The primitives of the Hopf algebra of noncommutative symmetric functions

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.

Full text

Page 1

Michiel Hazewinkel
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1
CWI
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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 Hazewinkel3 Primitives 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
1126576 69120
,
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 Hazewinkel4Primitives 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)