On Higher Frobenius-Schur Indicators

0.49 MB·English

by  Yevgenia Kashina

On Higher Frobenius-Schur Indicators
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Yevgenia Kashina Yorck Sommerh¨auser

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Yongchang Zhu
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Abstract
]
A
We study the higher Frobenius-Schur indicators of modules over semi-
R
simpleHopfalgebras,andrelatethemtootherinvariantsastheexponent,
. theorder, and theindex.Weprovevariousdivisibility and integrality re-
h
t sults for these invariants. In particular, we prove a version of Cauchy’s
a
theorem for semisimple Hopf algebras. Furthermore,we give some exam-
m
ples that illustrate thegeneral theory.
[

2
v Introduction
9
9
1
For a finite group, one can evaluate a character on the sum of all m-th powers
1
ofthegroupelements.Theresultingnumber,dividedbytheorderofthegroup,
1
3 is called the m-th Frobenius-Schur indicator of the character. The first use of
0 these indicators was made by F. G. Frobenius and I. Schur (cf. [13]) to give a
/ criterionwhenarepresentationofafinitegroupcanberealizedbymatriceswith
h
t realentries—forthis question,it isthe secondindicatorthatis relevant.This is
a
also meaningful for other fields than the complex numbers: Here the indicator
m
tells whether or not a given module is self-dual.
:
v
i Higherindicators,i.e.,indicatorswithm>2,arisewhenoneconsidersthe root
X
number function in a finite group.This function assignsto a groupelement the
r number of its m-th roots,i.e., the number of groupelements whosem-thpower
a
is equal to the given element. It is clear that this number depends only on the
conjugacy class, and therefore defines a class function that can be expanded in
terms of the irreducible characters. Using the orthogonality relations for char-
acters,itisnothardtoseethatthecoefficientofanirreduciblecharacterinthis
expansion is its m-th Frobenius-Schur indicator (cf. [19], Lem. (4.4), p. 49).
For Hopf algebras, Frobenius-Schur indicators were first considered by V. Lin-
chenko and S. Montgomery on the one hand (cf. [27]) and by J. Fuchs, A. Ch.
Ganchev,K.Szlacha´nyi,andP.Vecserny´esontheotherhand(cf.[14]).Here,the
sumofthe m-thpowersofthe groupelements is replacedby the m-thSweedler
1
poweroftheintegral.Theauthorsthenusetheindicators,oratleastthesecond
indicator,to provean analogueof the criterionofFrobenius and Schurwhether
or not a representation is self-dual: The Frobenius-Schur theorem asserts that
this depends on whether the second indicator is 0, 1, or −1.
The topic of the present writing are the higher Frobenius-Schur indicators for
semisimple Hopf algebras and their relation to other invariants of irreducible
characters.These other invariants are the order,the multiplicity, the exponent,
andthe index.Letusbriefly describethe natureoftheseinvariants.Thenotion
of the order of an irreducible character is a generalization of the notion of the
orderofanelementinafinitegroup:Itisthesmallestintegersuchthatthecor-
responding tensor power contains a nonzero invariantsubspace. The dimension
of this invariant subspace is called the multiplicity of the irreducible character.
An irreducible character has order 1 if and only if it is trivial, and has order 2
if andonly if itis self-dual.Inthese cases,the multiplicity ofthe characteris 1.
The exponent of a semisimple Hopf algebra is another invariant, which gener-
alizes the exponent of a group (cf. [21]). The exponent of a module is a slight
generalizationofthis concept:Inthe groupcase,itistheexponentofthe image
of the group in the representation. There are various ways to generalize this
concept to semisimple Hopf algebras; we will use Sweedler powers on the one
hand and a certain canonical tensor on the other hand.
Thenextinvariantthatwestudy,theindexofimprimitivity,arisesfromPerron-
Frobenius theory. It is clear that the matrix representation of the left multipli-
cation by the character of a module with respect to the basis consisting of all
irreducible characters has nonnegative integer entries. As we will explain be-
low,thecorrespondingPerron-Frobeniuseigenvalue,i.e.,thepositiveeigenvalue
thathasthelargestpossibleabsolutevalue,isthe degreeofthecharacter.How-
ever, since the entries of the above matrix are in general not strictly positive,
this eigenvalue is notnecessarilystrictly greaterthanthe absolutevalues of the
other characters, so that there can be other eigenvalues which are not positive,
but have the same absolute value. As we will see, in the most interesting cases
theabovematrixisindecomposable;inthiscase,thenumberofsucheigenvalues
is called the index of imprimitivity.
The text is organized as follows: In Section 1, we discuss the formalism of
Sweedler powers in an arbitrary bialgebra. A Sweedler power of an element
in a bialgebra is constructed by applying the comultiplication several times,
permuting the arisingtensor factorsand multiplying them together afterwards.
This notion is a slight modification of the original notion (cf. [21]), where the
tensorfactorswerenotpermuted,andhastheadvantagethatiteratedSweedler
powers are still Sweedler powers. We then consider the Sweedler powers that
arise from a certain special kind of permutations. This is motivated by the fact
thatthevaluesofcharactersontheSweedlerpowersoftheintegrallieincertain
cyclotomic fields, and this special kind of Sweedler powers is well adapted to
describe the action of the Galois group on these values.
2
From Section 2 on, we consider semisimple Hopf algebras over algebraically
closed fields of characteristic zero. We prove a first formula for the higher
Frobenius-Schur indicators that should be understood as a generalization of
the Frobenius-Schur theorem for these indicators—in particular, it implies the
Frobenius-Schur theorem for the second indicators. This first formula describes
Frobenius-Schur indicators in terms of a certain operator on the corresponding
tensor power of the module, and we establish several other properties of this
operator as well.
In Section 3, we then consider the exponent of a module and prove a second
formula for the higher Frobenius-Schur indicators that uses a certain canonical
tensor. Combining this with the first formula, we prove a version of Cauchy’s
theorem for Hopf algebras:A prime that divides the dimension of a semisimple
Hopf algebra must also divide its exponent. This result was conjectured by
P. Etingof and S. Gelaki (cf. [10]); it was known in the case of the prime 2
(cf. [23]). Furthermore, we prove that the higher indicators are integers if the
exponent is squarefree.
In Section 4, we define the notion of the order and the multiplicity of a module
andprovethattheorderofamoduledividesitsmultiplicitytimesthedimension
ofthe Hopfalgebra.This resultgeneralizesthe theoremthatasemisimple Hopf
algebrathathasanontrivialself-dualsimple modulemusthaveevendimension
(cf.[23])tomodulesofarbitraryorders—nontrivialself-dualsimplemodulesare
oforder2,andthemultiplicityis1inthiscase.Inparticular,wegetinthisway
a fully independent new proof of the old theorem.
In Section 5, we study the index of imprimitivity, or briefly the index, of the
matrix that represents the left multiplication by a character with respect to
the canonical basis that we have in the character ring—the basis consisting of
the irreducible characters. The main result of this section is a precise formula
for the index in terms of central grouplike elements. Essentially, the result says
that the eigenvalues of the above matrix that have the same absolute value as
the degreeare obtainedby evaluating the characteratcertaincentralgrouplike
elements. As a consequence of this formula, we see that the index divides the
order as well as the exponent.
InSection6,weapplyanewtool—theDrinfel’ddoubleoftheHopfalgebra.We
provethat,byrestrictingmodulesovertheDrinfel’ddoubletotheHopfalgebra,
we get a map from the characterring of the Drinfel’d double onto the center of
the character ring of the Hopf algebra. From this, we deduce that the center of
the rational character ring of the Hopf algebra, i.e., the span of the irreducible
characters over the rational numbers, is isomorphic to a product of subfields of
thecyclotomicfieldwhoseorderistheexponentoftheHopfalgebra.Finally,we
deduceathirdformulafortheFrobenius-Schurindicatorsintermsoftheaction
of the Drinfel’d element on the induced module over the Drinfel’d double.
In Section 7, we finally compute explicitly a number of examples. In this way,
we can limit the possible generalizations of the results that we have obtained.
3
The class of examples that we study are certain extensions of group rings by
dual group rings. In particular, the Drinfel’d doubles of finite groups belong to
this class.
Throughoutthewholeexposition,weconsiderabasefieldthatisdenotedbyK.
AllvectorspacesconsideredaredefinedoverK,andalltensorproductswithout
subscriptsaretakenoverK.Unlessstatedotherwise,amoduleisaleftmodule.
The set of naturalnumbers is the set N:={1,2,3,...}; in particular,0 is not a
natural number. The symbol Q denotes the n-th cyclotomic field, and not the
n
fieldofn-adicnumbers,andZ denotesthe setZ/nZofintegersmodulon,and
n
not the ring of n-adic integers.
From Section 2 on, except for Corollary3.4, we assume that the base field K is
algebraicallyclosedofcharacteristiczero.H denotes a semisimple Hopfalgebra
with coproduct ∆, counit ε, and antipode S. We will use the same symbols
to denote the corresponding structure elements of the dual Hopf algebra H∗.
NotethatasemisimpleHopfalgebraisautomaticallyfinite-dimensional(cf.[41],
Chap.V,Exerc.4,p.108).ByresultsofR.G.LarsonandD.E.Radford(cf.[26],
Thm. 3.3, p. 276; [25], Thm. 4, p. 195), H is also cosemisimple and involutory,
i.e.,H∗issemisimpleandtheantipodeisaninvolution.FromMaschke’stheorem
(cf. [28], Thm. 2.2.1, p. 20), we get that there is a unique two-sided integral
Λ∈H such that ε(Λ)=1; this element will be used heavily throughout.
Furthermore, we use the convention that propositions, definitions, and simi-
lar items are referenced by the paragraph in which they occur; they are only
numbered separately if this reference is ambiguous.
1 The Calculus of Sweedler Powers
1.1 Let us begin by introducing some notation that will be used throughout
thissection.Foranaturalnumbern,whichisbydefinitionatleast1,weusethe
notationI :={1,...,n}.Supposenowthatm ,m ,...,m areseveralnatural
n 1 2 k
numbers. On the product I ×I × ...×I , we introduce the so-called
m1 m2 mk
lexicographical ordering. This means that, for two k-tuples (i ,i ,...,i ) and
1 2 k
(j ,j ,...,j ) in this set, we define (i ,i ,...,i ) < (j ,j ,...,j ) if and only
1 2 k 1 2 k 1 2 k
if there is an index l ≤k such that i =j ,i =j ,...,i =j , but i <j .
1 1 2 2 l−1 l−1 l l
As this is a total ordering, there is a unique strictly monotone map
ϕ :I ×I ×...×I −→I
m1,...,mk m1 m2 mk n
where n:=m ·m ·...·m . Explicitly, this map is given by the formula
1 2 k
ϕ (i ,...,i )=(i −1)n +(i −1)n +...+(i −1)n +i
m1,...,mk 1 k 1 2 2 3 k−1 k k
where n :=m ·m ·...·m , so that n =m and n =n. This holds since
i i i+1 k k k 1
if i =j for r=1,...,l−1, but i <j , then we have
r r l l
(i −1)n +...+(i −1)n +i <(j −1)n +...+(j −1)n +j
l l+1 k−1 k k l l+1 k−1 k k
4
as the maximal value that (i −1)n +...+(i −1)n +i can attain is
l+1 l+2 k−1 k k
(m −1)n +...+(m −1)n +m =m n =n
l+1 l+2 k−1 k k l+1 l+2 l+1
and j ≥ 1. Considering this calculation for l = 0, we also see that the map is
k
well-defined. Obviously, we have ϕ =id if k=1.
n In
Now suppose that, for a different l, we have another set of natural numbers
m′,m′,...,m′, and let n′ := m′ ·m′ ·...·m′. On the one hand, we can then
1 2 l 1 2 l
look at the composition
(I ×...×I )×(I ×...×I )−ϕ−m−1−,.−..,−m−k−×−ϕ−m−′1−,.−..,−m→′l I ×I −ϕ−n−,n→′ I
m1 mk m′1 m′l n n′ nn′
On the other hand, we have the map
ϕ :I ×...×I ×I ×...×I −→I
m1,...,mk,m′1,...,m′l m1 mk m′1 m′l nn′
It is clear from the definition of the lexicographical ordering that we have
(i ,...,i ,i′,...,i′)<(j ,...,j ,j′,...,j′)
1 k 1 l 1 k 1 l
if and only if we have (i ,...,i ) < (j ,...,j ) or (i ,...,i ) = (j ,...,j )
1 k 1 k 1 k 1 k
and (i′,...,i′)<(j′,...,j′). This shows that, if all appearing product sets are
1 l 1 l
ordered lexicographically, both maps considered above are strictly monotone,
and therefore must be equal. This proves the following statement:
Lemma For natural numbers m ,...,m and m′,...,m′, we have
1 k 1 l
ϕ =ϕ ◦(ϕ ×ϕ )
m1,...,mk,m′1,...,m′l n,n′ m1,...,mk m′1,...,m′l
where n:=m ·m ·...·m and n′ :=m′ ·m′ ·...·m′.
1 2 k 1 2 l
Of course, this can also be seen from the explicit formulas, as we have
ϕ (ϕ (i ,...,i ),ϕ (i′,...,i′))
n,n′ m1,...,mk 1 k m′1,...,m′l 1 l
=n′(ϕ (i ,...,i )−1)+ϕ (i′,...,i′)
m1,...,mk 1 k m′1,...,m′l 1 l
k l
= (i −1)n′n + (i′ −1)n′ +1
X r r+1 X s s+1
r=1 s=1
=ϕ (i ,...,i ,i′,...,i′)
m1,...,mk,m′1,...,m′l 1 k 1 l
where we have set n =n′ =1.
k+1 l+1
1.2 We now consider the union
∞
Sˆ:= S
[ n
n=1
5
of all permutation groups. Note that the different permutation groups are dis-
joint, which implies that for an element σ ∈ Sˆ we can say exactly to which
permutation group S it belongs. This unique n will be called the degree of σ.
n
On the set Sˆ, we introduce a product as follows: For σ ∈ S and τ ∈ S ,
n m
we define σ·τ ∈ S to be the unique permutation that makes the following
mn
diagram commutative:
ϕm,n //
I ×I I
m n mn
(i,j)7→
σ·τ
(σ(j),τ(i))
(cid:15)(cid:15) (cid:15)(cid:15)
//
I ×I I
n m ϕn,m mn
Explicitly, this permutation is given by the formula
σ·τ((i−1)n+j)=(σ(j)−1)m+τ(i)
This product turns Sˆ into a monoid:
Proposition Sˆ is a monoid with unit element id ∈S .
I1 1
Proof. Suppose that n, m, and p are natural numbers. For ρ ∈ S , σ ∈ S ,
p n
and τ ∈S , we have to show that (ρ·σ)·τ =ρ·(σ·τ). For this, note that the
m
following diagram is commutative:
I ×I ×I ϕm,n×id //I ×I ϕmn,p //I
m n p mn p mnp
(i,j,k)7→ (l,k)7→
ρ·(σ·τ)
(ρ(k),σ(j),τ(i)) (ρ(k),(σ·τ)(l))
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
I ×I ×I // I ×I //I
p n m id×ϕn,m p mn ϕp,mn mnp
In addition, the following diagram is also commutative:
I ×I ×I id×ϕn,p // I ×I ϕm,np //I
m n p m np mnp
(i,j,k)7→ (i,l)7→
(ρ·σ)·τ
(ρ(k),σ(j),τ(i)) ((ρ·σ)(l),τ(i))
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
I ×I ×I // I ×I //I
p n m ϕp,n×id np m ϕnp,m mnp
Since the horizontal compositions in both diagrams are equal by Lemma 1.1,
the associative law follows.
6
For the discussion of the unit element, note that the diagrams
I ×I ϕm,1 // I I ×I ϕ1,n // I
m 1 m 1 n n
(i,1)7→ (1,j)7→
τ σ
(1,τ(i)) (σ(j),1)
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
// //
I ×I I I ×I I
1 m ϕ1,m m n 1 ϕn,1 n
commute, since ϕ (i,1) = i = ϕ (1,i). This shows that id ·τ = τ and
n,1 1,n I1
σ·id =σ. 2
I1
1.3 Suppose now that H is a bialgebra with coproduct ∆ and counit ε. We
define certain powers that will play an important role in the sequel:
Definition For σ ∈S and h∈H, we define the σ-th Sweedler power of h to
n
be
hσ :=h ·h ·...·h
(σ(1)) (σ(2)) (σ(n))
Here,wehaveusedavariantoftheso-calledSweedlernotation(cf.[41],Sec.1.2,
p. 10)to denote the images under the iterated comultiplication∆n :H →H⊗n
as
∆n(h)=h ⊗h ⊗...⊗h
(1) (2) (n)
If n is the degreeof σ, we shall alsosay that hσ is an n-thSweedler power ofh.
This definition of Sweedler powers deviates slightly from the one in [20], [21],
where the permutation σ is always the identity. Of course, this permutation
is not relevant if H is commutative or cocommutative, which is the case in
the theory of algebraic groups, the setting in which Sweedler powers were first
considered (cf. [15], Par. 8.5, p. 474; [42], Sec. 1, p. 3). But the introduction of
theadditionalpermutationcausesSweedlerpowerstobeclosedunderiteration,
as the following power law for Sweedler powers asserts:
Proposition For σ ∈S , τ ∈S , and h∈H, we have (hσ)τ =hσ·τ.
n m
Proof. We givea simplificationofthe originalargumentthatwaspointedout
by P. Schauenburg. Recall that σ·τ satisfies by definition the equation
σ·τ((i−1)n+j)=(σ(j)−1)m+τ(i)
We have in general that
n n m
∆m(h )=h ⊗...⊗h = h
O (j) (1) (mn) OO ((j−1)m+i)
j=1 j=1 i=1
7
We therefore get
n n n m
∆m(hσ)=∆m( h )= ∆m(h )= h
Y (σ(j)) Y (σ(j)) YO ((σ(j)−1)m+i)
j=1 j=1 j=1i=1
m n
= h
OY ((σ(j)−1)m+i)
i=1j=1
Permuting the tensor factors and multiplying, we get
m n m n mn
(hσ)τ = h = h = h =hσ·τ
YY ((σ(j)−1)m+τ(i)) YY (σ·τ((i−1)n+j)) Y (σ·τ(i))
i=1j=1 i=1j=1 i=1
as asserted. 2
1.4 We proceed to construct certain Sweedler powers from special permuta-
tions that are based on finite sequences of positive integers. We denote such
sequences, using square brackets, in the form [n ,n ,...,n ]. We call such a
1 2 k
sequence normalized if every entry divides its predecessor, so that we have
n /n /.../n /n . Given a sequence [n ,n ,...,n ], we define its normal-
k k−1 2 1 1 2 k
ization [n′,n′,...,n′] recursively as follows: We set n′ := n and n′ :=
1 2 k 1 1 i+1
gcd(n ,n′), the greatest common divisor of n and n′.
i+1 i i+1 i
We define a product of two such sequences by the formula
[n ,n ,...,n ][m ,m ,...,m ]=[m n ,m n ,...,m n ,m ,m ,...,m ]
1 2 k 1 2 l 1 1 1 2 1 k 1 2 l
In addition, we introduce a unique element, called the empty sequence and
denotedby[],that isby definitionnormalizedandaunit forthis product.Ifwe
denotethesetofallsuchsequencesbyM andthesetofallnormalizedsequences
by M , we have the following result:
N
Proposition M is a monoid and M is a submonoid. Normalization defines
N
a monoid homomorphism from M to M .
N
Proof. The product is associative since we have
([n ,n ,...,n ][m ,m ,...,m ])[p ,p ,...,p ]=
1 2 k 1 2 l 1 2 q
[p m n ,p m n ,...,p m n ,p m ,p m ,...,p m ,p ,p ,...,p ]=
1 1 1 1 1 2 1 1 k 1 1 1 2 1 l 1 2 q
[n ,n ,...,n ]([m ,m ,...,m ][p ,p ,...,p ])
1 2 k 1 2 l 1 2 q
It is obvious that M is a submonoid. If [n ,...,n ] and [m ,...,m ] are two
N 1 k 1 l
sequences with normalizations [n′,...,n′] and [m′,...,m′], then the normal-
1 k 1 l
ization of the product sequence [m n ,m n ,...,m n ,m ,m ,...,m ] is
1 1 1 2 1 k 1 2 l
[m′n′,m′n′,...,m′n′,m′,m′,...,m′]
1 1 1 2 1 k 1 2 l
sincem′n′ =gcd(m′n ,m′n′).Thisprovesthatnormalizationisamonoid
1 i+1 1 i+1 1 i
homomorphism. 2
8
1.5 Suppose now that [n ,n ,...,n ] is a finite sequence of positive integers,
1 2 k
and denote its normalization by [n′,n′,...,n′], so that n′ := gcd(n ,n′).
1 2 k i+1 i+1 i
We define the numbers m ,m ,...,m by n′ = m n′ and put m := n′.
1 2 k−1 i i i+1 k k
Similarly, we define l ,l ,...,l by n = l n′ and put l := 1. Note that
1 2 k−1 i+1 i i+1 k
m and l are relatively prime by construction, which implies that the map
i i
ρ :I →I , j 7→l (j−1)+1
i mi mi i
isbijective,wheretherighthandsidereallyshouldbeunderstoodastheunique
element of I that is congruent to l (j −1)+1 modulo m . The subtraction
mi i i
andadditionof1inthedefinitionofρ stems fromthe factthatwehavechosen
i
the set I to consist of the numbers from 1 to m , and not from 0 to m −1.
mi i i
Up to this shift, ρ is just multiplication by l . Note that ρ =id .
i i k Imk
Definition We define P(n ,...,n )∈S to be the unique permutation that
1 k n1
makes the following diagram commutative:
I ×...×I ϕmk,...,m1 //I
mk m1 n1
(ρ1((iik1,).,....,.i,1ρ)k7→(ik)) P(n1,...,nk)
(cid:15)(cid:15) (cid:15)(cid:15)
I ×...×I // I
m1 mk ϕm1,...,mk n1
For the empty sequence [], we define P():=id .
I1
Sincewehaven =m ·...·m ,thisdefinitionmakessense.Therelationbetween
1 1 k
sequences and permutations becomes clear through the following fact:
Proposition The map
M →Sˆ, [n ,...,n ]7→P(n ,...,n )
1 k 1 k
is a monoid homomorphism.
Proof. Suppose that, besides the sequence [n ,n ,...,n ] considered above,
1 2 k
we have another sequence [p ,p ,...,p ] with normalization [p′,p′,...,p′], so
1 2 l 1 2 l
that p′ =gcd(p ,p′). For i=1,...,l−1,define q by p′ =q p′ and r by
i+1 i+1 i i i i i+1 i
p =r p′ . In addition, we set q :=p′ and r :=1. For i=1,...,l, we then
i+1 i i+1 l l l
have a permutation
π :I →I , j 7→r (j−1)+1
i qi qi i
9
and P(p ,...,p ) is defined via the commutative diagram
1 l
I ×...×I ϕql,...,q1 //I
ql q1 p1
(π1((jjl1,).,....,.j,1π)l7→(jl)) P(p1,...,pl)
(cid:15)(cid:15) (cid:15)(cid:15)
I ×...×I //I
q1 ql ϕq1,...,ql p1
Consider now the sequence
[n ,n ,...,n ][p ,p ,...,p ]=[n p ,n p ,...,n p ,p ,p ,...,p ]
1 2 k 1 2 l 1 1 2 1 k 1 1 2 l
By Proposition 1.4, its normalization is [n′p′,n′p′,...,n′p′,p′,p′,...,p′].
1 1 2 1 k 1 1 2 l
Therefore, forming quotients of adjacent elements as above, we arrive at the
sequence
[m ,m ,...,m ,q ,q ,...,q]
1 2 k 1 2 l
By dividing every element of the sequence by its normalized counterpart and
shifting the result by 1, we arrive at the sequence
[l ,l ,...,l ,l ,r ,r ,...,r ,r ]
1 2 k−1 k 1 2 l−1 l
We then have the commutative diagram
I ×...×I ×I ×...×I ϕql,...,q1×ϕmk,...,m1//I ×I ϕp1,n1 //I
ql q1 mk m1 p1 n1 n1p1
(jl,...,j1,ik,...,i1)7→ (j,i)7→
(ρ1(i1),...,ρk(ik), (σ(i),σ′(j)) σ′′
π1(j1),...,πl(jl))
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
I ×...×I ×I ×...×I // I ×I // I
m1 mk q1 qlϕm1,...,mk×ϕq1,...,ql n1 p1 ϕn1,p1 n1p1
where we have used the abbreviations
σ :=P(n ,...,n ) σ′ :=P(p ,...,p ) σ′′ :=P(n p ,...,n p ,p ,...,p )
1 k 1 l 1 1 k 1 1 l
In this diagram, the left rectangle is commutative by definition, and the large
rectangleiscommutativesincethehorizontalarrowsare,byLemma1.1,equalto
ϕ andϕ respectively.Therefore,therightrectangle
ql,...,q1,mk,...,m1 m1,...,mk,q1,...,ql
isalsocommutative,which,bycomparisonwiththedefinitioninParagraph1.2,
shows that σ′′ =σ·σ′. 2
1.6 Suppose now again that H is a bialgebra and that [n ,n ,...,n ] is a
1 2 k
finite sequence of positive integers. For h∈H, we define
h[n1,n2,...,nk] :=hP(n1,n2,...,nk)
10