A-infinity Bialgebras of Type (m,n)

0.24 MB·English

by  Ainhoa Berciano

A -BIALGEBRAS OF TYPE (m,n)
∞
AINHOABERCIANO1,SEANEVANS,ANDRONALDUMBLE2
0
1
0 Abstract. AnA∞-bialgebraoftype(m,n)isaHopfalgebraHequippedwith
2 a“compatible”operationωmn :H⊗m→H⊗nofpositivedegree. Wedetermine
thestructurerelationsforA∞-bialgebrasoftype(m,n)andconstructapurely
n algebraicexampleforeachm≥2andm+n≥4.
a
J

9

]
T
A
1. Introduction
.
h
Let Λ be a (gradedor ungraded)commutative ring with unity, and let m,n∈N
t
a with m+n ≥ 4. An A -bialgebra of type (m,n) is a graded Λ-Hopf algebra H
∞
m equippedwitha“compatible”multilinearoperationωn ∈Homm+n−3(H⊗m,H⊗n).
m
[ Thefirstexamplesoftype (1,3)weregivenby H.J.Bauesin[2]. Recently, the first

and third authors observed that when p is an odd prime and n ≥ 3, examples
2
v of type (1,p) appear as a pair of tensor factors E ⊗Γ in the mod p homology of
4 an Eilenberg-Mac Lane space K(Z,n) [1]. Examples of type (m,1) and the first
7 “non-operadic” example of type (2,2) were constructed by the third author in [8];
6
the firstexample of type (2,3), whichappears here as Example 2, was constructed
0
by the second author in his undergraduate thesis [3]. In this paper we determine
.
8 the structure relations for A -bialgebras of type (m,n) and construct a purely
∞
0
algebraic example for each (m,n) with m≥2 and m+n≥4.
9
The paper is organized as follows: In Section 2 we construct the components of
0
: the particular biderivative we need. In Section 3 we review M. Markl’s “fraction
v
product”definedin[5]anduseittocomputethe A -bialgebrastructurerelations.
i ∞
X Finally, as a consequence of Theorem 1 in Section 4, we obtain the following ex-
r amples: Let R be a commutative ring with unity, let m,n ∈ N with m ≥ 2 and
a
m + n ≥ 4, and let Λ = E(x) be an exterior R-algebra generated by x of de-
gree 2m−3. Let H = E(y) be an exterior Λ-algebra generated by y of degree 1,
let µ and ∆ denote the trivial product and primitive coproduct on H, and define
ωn ∈ Homm+n−3(H⊗m,H⊗n) by ωn (y|···|y) = xy|y|···|y. Then (H,µ,∆,ωn)
m m m
is an A -bialgebra of type (m,n).
∞
Date:January6,2010.
Key words and phrases. A∞-algebra, A∞-bialgebra, A∞-coalgebra, associahedron, bar and
cobarconstructions,Hopfalgebra,S-Udiagonal.
MSC2010:Primary18D50(operads); 57T30(barandcobarconstructions).
1Thisworkwaspartiallysupported by“Computational Topology andAppliedMathematics”
PAICYTresearchprojectFQM-296,SpanishMECprojectMTM2006-03722.
2ThisresearchwasfundedinpartbyaMillersvilleUniversityfacultyresearchgrant.
1
2 AINHOABERCIANO1,SEANEVANS,ANDRONALDUMBLE2
2. The Biderivative
Given a graded R-module H of finite type, consider the bigraded module M =
{Mn =Hom(H⊗m,H⊗n)} and a family of R-multilinear operations
m m,n≥1
ωn ∈Homm+n−3 H⊗m,H⊗n .
m n,m≥1
(cid:8) (cid:0) (cid:1)(cid:9)
Thesumω = ωn extendsuniquelytoitsbiderivative d ∈TTM,constructedby
m ω
SaneblidzeandPthe secondauthorin[7], andthereisa(non-bilinear)⊚-producton
TM ⊂TTM, which extends Gerstenhaber’s ◦ -products on M1⊕M∗. The family
i ∗ 1
{ωn} defines an A -bialgebra structure on H whenever d ⊚ d = 0, and the
m ∞ ω ω
structure relations are the homogeneous components of the equation d ⊚d =0.
ω ω
Consider a (biassociative) Hopf algebra (H,∆,µ) together with an operation
ωn ∈ Homm+n−3(H⊗m,H⊗n) with m+n ≥ 4. Let us construct the component
m
of the biderivative d in TM when ω = ∆ − µ + ωn. Let ↑ and ↓ denote the
ω m
suspension and desuspension operators, which shift dimension +1 and −1, respec-
tively. Let H¯ = H/H . For purposes of computing signs, we assume that H is
0
simply-connected; once the signs have been determined, the simple-connectivity
assumption will be dropped. The bar construction of H is the tensor coalgebra
BA = T ↑H¯ with differential induced by µ; the cobar construction of H is the
tensor alg(cid:0)ebra(cid:1)ΩH = T ↓H¯ with differential induced by ∆ (for details see [4]).
Given a map f : M →(cid:0)N of(cid:1)graded Λ-module modules, let fi,j denote the map
1⊗i⊗f ⊗1⊗j :N⊗i⊗M ⊗N⊗j →N⊗i+j+1.
Freely extend the map ↓⊗2 ∆ ↑:↓ H¯ → ΩH as a derivation of ΩH, and dually,
cofreelyextendthemap−↑µ↓⊗2:↑⊗2 H¯⊗2 →BH asacoderivationofBH.These
extensions contribute the components
k k
δk = (−1)i+1∆ and −∂ = (−1)i+1µ
i−1,k−i k i−1,k−i
Xi=1 Xi=1
to the biderivative, where k ≥ 1 and the signs are the Koszul signs that appear
when factoring out suspension and desuspension operators.
Next, cofreely extend the map ↑ ∆ ↓:↑ H¯ → B H⊗2 as a coalgebra map; this
extension (cid:0) (cid:1)
(↑∆↓)⊗k :BH →B H⊗2
kX≥1 (cid:0) (cid:1)
contributes the components
(−1)⌊k/2⌋∆⊗k.
kX≥1
Dually, freely extend the map − ↓ µ ↑:↓ H¯⊗2 → ΩH as an algebra map; this
extension
(−↓µ↑)⊗k :Ω H⊗2 →ΩH
kX≥1 (cid:0) (cid:1)
contributes the components
(−1)⌊(k+1)/2⌋µ⊗k.
kX≥1
Additional terms arise from various extensions involving ωn. When n = 2, the
m
components
ω2 −ω2 ⊗∆+(−1)m∆⊗ω2 −ω2 ⊗ω2 +··· .
m m m m m
A∞-BIALGEBRAS OF TYPE (m,n) 3
arise from the cofree extension of ↑∆↓+↑ω2 ↓⊗m:↑H¯⊕↑⊗m H¯⊗m →B H⊗2
m
as a coalgebra map BH →B H⊗2 ; and when m=2, the components (cid:0) (cid:1)
ωn+µ⊗ω(cid:0)n−(−(cid:1)1)nωn⊗µ−ωn⊗ωn+···
2 2 2 2 2
arise from the free extension of ↓⊗n ωn ↑−↓µ↑:↓H¯⊗2 →ΩH as an algebra map
2
Ω H⊗2 →ΩH. In particular, when m=n=2, the component of d in TM is
ω
(cid:0) (cid:1)
d =δ∗−∂ +ω2−ω2⊗ω2+···+ (−1)⌊(k+1)/2⌋µ⊗k+(−1)⌊k/2⌋∆⊗k
ω ∗ 2 2 2
Xk≥2h i
+µ⊗ω2−ω2⊗µ+···−ω2⊗∆+∆⊗ω2+··· .
2 2 2 2
When m 6= 2 or n 6= 2, additional terms arise from extensions governed by the
following generalization of an (f,g)-(co)derivation:
Definition 1. Let (A,µ ) and (B,µ ) be graded algebras and let f,g : A → B
A B
be algebra maps. A map h : A → B of degree p is an (f,g)-derivation of degree
p if hµ = µ (f ⊗h+h⊗g). There is the completely dual notion of an (f,g)-
A B
coderivation of degree p.
Thus a (1,1)-(co)derivation of degree 1 is a classical (co)derivation and an (f,g)-
(co)derivation of degree 1 is a classical (f,g)-(co)derivation. Define f1 = g = 1,
1
and for m,n≥2 define
fn = ∆⊗1⊗n−2 ···(∆⊗1)∆= 1⊗n−2⊗∆ ···(1⊗∆)∆ and
g =µ(cid:0) (µ⊗1)···(cid:1)µ⊗1⊗m−2 =µ(cid:0)(1⊗µ)··· (cid:1)1⊗m−2⊗µ .
m
(cid:0) (cid:1) (cid:0) (cid:1)
Ifn>2,cofreelyextend(−1)⌊n/2⌋+1 ↑fn ↓:↑H¯ →B(H⊗n)asacoalgebramap
BH →B(H⊗n) and obtain
Fn = (−1)(⌊n/2⌋+1)k(↑fn ↓)⊗k;
kX≥1
then cofreelyextend ↑ωn ↓⊗m:↑⊗m H¯⊗m →B(H⊗n) as an(Fn,Fn)-coderivation
m
BH →B(H⊗n) of degree n−2. This extension contributes the components
ωn +(−1)⌊(n+1)/2⌋(ωn ⊗fn−(−1)mfn⊗ωn)+···
m m m
to the biderivative, where we have applied the identity ⌊n/2⌋+n ≡ ⌊(n+1)/2⌋
(mod2). In particular, when m=2 we have
d =δ∗−∂ +ωn−ωn⊗ωn+···+ (−1)⌊(k+1)/2⌋µ⊗k+(−1)⌊k/2⌋∆⊗k
ω ∗ 2 2 2
kX≥2h i
+(−1)⌊(n+1)/2⌋(ωn⊗fn−fn⊗ωn)+···+µ⊗ωn−(−1)nωn⊗µ+··· .
2 2 2 2
Dually,ifm>2,freelyextend(−1)⌊(m+1)/2⌋ ↓g ↑:↓H¯⊗m →ΩH asanalgebra
m
map Ω(H⊗m)→ΩH and obtain
G = (−1)⌊(m+1)/2⌋k(↓g ↑)⊗k;
m m
kX≥1
then freely extend the map ↓⊗n ωn ↑:↓ H¯⊗m → ΩH as a (G ,G )-derivation
m m m
Ω(H⊗m)→ΩH of degree m−2. This extension contributes the components
ωn −(−1)⌊m/2⌋(g ⊗ωn −(−1)nωn ⊗g )+··· .
m m m m m
4 AINHOABERCIANO1,SEANEVANS,ANDRONALDUMBLE2
In particular, when n=2 we have
d =δ∗−∂ +ω2 −ω2 ⊗ω2 +···+ (−1)⌊(k+1)/2⌋µ⊗k+(−1)⌊k/2⌋∆⊗k
ω ∗ m m m
kX≥2h i
−ω2 ⊗∆+(−1)m∆⊗ω2 +···−(−1)⌊m/2⌋ g ⊗ω2 −ω2 ⊗g +··· .
m m m m m m
(cid:0) (cid:1)
And finally, if m,n6=2 we have
d =δ∗−∂ +ωn + (−1)⌊(k+1)/2⌋µ⊗k+(−1)⌊k/2⌋∆⊗k
ω ∗ m
kX≥2h i
+(−1)⌊(n+1)/2⌋(ωn ⊗fn−(−1)mfn⊗ωn)+···
m m
−(−1)⌊m/2⌋(g ⊗ωn −(−1)nωn ⊗g )+··· .
m m m m
This completes the construction of the component of d in TM.
ω
TodeterminetheA -bialgebrastructurerelations,wemustdefinethe⊚-product
∞
and extract the homogeneous components of the equation d ⊚d =0.
ω ω
3. The ⊚-Product
As mentioned above, a family of operations ω ={ωn} defines an A -bialgebra
m ∞
structureonH wheneverd ⊚d =0,andthe structurerelationsappearashomo-
ω ω
geneouscomponentsofthe equationd ⊚d =0. The ⊚-productis arestrictionof
ω ω
thefraction product (cid:30):TM×TM →TM definedbyM.Marklin[5]andreviewed
below. For α,β ∈TM define
α(cid:30)β, if α and β are components of d
α⊚β = ω
(cid:26) 0, otherwise.
Consider the submodule S of special elements in the free PROP M whose ad-
ditive generators are either indecomposables θn ∈ Mn of dimension m+n−3 or
m m
monomials αy ∈TM ×TM expressed as “elementary fractions” of the form
x
αy1···αyq
(3.1) αy = p p ,
x αq ···αq
x1 xp
where αq and αyj are additive generators of S and the jth output of αq is linked
to the itxhiinput opf αyj. All elementary fractions define the fraction prodxuict.
p
In terms of a composition, let σ : (H⊗q)⊗p →≈ (H⊗p)⊗q be the standard per-
q,p
mutationoftensorfactorsandnotethatαyp1···αypq ∈Hom (H⊗p)⊗q,H⊗Σyi and
(cid:16) (cid:17)
αqx1···αqxp ∈Hom H⊗Σxj,(H⊗q)⊗p ;thenαyx =αyp1···αypq ◦σq,p ◦αqx1···αqxp ∈
(cid:16) (cid:17)
Hom H⊗Σxj,H⊗Σyi .Whereasjuxtapositioninthe“numerator”and“denomina-
tor” d(cid:0)enotes tensor p(cid:1)roduct in TM, dimαy = dimαq +dimαyj.
x i,j xi p
WepictureindecomposablesandfractionproPductstwoways. First,anindecom-
posable generator θn is pictured as a double corolla with m inputs and n outputs
m
as in Figure 1, and a general elementary fraction αy is pictured as a connected
x
non-planar graph as in Figure 2.
n
θn =
m
m
A∞-BIALGEBRAS OF TYPE (m,n) 5
Figure 1. An indecomposable pictured as a double corolla.
α1α11
α12 = 3 12 = =
111 α2α2α2
1 1 1
Figure 2. A fraction product pictured as a non-planar graph.
Second, identify the pair of isomorphic modules (H⊗q)⊗p ≈ (H⊗p)⊗q with the
point (p,q)∈N2 and think of indecomposables and fraction products as operators
on N2. An indecomposible ωn is pictured as a “transgressive arrow” (m,1) −→
m
(1,n)andafractionproduct αy1···αyq (cid:30) αq ···αq asa“transgressivepath”
(Σx ,1) −→ (q,p) −→ (1,Σ(cid:0)yp) (seepin(cid:1)Fig(cid:16)urxe13). xWp(cid:17)hile this representation is
i j
helpful conceptually, it is not faithful. For example, both α1 =ω1(cid:30) ω1⊗1 and
21 2 2
α1 =ω1(cid:30) 1⊗ω1 are represented by the path (3,1)→(2,1)→(1,(cid:0)1). (cid:1)
12 2 2
(cid:0) (cid:1)
Hsn (1, n )
( q , p )B (p , q ) (Hsq)spB (Hsp)sq
(m,1) Hsm
Figure 3. A fraction product as a composition of operators on N2.
Uptosign,thehomogeneouscomponentofd ⊚d withminputsandnoutputs
ω ω
consists of ωn plus the sum of all fractions α⊚β thought of as paths (m,1) →β
m
α
(q,p)→(1,n).
4. A -Bialgebras of Type (m,n)
∞
EverythingweneedtodeterminethedesiredA -bialgebrastructurerelationsis
∞
nowinplace. Inadditiontothe(co)associativityrelationsµ(µ⊗1)=µ(1⊗µ)and
(∆⊗1)∆=(1⊗∆)∆,andtheclassicalHopfrelation∆µ=(µ⊗µ)σ (∆⊗∆),
2,2
the homogeneous components of d ⊚d = 0 with m+1 inputs and n outputs
ω ω
produce Structure Relation 1:
δnωn =(g ⊗ωn −(−1)nωn ⊗g )σ ∆⊗m.
m m m m m 2,m
6 AINHOABERCIANO1,SEANEVANS,ANDRONALDUMBLE2
Figure 4. Structure relation 1.
Similarly, the homogeneous components of d ⊚d = 0 with m inputs and n+1
ω ω
outputs produce Structure Relation 2:
ωn∂ =(−1)⌊(n+1)/2⌋µ⊗nσ (ωn ⊗fn−(−1)mfn⊗ωn).
m m n,2 m m
Figure 5. Structure relation 2.
AdditionalrelationsappearinDefinition2below. Whereasourexamplesdonot
dependonthesignsinRelations3,4(a-c),and5(a-c),weomitthecomplicatedsign
formulas. Note that Relation 3 is a classical A -(co)algebra relation when n = 1
∞
(or m=1).
Definition2. Given m,n∈Nwith m+n≥4,an A -bialgebraoftype (m,n)is a
∞
Λ-Hopfalgebra(H,∆,µ)togetherwithanoperationωn ∈Homm+n−3(H⊗m,H⊗n)
m
such that
A∞-BIALGEBRAS OF TYPE (m,n) 7
1. δnωn =(g ⊗ωn −(−1)nωn ⊗g )σ ∆⊗m
m m m m m 2,m
2. ωn∂ =µ⊗nσ (fn⊗ωn −(−1)mωn ⊗fn)
m m n,2 m m
3. ωn ⊗g⊗n−1±g ⊗ωn ⊗g⊗n−2±···±g⊗n−1⊗ωn σ
m m m m m m m n,m
(cid:0) ωn ⊗(fn)⊗m−1±fn⊗ωn ⊗(fn)⊗m−2±···±((cid:1)fn)⊗m−1⊗ωn =0
m m m
(cid:16) (cid:17)
4. If m=2, then
a. (ωn⊗ωn)σ (∆⊗∆)=0
2 2 2,2
b. (ωn)⊗nσ (fn⊗ωn−ωn⊗fn)=0
2 n,2 2 2
c. µ⊗(ωn)⊗n−1±ωn⊗µ⊗(ωn)⊗n−2
2 2 2
h
±···±(ωn)⊗n−1⊗µ σ (fn⊗ωn−ωn⊗fn)=0
2 n,2 2 2
i
.
.
.
d. µ⊗n−1⊗ωn+µ⊗n−3⊗ωn⊗µ⊗2+···
2 2
(cid:2) −(−1)n µ⊗n−2⊗ωn⊗µ+µ⊗n−4⊗ωn⊗µ⊗3+···
2 2
σn,2(cid:0)(fn⊗ω2n−ω2n⊗fn)=0 (cid:1)(cid:3)
5. If n=2, then
a. (µ⊗µ)σ ω2 ⊗ω2 =0
2,2 m m
b. g ⊗ω2 −(cid:0)ω2 ⊗g (cid:1)σ ω2 ⊗m =0
m m m m 2,m m
c. (cid:0)g ⊗ω2 −ω2 ⊗g (cid:1)σ (cid:0)∆⊗(cid:1) ω2 ⊗m−1±ω2 ⊗∆⊗ ω2 ⊗m−2
m m m m 2,m m m m
(cid:0) ±···± ω2 ⊗m(cid:1)−1⊗∆h =0(cid:0) (cid:1) (cid:0) (cid:1)
m
(cid:0) .(cid:1) i
.
.
d. g ⊗ω2 −ω2 ⊗g σ ∆⊗m−1⊗ω2 +∆⊗m−3⊗ω2 ⊗∆⊗2+···
m m m m 2,m m m
(cid:0) −(−1)m ∆⊗m−2(cid:1)⊗ω2(cid:2)⊗∆+∆⊗m−4⊗ω2 ⊗∆⊗3+··· =0
m m
6. If m=n=2, then ω2⊗ω2 σ ω2⊗ω2 =0. (cid:3)
2 2 2,2 2 2
(cid:0) (cid:1) (cid:0) (cid:1)
Example 1. When (m,n) = (2,3), the relations in Definition 2 are represented
graphically as follows:
Relation1: ∆⊗1⊗2−1⊗∆⊗1+1⊗2⊗∆ ω3 = µ⊗ω3+ω3⊗µ σ (∆⊗∆)
2 2 2 2,2
(cid:0) (cid:1) (cid:0) (cid:1)
Relation 2: ω3(µ⊗1−1⊗µ)=µ⊗3σ (∆⊗1)∆⊗ω3−ω3⊗(1⊗∆)∆ .
2 3,2 2 2
(cid:0) (cid:1)
Relation 3 (4d):
ω3⊗µ⊗2+µ⊗ω3⊗µ+µ⊗2⊗ω3 σ (∆⊗1)∆⊗ω3−ω3⊗(1⊗∆)∆ =0.
2 2 2 3,2 2 2
(cid:0) (cid:1) (cid:0) (cid:1)
8 AINHOABERCIANO1,SEANEVANS,ANDRONALDUMBLE2
Example 2. Let H = hyi be a graded Z -module with |y| = −2. Let µ and ∆ be
2
the trivial product and primitive coproduct on H, i.e., µ(1|y) = µ(y|1) = y and
∆(y)=1|y+y|1. Then H is a Hopf algebra. Definean operation ω3 :H⊗2 →H⊗3
2
by ω3(y|y) = y|1|1 + 1|1|y and zero otherwise. Then ω3 satisfies the structure
2 2
relations in Definition 2 and H,∆,µ,ω3 is an A -bialgebra of type (2,3). We
2 ∞
verify Relation 1; verification o(cid:0)f the other(cid:1)relations is left to the reader.
(µ⊗ω3+ω3⊗µ)σ (∆⊗∆)(y|y)
2 2 2,2
=(µ⊗ω3+ω3⊗µ)(y|y|1|1+y|1|1|y+1|y|y|1+1|1|y|y)
2 2
=(y|1|1+1|1|y)⊗1+1⊗(y|1|1+1|1|y)
=y|1|1|1+1|1|y|1+1|y|1|1+1|1|1|y
=(∆⊗1⊗2+1⊗∆⊗1+1⊗2⊗∆)ω3(y|y)=δ3ω3(y|y).
2 2
Before we present our general family of examples, we determine all possible
dimensions for our generators. Let R be a commutative ring with unity.
Proposition 1. Let p,q ∈N, let Λ =E(x) be an exterior R-algebra generated by
x of degree p, and let H = hyi be a graded Λ-module generated by y of degree q.
The operation ωn :H⊗m →H⊗n defined by
m
ωn (y|···|y)=xy|y|···|y
m
has degree m+n−3 if
(i) m=n=2, p=1, and q ≥2;
(ii) m≥2, n≥1, p=2m−3, and q =1;
(iii) m≥3,m+1≤n≤3m−p−3,1≤p≤2m−4,andq =(p−m−n+3)/(m−n).
Proof. In general, |ωn|=m+n−3 if and only if
m
(4.1) m(q+1)=n(q−1)+p+3.
Thus p,q ≥1 implies m≥2.
(i) If m=n=2, p=1, and q ≥2, equation 4.1 implies ω2 =1.
2
(cid:12) (cid:12)
(ii) If m ≥ 2, n ≥ 1, p = 2m−3, and q = 1, equation(cid:12) 4.(cid:12)1 implies that |ωn| =
m
m+n−3.
(iii) If m≥3, 1≤p≤2m−4, and q ≥2, equation 4.1 defines a rational function
p−m−n+3
q(n)= ,
m−n
which decreases from 2m−p−2 to 2 as n increases from m+1 to 3m−p−3 (see
Figure6below). Consequently|ωn|=m+n−3 wheneverm+1≤n≤3m−p−3
m
and q(n)∈N. (cid:3)
A∞-BIALGEBRAS OF TYPE (m,n) 9
Figure 6.
Our main result now follows. Note that when H is a module over an exterior
R-algebraΛ=E(x),therelationyx|y =y|xy =(−1)|x||y|x(y|y)holdsinTH since
we tensor over Λ. Thus x2 =0 implies yx|yx=0.
Theorem 1. Let m, n, p, and q satisfy the hypotheses of Proposition 1, and let
Λ=E(x) be an exterior R-algebra generated by x of degree p. Let H =E(y) be an
exterior Λ-algebra generated by y of degree q, let µ and ∆ denote the trivial product
and primitive coproduct on H, and define ωn : H⊗m → H⊗n by ωn (y|···|y) =
m m
xy|y|···|y. Then (H,∆,µ,ωn) is an A -bialgebra of type (m,n).
m ∞
Proof. Theoperationωn hastherequireddegreebyProposition1. Wemustverify
m
that the structure relations in Definition 2 hold:
Relation 1 is non-trivial on y|···|y:
(g ⊗ωn −(−1)nωn ⊗g )σ ∆⊗m(y|···|y)
m m m m 2,m
=(g ⊗ωn −(−1)nωn ⊗g )σ [(1|y+y|1)|···|(1|y+y|1)]
m m m m 2,m
=g (1|···|1)⊗ωn (y|···|y)−(−1)nωn (y|···|y)⊗g (1|···|1|)
m m m m
=1|xy|y|···|y+(−1)n+1xy|y|···|y|1=δn(xy|y|···|y)=δnω(y|···|y).
Relation 2 is non-trivial on y|y|···|y|1 and 1|y|···|y, for example,
µ⊗nσ (fn⊗ωn −(−1)mωn ⊗fn)(y|y|···|y|1)
n,2 m m
=(−1)m+1µ⊗nσ (xy|y|···|y|1|···|1)
n,2
=(−1)m+1xy|y|···|y =ωn∂ (y|y|···|y|1).
m m
Relations3,4,5and6holdtriviallysinceeachtermisacompositionoftwotensors,
each with a factor of ωn whose non-vanishing values have coefficient x. (cid:3)
m
Since odd dimensional spheres are rational H-spaces, the module H =
H S1;H S2m−3;Q admitsanon-trivialA -bialgebrastructureoftype(m,n).
∗ ∗ ∞
Thu(cid:0)s we ask(cid:0): Does th(cid:1)e(cid:1)re exist an H-space X, a coefficient ring Λ, and an induced
operation ωn such that (H (X;Λ),∆,µ,ωn) is an A -bialgebra?
m ∗ m ∞
Acknowledgement. We wish to thank Jim Stasheff for reading an early draft of
this paper and offering many helpful suggestions.
10 AINHOABERCIANO1,SEANEVANS,ANDRONALDUMBLE2
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Departamento de Dida´ctica de la Matema´tica y de las Ciencias Experimentales,
Universidad del Pa´ıs Vasco-Euskal Herriko Unibertsitatea, Ramo´n y Cajal, 72., 48014-
Bilbao(Bizkaia). SPAIN
E-mail address: [email protected]
Departmentof Mathematics,University ofPittsburgh,Pittsburgh,PA15260
E-mail address: [email protected]
DepartmentofMathematics,MillersvilleUniversityofPennsylvania,Millersville,
PA.17551
E-mail address: [email protected]