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
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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.
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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
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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
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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