymm)n be the piece of
weight n. By Corollary 4.13, the P with Lyndon of weight n , i.e.
rank subgroup of Prim(NSymm)n⊂ NSymmn. It therefore only remains to show that that it is a
pure subgroup of NSymmn.
∈LYNn, span a full
Page 21
Michiel Hazewinkel21Primitives of NSymm
( A pure subgroup of a free Abelian group A of finite rank is a subgroup G such that if
for a prime number p and a ∈A, pa ∈G , then a ∈G . Now Prim(NSymm)n⊂ NSymmn is a
pure subgroup, because NSymm is torsion free. Finally if G1⊂G2, are subgroups of the torsion
free Abelian finite rank group A, G1 is pure, and G1 and G2 have the same rank than they are
equal).
The proof that (Subgroup generated by the P ,
induction on weight. The case of weight 1 is trivial
∈LYNn)⊂ NSymmn is pure), is done by
NSymm1=ZZ1= Prim(NSymm)1=ZP[1]
(and the case of weight two is easy with Prim(NSymm)2 of rank 1 and ZP[2]=Z(2Z2− Z1
manifestly a pure subgroup of NSymm2= ZZ2⊕ZZ1
weight 3,4,5 by direct inspection; after that things become rapidly too complicated for direct
hand calculation).
So take an n ≥2 and suppose that pureness has been proved for all smaller weights. When
tested against the Lyndon monomials of weight n, the P with of weight n, yield a
triangular matrix with the g( )‘s on the diagonal. It follows that the only primes to worry about
are the divisors of the determinant
g( )
∈LYNn
∏
∑
2)
2; one can also easily deal with the cases
. These are all divisors of n. So let
Q =
x P
∈LYNn
∈pNSymmn
(4.25)
where the prime number may be assumed to be a divisor of n. There is to prove that all the
integer coefficients x are divisible by p. By the nature of the primitiveness condition
p−1Q =′
Q ∈Prim(NSymm)n (as NSymm is torsion free). Hence, by Lemma 4.17,
VpQ ∈p2NSymm. Now apply Vp to equation (4.25) to get (using Theorem 4.14)
Vp(
x P
∈LYNn
∑
) =
px Pp−1
∈LYNn
p divides g( )
∑
= VpQ∈ p2NSymmn
So that
x Pp−1
∈LYNn
p divides g( )
∑
∈pNSymmn
(4.26)
Now the p−1
by induction, we see from (4.26) that the coefficients x in (4.25) for which p divides g( )
are divisible by p. Thus these terms can be moved to the aother side of (4.25) to give a relation
∑
with
∈LYNn and p | g( ) are precisely the elements of LYNp−1n. And thus,
x P
∈LYNn
gcd( p,g( ))=1
∈pNSymmn
(4.27)
Let 〈P , 〉 be the coefficient of the monomial Z in P . (This is just the duality pairing
mapping (4.18)). Then the matrix (〈P , 〉) for ,
gcd(p,g( )) = 1= gcd(p,g( )) is triangular with determinant prime to p (Corollary 4.12).
Together with (4.27) this proves that the prime number p divides all the coefficients x in
(4.27). This finishes the proof.
∈LYNn and
Page 22
Michiel Hazewinkel22 Primitives of NSymm
4.28. Comment. In the proof above rather heavy use has been made of the notion of a
V-curve (via Theorem 4.14). This can be avoided, as in [12], at the price of an additional double
induction (with respect to the higher commutator ideals of NSymm and with respect to the
wll-ordering). The essential thing being to show that the top term (wll-ordering) in (4.25) does
not go to zero under Vp (if p divides its index).
4.29. Corollary (of Theorem 4.24 together with Theorem 4.14).
VrPrim(NSymm) =rPrim(NSymm) (4.30)
This has been proved in [1] under the assumption that the graded dual of NSymm is free over
the integers.
The P ,
power sequence, viz d . For the P ,
which it is the first term. Indeed, consider the curve
∈LYN for which g( ) =1 are by construction the first term of a (curve) divided
∈LYN with g( ) >1, there is also a (natural) curve of
1+ N1,g( )−1(d )t +N2,2g( )−2(d )t
where the Nu,v(Z) are the noncommutative polynomials in NSymm defined by the second
isobaric decomposition theorem 3.22. Thus
2+ N3,3g( )−3+(d )t3+L
(4.31)
4.32. Theorem. The Lie algebra of primitives of NSymm has a basis consisting of
homogenous elements that each extend to an isobaric curve. There are
weight n.
n basis elements of
Here an isobaric curve over a homogeneous primitive of weight n is a curve
d =(1,P,d(2),d(3),L) such that wt(d(k)) = kn, and
Theorem 4.32 has a most important consequence
n= #LYNn.
4.33. Theorem. The graded dual Hopf algebra of NSymm, the Hopf algebra QSymm of
quasisymmetric functions, is free (as a commutative algebra) over the integers with
generators of weight n.
n
There is a proof of the implication 4.32 ⇒ 4.33 in [12]. One can also adapt the proof in [13],
Chapter XIII, p.273ff of the same result for Hopf algebras over a field of characteristic zero. For
completeness sake there is a simplified proof (which has ingredients from both) in the appendix,
adapted to this special case.
For another, completely different, proof of the freeness theorem 4.33, see [5].
References
1.E J Ditters, 0n the structure of P(Z(Z)), Preprint, University of Nijmegen, 1971.
2.E J Ditters, Curves and formal (co)groups, Inv. Math. 17(1972), 1-20.
3.
Jean-Yves-Thibon, Noncommutative symmetric functions, Adv. Math. 112(1995), 218-348.
Israel M Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir S Retakh,
4.
Amsterdam, 1996.
Michiel Hazewinkel, The Leibniz-Hopf algebra and Lyndon words, preprint, CWI,
Page 23
Michiel Hazewinkel23 Primitives of NSymm
5.
Preprint, CWI, Amsterdam and ICTP, Trieste, 1999. To appear Adv. Math.
Michiel Hazewinkel, The algebra of quasi-symmetric functions is free over the integers,
6.
Mikhalev (ed.), Formal series and algebraic combinatorics. Proc. of the 12-th international
conference, Moscow, June 2000, Springer, 2000, 30-44.
Michiel Hazewinkel, Quasisymmetric functions. In: D Krob, A A Mikhalev and A V
7.
Math. 155(2000), 181-238.
Florent Hivert, Hecke algebras, difference operators and quasi symmetic functions, Adv.
8.
of alphabeths, Int. J. Algebra and Computation 7:2(1997), 181-264.
D Krob, B Leclerc, J-Y Thibon, Noncommutative symmetric functions II: transformations
9.
groups and Hecke algebras at q=0, J. of Algebraic Combinatorics 6(1997), 339-376.
Daniel Krob, Jean-Yves Thibon, Noncommutative symmetric functions IV: quantum linear
10.
Uq(glN), Int. J. Algebra and Computation 9:3&4(1997), 405-430.
D Krob, J-Y Thibon, Noncommutative symmetric functions V: a degenerate version of
11.Chr Reutenauer, Free Lie algebras, Oxford University Press, 1993.
12.
College, New York City Univ., undated, probably 1974.
P. Brian Shay, An obstruction theory for smooth formal group structure, preprint, Hunter
13.Moss E. Sweedler, Hopf algebras, Benjamin, 1969, 336pp.
14.
A: Math. Gen. 29(1996), 7337-7348.
J-Y Thibon, B-C-V Ung, Quantum quasisymmetric functions and Hecke algebras, J. Phys.
Appendix
A.1. The graded cocommutative cofree coalgebra over a graded set.
Let S be a graded set; i.e. each element of S has a weight attached to it that is a positive
integer. Only those graded sets are considered for which there are only finitely many elements of
each weight. Let
The graded commutative free algebra over Z on S is the algebra of polynomials
n be the number of elements of S of weight n.
CF(S) = Z[xs:s ∈S], wt(xs) = wt(s) (A.1.1)
Give the set S a total ordering; it is natural to do this in such a way that the elements of weight 1
come first, then those of weight 2, etc.
If f:S → N ∪{0} is a function of finite support to the nonnegative integers, let
xf=
xs
f (s)
s∈S, f(s)> 0
→
∏
(A.1.2)
where the product is ordered by the ordering of S. These monomials form an Abelian group
basis for CF(S). The number of monomials in the xs of weight n is the coefficient of tn in
the power series expression
1
(1−t)
1(1−t2)
2(1−t3)
3L
(A.1.3)
Page 24
Michiel Hazewinkel24Primitives of NSymm
The graded cofree cocommutative coalgebra over S , CCoF(S), is the graded dual of
CF(S). The rank of its homoneous part of weight n is of course again the coefficient of tn in
(A.1.3). By duality it is easy to figure out the comultiplication on CCoF(S). By definition a basis
(as an Abelian group) is given by the finite support functions g:S →
pairing is
N∪{0} and the duality
〈g,xf〉 =
f
g
(A.1.4)
(Kronecker delta). It follows immediately that the comultiplication is
∑
(f ) =
f1⊗ f2
f1+ f2=f
(A.1.5)
The counit takes each f that is not identically zero to zero and the identically zero function on
S to 1. The graded cofree cocommutative coalgebra over S , CCoF(S), has an appropriate
universality property as follows. Let M(S) be the graded free module spanned by S and
M(S)∗ its graded dual module. Then CCoF(S) comes with a natural morphism of Abelian
groups CCoF(S) → M(S)∗ (the dual of the natural inclusion M(S) ⊂CF(S) ), and satisfies
the following universality property: for each graded cocommutative coalgebra C and morphism
of graded Abelian groups C+ → M(S)∗ there is a unique lift C →
morphism of coalgebras. Here C+= Ker( ) where is the counit of CCoF(S). This
universality property will not be used here.
For a construction via the tensor power sum Z⊕ M ⊕ M⊗2⊕M⊗3⊕L, where M is short
for M(S)∗, see [13], Chapter XII, p.243ff.
CCoF(S) that is a
A.2. Proof of the freeness theorem (see Theorem 4.33 above).
For each Lyndon word over the integers let c be the curve over the primitive P
constructed in section 4. I.e.
c =
d if g( )=1
the curve (4.28) if g( ) >1
(A.2.1)
For each finite support function f:LYN → N∪{0} let P
f be the ordered product
P
f=
c (f ( ))
∈LYN
f()>0
→
∏
(A.2.2)
where the wll-ordering is used on LYN . It is an easy exercise to show that
∑
(P
f) =
P
f1⊗ P
f2
f1+f2= f
(A.2.3)
A.2.4. Lemma. The P
f form a basis of NSymm (as an Abelian group).
Proof (following [12]). First independence. The idea is that by applying the comultiplication a
suitable number of times the c (i),i >1 finally break up into tensor products of the c (1)= P
(in the ‘middle terms’) and then independence of the Pa finishes the job.
Define
NSymm, where e: Z →
1= id− e : NSymm → NSymm is the unit
Page 25
Michiel Hazewinkel25 Primitives of NSymm
morphism and : NSymm →
and is the identity on the weight >0 part. Let
Z is the counit. Note that
1 kills the constants of NSymm
2= (1⊗
1) : NSymm → NSymm
⊗2
(comultiplication without the ‘trivial parts’ 1⊗ ? and ?⊗1) and recursively
n= (1
⊗(n−2)⊗
2)vn−1: NSymm →
NSymm⊗n.
For instance, if c(1),c(2),c(3) are the first, second, and third term of a c , then
2(c(3))= c(1)⊗c(2)+c(2)⊗c(1),
Let Sn be the group of permutations on n letters acting on NSymm⊗n by permuting the
factors. For a function of finite support f:LYN →
2(c(1)) = 0,
3(c(3)) =c(1)⊗ c(1)⊗c(1),
3(c(2)) =0.
N∪{0}, let f =
f ( )
∑
and let
Sf=
Sf ( )
f ( )>0
→
∏
⊂ Sf be the corresponding Young subgroup. Now suppose that the P
f with
f < n, n ≥ 2, have already been proved to be independent and let
afP
f=0
f ≤n
∑
. Then
0 =
n(
afPf)
f ≤n
∑
=
af
(⊗
∈Sn/Sf
∑
f =n
∑
P⊗ f( ))
so that all the af with f = n are zero. By the induction hypothesis all the other af are then
also zero, proving independence of the P
f for all f < n+1.
Generation is proved by induction on weight. The cases weight zero and weight 1 being obvious.
So suppose that NSymmk is in the linear span of the P
∑
f for all k < n and let x ∈NSymmn. Then
(x) =1⊗ x + x ⊗1+
bf,gPf⊗Pg
f, g≠0
(A.2.5)
For certain unique coefficients bf,g. (Existence because wt( (x))= wt(x) and the induction
hypothesis; uniqueness because of independence). Cocommutativity now ensures
bf,g= bg, f
(A.2.6)
Writing out coassociativity (using the expression (A.2.5) and using (A.2.3)), gives
bf +g,h= bf ,g+h
If follows that there are unique coefficients ch such that
bf,g= cf +g
and this in turn gives that x −
chPh
h∑
is a primitive (of weight n), showing that x is in the
linear span of the P
f. This proves the lemma.
Now consider the graded cofree cocommutative coalgebra on LYN and consider the following
morphism of Abelian groups
Page 26
Michiel Hazewinkel26Primitives of NSymm
CCoF(LYN) → NSymm, f a P
f
(A.2.7)
By (A.1.5) and (A.2.3) this is a morphism of coalgebras. By Lemma A.2.4 it is surjective. Finally
the
n= #LYNn satisfy the power series relation (see e.g. [11])
(1−t)(1− tn)
n
n=1
∞
∏
=1− 2t
and it follows from the statements above just below (A.1.3) that the rank of CCoF(LYN)n is the
coefficient of tn in (1−t)(1− 2t)−1 which is 2n−1= rank(NSymmn). Thus (A.2.7) is a
homogeneous surjective morphism of graded coalgebras over the integers between graded
coalgebras whose free weight n components have the same finite rank. That makes (A.2.7) an
isomorphism of coalgebras and hence the graded dual of NSymm a free commutative algebra
over the integers.
Download full-text