A DIAGONAL ON THE ASSOCIAHEDRA
SAMSON SANEBLIDZE AND RONALD UMBLE1
To Jim Stasheff on the occasion of his 65th birthday
1. Introduction
An associahedral set is a combinatorial object generated by Stasheff associahedra
{Kn } and equipped with appropriate face and degeneracy operators. Associahedral
sets are similar in many ways to simplicial or cubical sets. In this paper we give a
formal definition of an associahedral set, discuss some naturally occurring examples
and construct an explicit geometric diagonal ∆ : C∗ (Kn ) −→ C∗ (Kn ) ⊗ C∗ (Kn ) on
the cellular chains C∗ (Kn ) . The diagonal ∆, which is analogous to the AlexanderWhitney diagonal on the simplices, gives rise to a diagonal on any associahedral set
and leads immediately to an explicit diagonal on the A∞ -operad. As an application
of this, we use the diagonal ∆ to define a tensor product in the A∞ category. This
tensor product will play a central role in our discussion of “A∞ -Hopf algebras” to
appear in the sequel.
We mention
P that Chapoton [1], [2] constructed a diagonal
P of the form ∆ :
C
(K
)
⊗
C
(K
)
on
the
direct
sum
C∗ (Kn ) →
∗
i
∗
j
i+j=n
n≥2 C∗ (Kn ) , which
coincides with the diagonal of Loday and Ronco [8] in dimension zero. Whereas
Chapoton’s diagonal is formally defined to be primitive on generators, our diagonal
is obtained by a purely geometrical decomposition of the generators and is totally
different from his.
The second author wishes to thank Millersville University for its generous financial support and the University of North Carolina at Chapel Hill for its hospitality
during the final stages of this project.
2. The Stasheff Associahedra
In his seminal papers of 1963, J. Stasheff [17] constructs the associahedra
{Kn+2 }n≥0 as follows: Let K2 = ∗; if Kn+1 has been constructed, let
[
(Kr × Ks )k
Ln+2 =
r+s=n+3
1≤k≤n−s+3
and define Kn+2 = CLn+2 , i.e., the cone on Ln+2 . The associahedron Kn+2 is
an n-dimensional polyhedron, which serves as a parameter space for homotopy
associativity in n + 2 variables. The top dimensional face of Kn+2 corresponds to
a pair of level 1 parentheses enclosing all n + 2 indeterminants; each component
(Kn−`+2 × K`+1 )i+1 of ∂Kn+2 corresponds to a pair of level 2 parentheses enclosing
Date: November 8, 2000.
Key words and phrases. Associahedra, A∞ -algebra, A∞ -coalgebra, tensor product.
1 This research funded in part by a Millersville University faculty research grant.
1
SAMSON SANEBLIDZE AND RONALD UMBLE1
2
st
` + 1 indeterminants beginning with the (i + 1) . We denote this parenthesization
by
d(i,`) = (x1 · · · (xi+1 · · · xi+`+1 ) · · · xn+2 )
and refer to the inner and outer parentheses as the first
Note that indices i and ` are constrained by

 0≤i≤n
1 ≤ ` ≤ n,
i=0

1 ≤ ` ≤ n + 1 − i, 1 ≤ i ≤ n
and last pair, respectively.



.
Thus, there is a one-to-one correspondence between (n − 1)-faces of Kn+2 and
parenthesizations d(i,`) of n + 2 indeterminants.
Alternatively, Kn+2 can be realized as a subdivision of the standard n-cube I n in
the following way: Let = 0, 1. Label the endpoints of K3 = [0, 1] via ↔ d(,1) . For
n
1 ≤ i ≤ n, let en−1
i, denote the (n − 1)-face (x1 , . . . , xi−1 , , xi+1 , . . . , xn ) ⊂ I and
obtain K4 from K3 × I = I 2 by subdividing the edge e11,1 as the union of intervals
1 × I0,1 ∪ 1 × I1,∞ . Label the edges of K4 as follows: e1i,0 ↔ d(0,i) ; e12,1 ↔ d(2,1) ;
1 × I0,1 ↔ d(1,1) ; and 1 × I1,∞ ↔ d(1,2) (see Figure 1).
(0, 1)
•
d(2,1)
(1, 1)
•
d(1,2)
d(0,1)
•
K4
d(1,1)
•
(0, 0)
d(0,2)
•
(1, 0)
Figure 1: K4 as a subdivision of K3 × I.
Now for 0 ≤ i ≤ j ≤ ∞, let Ii,j denote the subinterval 2i − 1 /2i , 2j − 1 /2j ⊂
I, where (2∞ − 1) /2∞ is defined to be 1. For n > 2, assume that Kn+1 has been
constructed and obtain Kn+2 from Kn+1 × I ≈ I n by subdividing the (n − 1)-faces
d(i,n−i) ×I as unions d(i,n−i) ×I0,i ∪d(i,n−i) ×Ii,∞ , 0 < i < n. Label the (n − 1)-faces
of Kn+2 as follows:
Face of Kn+2
Label
en−1
`,0
d(0,`) ,
en−1
n,1
d(n,1),
d(i,`) × I
d(i,`),
1 ≤ ` < n − i, 0 < i < n − 1
d(i,n−i) × I0,i
d(i,n−i),
0<i<n
d(i,n−i) × Ii,∞
d(i,n−i+1) , 0 < i < n
1≤`≤n
A DIAGONAL ON THE ASSOCIAHEDRA
cc
cc•
c
(0, 0, 1)
•
•
cc
cc•
d(2,2)
d(1,3)
cc
c
cc
•
d(3,1)
cc
cc•
c
•
•
•
(1, 0, 0)
cc
c
••
d(1,1)
3
c•
cc
c
•
d(1,2)
d(2,1)
cc•
cc
cc•
(0, 1, 0)
Figure 2: K5 as a subdivision of K4 × I.
In Figure 2 we have labeled the 2-faces of K5 that are visible from the viewpoint
of the diagram.
Compositions d(im ,`m ) · · · d(i2 ,`2 ) d(i1 ,`1 ) denote a successive insertion of m + 1
pairs of parentheses into n + 2 indeterminants as follows: Given d(ir ,`r ) · · · d(i1 ,`1 ) ,
1 ≤ r < m, regard each pair of level 2 parentheses and its contents as a single indeterminant and apply d(ir+1 ,`r+1 ) . Conclude by inserting a last pair enclosing everything. Note that each parenthesization can be expressed as a unique composition
d(im ,`m ) · · · d(i1 ,`1 ) with ir+1 ≤ ir for 1 ≤ r < m, in which case the parentheses inserted by d(ir+1 ,`r+1 ) begin at or to the left of the pair inserted by d(ir ,`r ) . Such compositions are said to have first fundamental form. Thus for 0 ≤ k < n, the k-faces
of Kn+2 lie in one-to-one correspondence with compositions d(in−k ,`n−k ) · · · d(i1 ,`1 )
in first fundamental form. The two extremes with m pairs of parentheses inserted
as far to the left and right as possible, are respectively denoted by
d(0,`m ) · · · d(0,`1 ) and d(im ,im−1 −im ) · · · d(i1 ,i0 −i1 ) ,
where i0 = n + 1 and ir+1 < ir , 0 ≤ r < m. In particular, the n-fold compositions
d(0,1) · · · d(0,1) and d(1,1) · · · d(n,1)
denote the extreme full parenthesizations of n + 2 indeterminants. When m = 0,
define d(im ,`m ) · · · d(i1 ,`1 ) = Id.
1
2
3 ··· n + 2
...
@@AA
@A
N0
•
Figure 3: The planar rooted tree Tn+2
Alternatively, each face of Kn+2 can be represented as a planar rooted tree (PRT)
with n+2 leaves; its leaves correspond to indeterminants and its nodes correspond to
pairs of parentheses. Let Tn+2 denote the PRT with n+2 leaves attached to the root
4
SAMSON SANEBLIDZE AND RONALD UMBLE1
at a single node N0 , called the root node (see Figure 3). The leaves correspond to a
single pair of parentheses enclosing all n + 2 indeterminants. Now consider a node
N of valence r + 1 ≥ 4 in an arbitrary PRT T . Choose a neighborhood U of N that
excludes the other nodes of T and note that Tr ⊆ U ∩ T. Labeling from left to right,
index the leaves of Tr from 1 to r as in Figure 3. Perform an (i, `)-surgery at node N
in the following way: Remove leaves i+1, . . . , i+`+1 of Tr , reattach them at a new
node N 0 6= N and graft in a new branch connecting N to N 0 (see Figure 4). Now
i+1
1 ···
···
i+`+1
@@ A . . . @A @A•
i
N0
i + ` + 2···n + 2
@@. . . AA
. . .
@@AA @A@A
N0
•
(i,`)
Figure 4: The planar rooted tree Tn+2
let n ≥ 1. Given a parenthesization d(i,`) of n + 2 indeterminants, obtain the PRT
(i,`)
Tn+2 from Tn+2 by performing an (i, `)-surgery at the root node N0 as shown
in Figure 4. Inductively, given a parenthesization d(im ,`m ) · · · d(i1 ,`1 ) of n + 2 indeterminants expressed as a composition in first fundamental form, construct the
(i1 ,`1 ),... ,(im ,`m )
(i1 ,`1 ),... ,(ir ,`r )
as follows: Assume that Tn+2
with
corresponding PRT Tn+2
nodes N0 , . . . , Nr has been constructed for some 1 ≤ r < m and note that the
root node N0 has valence n + 3 − `1 − · · · − `r . Perform an (ir+1 , `r+1 )-surgery at
(i1 ,`1 ),... ,(ir+1 ,`r+1 )
containing a new node Nr+1 and a new branch
N0 and obtain Tn+2
(i1 ,`1 ),... ,(im ,`m )
= Tn+2 when m = 0 and
connecting N0 to Nr+1 . Finally, define Tn+2
obtain a one-to-one correspondence between k-faces of Kn+2 , 0 ≤ k ≤ n and PRT’s
(i1 ,`1 ),... ,(in−k ,`n−k )
consisting of n − k + 1 nodes and n + 2 leaves. In particular,
Tn+2
(i1 ,`1 ),... ,(in ,`n )
each vertex of Kn+2 corresponds to a planar binary rooted tree Tn+2
(see Figure 5).
Now given a k-face ak ⊆ Kn+2 , k > 0, consider the two vertices of ak at which
parentheses are shifted as far to the left and right as possible; we refer to these
and
vertices as the minimal and maximal vertices of ak , and denote them by amin
k
,
respectively.
In
particular,
the
minimal
and
maximal
vertices
of
K
amax
n+2 are
k
the origin and the vertex of I n diagonally opposite to it, i.e.,
min
max
↔ (0, 0, . . . , 0) and Kn+2
↔ (1, 1, . . . , 1) ;
Kn+2
the respective binary trees in Figure 5 correspond to K4min and K4max .
A DIAGONAL ON THE ASSOCIAHEDRA
@@•
@•@
@@ @@ @@•
@@ @•
•
•
(0,1),(0,1)
T4
5
(2,1),(1,1)
↔
T4
d(0,1) d(0,1) = (((••)•)•)
↔
d(1,1) d(2,1) = (•(•(••)))
Figure 5: Planar binary rooted trees.
(i ,` ),... ,(i
,`
)
1 1
n−k n−k
of ak , construct the minimal (resp.,
Given a representation Tn+2
maximal) tree of ak by replacing each node of valence r ≥ 4 with the planar binary
min
max
(resp., Kr−1
). Note that amin
and amax
determine
rooted tree representing Kr−1
k
k
ak since their convex hull is a diagonal of ak . But we can say more.
When a composition of face operators d(im ,`m ) · · · d(i1 ,`1 ) is defined we refer to
the sequence of lower indices I = (i1 , `1 ) , . . . , (im , `m ) as an admissible sequence
of length m; if d(im ,`m ) · · · d(i1 ,`1 ) has first fundamental form we refer to the the
sequence I as a type I sequence of length m. The set of all planar binary rooted
trees
I
| I is type I sequence of length n
Yn+2 = Tn+2
I
I
p
q
≤ Tn+2
if there
is a poset with partial ordering defined as follows: Say that Tn+2
Ip
Iq
is an edge-path in Kn+2 from vertex Tn+2 to vertex Tn+2 along which parentheses
shift strictly to the right. This partial ordering can be expressed geometrically in
I
and
terms of the following operation on tress: Let N0 denote the root node of Tn+2
I
.
let N be a node joining some left branch L and a right branch or leaf R in Tn+2
Let NL denote the node on L immediately above N . A right-shift through node N
repositions NL either at the midpoint of leaf R or midway between N and the node
Ip
Iq
immediately above it.
o Tn+2 ≤ Tn+2 if there is a right-shift sequence of planar
n Then
Ir
binary rooted trees Tn+2
I
p≤r≤q
Ir
r+1
, i.e., for each r < q, Tn+2
is obtained from Tn+2
Ir
by a right-shift through some node in Tn+2
(see Figure 6).
@@•
@•
@•
≤
@@•
@@•
@@
•
≤
@@ @@ @@•
@@ @•
•
Figure 6: A right-shift sequence of planar binary trees
Let I 0 = (i01 , `01 ), ..., (i0m , `0m ) be an admissible sequence of length m > 0 and
I0
. Let N denote
consider a node N 0 distinct from the root node N0 in the PRT Tn+2
the node immediately below N 0 and let N N 0 denote the branch from N to N 0 ;
I
I0
= Tn+2
/N N 0 as the (N, N 0 )-contraction of
we refer to the quotient space Tn+2
SAMSON SANEBLIDZE AND RONALD UMBLE1
6
0
I
Tn+2
. Now given a type I sequence J 0 of length n, consider the planar binary
J0
J
J0
and let Tn+2
be the PRT obtained from Tn+2
by some sequence of k
tree Tn+2
J
⊆ Yn+2 of all planar binary
successive (N, N 0 )-contractions. The subposet Yn+2
J
can be so obtained is exactly the poset of vertices of
rooted trees from which Tn+2
J
. In this way, we may regard ak as the
the k-face ak ⊆ Kn+2 represented by Tn+2
J
geometric realization of Yn+2 just as we regard a k-face of the standard n-simplex
as the geometric realization of a (k + 1)-subset of a linearly ordered (n + 1)-set. In
particular, Kn+2 is the geometric realization of Yn+2 .
We summarize the discussion above as a proposition:
Proposition 1. For 0 ≤ k ≤ n, there exist one-to-one correspondences
(n − k) -fold compositions of face
{k-faces of Kn+2 } ↔
operators in first fundamental form
Planar rooted trees with
↔
n − k + 1 nodes and n + 2 leaves
J
Subposets of planar binary rooted trees Yn+2
↔
where J is a type I sequence of length n − k
3. Associahedral Sets
Definition 1. An associahedral set is a graded set
Xk
n1 ,... ,nk nj ≥ 0 and
nj = n − k + 1
K = Kn−k+3
j=1
n≥0; k≥1
together with face operators
n
n1 ,... ,nq−1 , `q −1, nq −`q ,nq+1 ,... ,nk
n1 ,... ,nk
→ Kn−k+2
dq(iq ,`q ) : Kn−k+3
o
1 ≤ q ≤ k; 0 ≤ iq ≤ nq ; 1 ≤ `q ≤ nq ; iq + `q ≤ nq + 1
n≥0; k≥1
and degeneracy operators
n
n ,... ,n
n1 ,... ,nk
1
q−1
→ Kn−k+4
sqj : Kn−k+3
,nq +1,nq+1 ,... ,nk
o
1 ≤ q ≤ k; 1 ≤ j ≤ nq + 3
n≥0; k≥1
that satisfy the following relations:
p
dp(ip ,`p ) dq(iq ,`q ) = dq+1
(iq ,`q ) d(ip ,`p ) ;
p<q
(1)
q
q
q
dq+1
(iq+1 ,`q+1 ) d(iq ,`q ) = d(iq −iq+1 ,`q ) d(iq+1 ,`q+1 +`q ) ;
iq+1 ≤ iq ≤ iq+1 + `q+1
(2)
q
q+1
q
dq+1
(iq+1 ,`q+1 ) d(iq ,`q ) = d(iq ,`q ) d(iq+1 +`q ,`q+1 ) ;
iq < iq+1
(3)
(continued)
A DIAGONAL ON THE ASSOCIAHEDRA
dp(i,`) sqj = sq+1
dp(i,`) ;
j
p<q
dp(i,`) sqj = sqj dp(i,`) ;
p>q
q
dq(i,`) sqj = sq+1
j−` d(i,`) ;
i+`+1<j
dq(i,`) sqj = sqj−i dq(i,`−1) ;
i < j < i + ` + 2, ` > 1
dq(i−1,`) ;
dq(i,`) sqj = sq+1
j
i ≥ j, ` ≤ nq
dq(i,`) sqj = 1;
(i, `) = (j − 1, 1), 1 ≤ j < nq + 3
dq(i,`) sqj = 1;
(i, `) = (j − 2, 1), 1 < j ≤ nq + 3
dq(i,`) sqj = 1;
(i, `) = (0, nq + 1), j = nq + 3
dq(i,`) sqj = 1;
spj sqj0 = sqj0 spj ;
(i, `) = (1, nq + 1), j = 1
p 6= q
sqj sqj0 = sqj0 +1 sqj ;
p = q, j ≤ j 0 .
7
n ,... ,n
n1 ,... ,nm
1
m+2
→ Kn−m+1
and
Remark 1. If the compositions dp(ip ,`p ) dq(iq ,`q ) : Kn−m+3
0
dp(i
p0 ,`p0 )
0
dq(i
q0 ,`q0 )
n ,... ,n
n0 ,... ,n0
n1 ,... ,nm
1
m+2
: Kn−m+3
→ Kn−m+1
are equal via relations (1) to (3), then
n0 ,... ,n0
1
m+2
1
m+2
∩ Kn−m+1
contains the image of these compositions. When subKn−m+1
n ,... ,n
n0 ,... ,n0
1
m+2
1
m+2
∩ Kn−m+1
,
sequently applying face operators to an element a ∈ Kn−m+1
where n1 , . . . , nm+2 > n01 , . . . , n0m+2 in the lexicographic ordering, think of a ∈
n ,... ,n
1
m+2
.
Kn−m+1
Example 1. Consider the set of all planar rooted trees
o
n
n1 ,...,nm+1
(i1 ,`1 ),...,(im ,`m )
∈ Kn−m+2
,
K = Tn+2
n≥0; 0≤m≤n
P
n
where nq = `q − 1 for 1 ≤ q ≤ m and nm+1 = n − m
j=1 `j , and note that Kn+2 =
{Tn+2 } consists of a single element. When n = 2, the sets in K are pairwise
disjoint; when n = 3 however, the sets in K intersect nontrivially:
K31,0,0 ∩ K30,1,0 = {((• • •) (••)) , ((••) (• • •))} .
(i ,` ),...,(i ,` )
1 1
m m
, 1 ≤ q ≤ m + 1,
To evaluate the face operator dq(i,`) on the element Tn+2
m
1
recall that the composition d(im ,`m ) · · · d(i1 ,`1 ) is given in first fundamental form.
1
Repeatedly apply face relations (1) to (3) to the composition dq(i,`) dm
(im ,`m ) · · · d(i1 ,`1 )
dm
to obtain the composition dm+1
· · · d1i0 ,`0 in first fundamental form
( 1 1)
(i0m+1 ,`0m+1 ) (i0m ,`0m )
and define
(i01 ,`01 ),...,(i0m+1 ,`0m+1 )
(i1 ,`1 ),...,(im ,`m )
.
= Tn+1
dq(i,`) Tn+2
Conceptually, given a parenthesization d(im ,`m ) · · · d(i1 ,`1 ) in first fundamental form,
the face operator dq(i,`) inserts a new pair of parentheses inside the q th pair while
st
treating the (q − 1) pair and its contents as a single indeterminant. In terms of the
SAMSON SANEBLIDZE AND RONALD UMBLE1
8
(i ,` ),...,(im ,`m )
, apply dq(i,`) by performing an (i, `)-surgery at Nq and relabel
(i01 ,`01 ),...,(i0m+1 ,`0m+1 )
.
the nodes as required by the first fundamental form to obtain Tn+2
When n = 2, the following five face operators relate T4 ∈ K42 to the edges of the
pentagon K4 :
1 1
tree Tn+2
d1(0,2) (T4 )
d1(1,2) (T4 )
d1(0,1) (T4 )
d1(1,1) (T4 )
d1(2,1) (T4 )
7−→ ((• • •) •)
7−→ (• (• • •))
7−→ ((••) • •)
7−→ (• (••) •)
7−→ (• • (••))
∈ K31,0
∈ K31,0
∈ K30,1
∈ K30,1
∈ K30,1
There are four compositions of face operators
d1(i2 ,1) d1(i1 ,2) : K42 → K31,0 → K20,0,0
with 0 ≤ i1 , i2 ≤ 1, and six compositions
d2(i2 ,1) d1(i1 ,1) : K42 → K30,1 → K20,0,0
with 0 ≤ i1 ≤ 2 and 0 ≤ i2 ≤ 1, which pair off via relations (1) to (3) and relate a2
to each of the five vertices of K4 as follows:
d2(0,1) d1(0,1) (T4 )
d2(0,1) d1(1,1) (T4 )
d2(1,1) d1(1,1) (T4 )
d2(1,1) d1(2,1) (T4 )
d2(0,1) d1(2,1) (T4 )
=
=
=
=
=
d1(0,1) d1(0,2) (T4 )
d1(1,1) d1(0,2) (T4 )
d1(0,1) d1(1,2) (T4 )
d1(1,1) d1(1,2) (T4 )
d2(1,1) d1(0,1) (T4 )
7−→
7−→
7−→
7−→
7−→
(((••) •) •)
((• (••)) •)
(• ((••) •))
(• (• (••)))
((••) (••))
These relations encode the fact that when inserting two pairs of parentheses into a
string of four variables either pair may be inserted first.
n1 ,... ,nk
n
by as follows: First, s1j : Kn+2
→
The degeneracy operator sqj acts on Kn−m+3
st
n+1
th
Kn+3 attaches a new leaf between the (j − 1) and j , i.e.,
s1j (x1 · · · xn+2 ) = (x1 · · · xj−1 e xj · · · xn+2 ) , 1 ≤ j ≤ n + 3,
where we think of e as a unit; thus for all j,
s1j (Tn+2 ) = Tn+3 .
(i ,` ),...,(i ,` )
st
n1 ,... ,nk
1 1
m m
∈ Kn−k+3
by treating the (q − 1) pair
In general, sqj acts on Tn+2
of parentheses and its contents in d(im ,`m ) · · · d(i1 ,`1 ) as a single indeterminant and
inserting the unit e inside the q th pair between the (j − 1)st and j th indeterminant.
Thus for all j,
(i1 ,`1 ),...,(iq ,`q +1),... ,(im ,`m )
(i1 ,`1 ),...,(im ,`m )
, 1 ≤ q ≤ m + 1.
= Tn+3
sqj Tn+2
It follows that K is an associahedral set.
Example 2. For n ≥ 0, let Yn+2 be the poset of all planar binary rooted trees with
n + 2 leaves discussed above. The set
o
n
n1 ,...,nm+1
(i1 ,`1 ),...,(im ,`m )
∈ Kn−m+2
,
K = Yn+2
n≥0; 0≤m≤n
Pm
where nq = `q − 1 for 1 ≤ q ≤ m and nm+1 = n − j=1 `j , is an associahedral set
q
q
under face and degeneracy operators d(i,`) and sj defined by formally identifying
A DIAGONAL ON THE ASSOCIAHEDRA
(i ,` ),...,(i ,`m )
1 1
m
Yn+2
Example 1.
(i ,` ),...,(im ,`m )
1 1
with the tree Tn+2
9
and applying the evaluation rules in
Example 2 is a universal example of an associahedral complex, which we now
define.
Definition 2. An associahedral complex is a poset K with distinguished finite subposets {Kα } such that
(a) Each element of K is distinguished;
(b) For each α, there is an ordering isomorphism
≈
(i1 ,`1 ),...,(imα ,`mα )
iα : Kα −→ Ynα
;
(i0 ,`0 ),...,(i0mα ,`0mα )
Ynα1 1
,
then Kα0 is distinguished.
(c) If Kα0 ⊆ Kα and iα (Kα0 ) =
(d) Face and degeneracy operators on Kα are determined by the face and degeneracy operators on iα (Kα ) defined in Example 2.
Let K be an associahedral set and let b = dq(imm ,`m ) · · · dq(i22 ,`2 ) dq(i11 ,`1 ) (a) for some
a ∈ K. For notational simplicity, we henceforth suppress upper indices q2 , . . . , qm
when qj+1 = qj + 1 for all j ≥ 1; we suppress all upper indices if, in addition,
q1 = 1.
Definition 3. Let m ≥ 2. A sequence of lower indices I = (i1 , `1 ) , . . . , (im , `m ) is
admissible whenever the composition of face operators dq(imm ,`m ) · · · dq(i11 ,`1 ) is defined.
The sequence I is a type I (resp. type II) sequence if I is admissible and ik ≥ ik+1
(resp. ik ≤ ik+1 + `k+1 ) for 1 ≤ k < m. The empty sequence (m = 0) and
sequences of length 1 (m = 1) are both type I and II sequences. A composition of
face operators d(im ,`m ) · · · d(i2 ,`2 ) ds(i1 ,`1 ) has first (resp. second) fundamental form
if (i1 , `1 ) , . . . , (im , `m ) is a type I (resp. type II) sequence. When m = 0, the
composition d(im ,`m ) · · · d(i2 ,`2 ) ds(i1 ,`1 ) is defined to be the identity. An element b =
dq(imm ,`m ) · · · dq(i22 ,`2 ) dq(i11 ,`1 ) (a) is expressed in first (resp. second) fundamental form
as a face of a if
dqm · · · dq2 dq1 = (d · · · ddsm ) · · · (d · · · dds2 ) (d · · · dds1 ) ,
s
where s1 < s2 < · · · < sm and each composition d(isj ,`sj ) · · · d(i2 ,`2 ) d(ij1 ,`1 ) has first
(resp. second) fundamental form.
Obviously, every composition of face operators can be uniquely transformed into
first or second fundamental form by successive applications of face relations (1) to
(3).
Let C∗ (Kn+2 ) denote the cellular chains on Kn+2 . For 0 ≤ m ≤ n, let Im =
(i1 , `1 ), (i2 , `2 ), . . . , (im , `m ) be a type I sequence and let dIm = d(im ,`m ) · · · d(i1 ,`1 ) .
Then
Im
= dIm (Tn+2 ) ∈ Cn−m (Kn+2 )
Tn+2
and
n
Im
| Im is a type I sequence
Tn+2
defines a system of generators for C∗ (Kn+2 ).
o
0≤m≤n
SAMSON SANEBLIDZE AND RONALD UMBLE1
10
Definition 4. For m ≥ 0, let Im = (i1 , `1 ), (i2 , `2 ), . . . , (im , `m ) be a type I seIm
) ∈ Cn−m−1 (Kn+2 ) is
quence. If 1 ≤ r ≤ m + 1, the sign of the face b = dr(i,`) (Tn+2
defined to be
P
sgn (b) = − (−1)e(b) , where e (b) = r + (i + 1) ` + r−1
j=1 `j .
Remark 2. When applying face relations (1) to (3) in Definition 1 to cellular
chains in C∗ (Kn+2 ), only (3) requires the sign (−1)(`q +1)(`q+1 +1) for commuting
factors.
Example 3. The set K defined in Example 1 generates the associahedral set of
cellular chains {C∗ (Kn+2 )}n≥0 . According to Definition 4, the cellular boundary
of Tn+2 ∈ Cn (Kn+2 ) is given by
X
(−1)(i+1)` d(i,`) (Tn+2 ) .
∂Tn+2 =
1≤`≤n
0≤i≤n−`+1
Example 4. Let X be a topological space. Define the singular associahedral complex Sing K X as follows: Let
K
] X)n1 ,...,nk = {continuous maps Kn1 +2 × · · · × Knk +2 → X},
(Sing
n−k+3
Pk
where j=1 nj = n − k + 1, nj ≥ 0 and Kn1 +2 × · · · × Knk +2 is a Cartesian product
of associahedra. Let
K
] X/∼
Sing K X = Sing
K
] X given by the permudenote the orbit set of the symmetric group action on Sing
tation isomorphisms
≈
σ : Kn1 +2 × · · · × Knk +2 → Kσ−1 (n1 )+2 × · · · × Kσ−1 (nk )+2 ,
where σ ∈ Sk , k ≥ 2. Let
q
: Kn1 +2 × · · · × Knq−1 +2 × K`q +1 × Knq −`q +2 × Knq+1 +2 × · · · × Knk +2
δ(i
q ,`q )
→ Kn1 +2 × · · · × Knq +2 × · · · Knk +2
be the inclusion and let
βiq : Kn1 +2 × · · · × Knq +3 × · · · Knk +2 → Kn1 +2 × · · · × Knq +2 × · · · × Knk +2
1 ,...,nk
be the projection (cf. [17]). Then for f ∈ (Sing K X)nn+2
, define
n ,...,nq−1 ,`q −1,nq −`q ,nq+1 ,...nk
1
1 ,...,nk
→ (Sing K X)n+1
dq(iq ,`q ) : (Sing K X)nn+2
and
n ,...,nq−1 ,nq +1,nq+1 ,...nk
1
1 ,...,nk
→ (Sing K X)n+3
sqi : (Sing K X)nn+2
as compositions
q
and sqi (f ) = βiq ◦ f.
dq(iq ,`q ) (f ) = f ◦ δ(i
q ,`q )
It is easy to check that (Sing K X, d(iq ,`q ) , sqi ) is an associahedral set.
A DIAGONAL ON THE ASSOCIAHEDRA
11
The singular associahedral complex Sing K X determines the singular (co)homology of X in the following way: Form the chain complex (C∗ (Sing K X), d) of
Sing K X and consider the quotient
C∗N (Sing K X) = C∗ (Sing K X)/D,
where D is the subcomplex of C∗ (Sing K X) generated by the degenerate elements
of Sing K X. The composition of canonical projections
Kn1 +2 × · · · × Knk +2 → I n1 × · · · × I nk = I n → ∆n
induces a sequence of chain maps
C∗ (SingX) → C∗ (Sing I X) → C∗ (Sing K X) → C∗N (Sing K X),
and consequently, a natural isomorphism H∗ (X) ≈ H∗ (C∗N (Sing K X), d). As in the
cubical setting (but unlike the simplicial setting), normalized chains are required
to obtain the singular homology of X. In the discussion that follows below, we
construct a diagonal on the associahedra, which is compatible under the projections above with the Alexander-Whitney diagonal on the standard simplex. Thus
H ∗ (C N (Sing K X), d) determines the singular cohomology ring of X as well.
4. A Diagonal ∆ on C∗ (Kn )
We begin with an overview of the geometric ideas involved. Let 0 ≤ q ≤ n and
In−q
is associated
let In−q be a type I sequence. The q-dimensional generator aq = Tn+2
with a face of Kn+2 corresponding to n + 2 indeterminants with n− q + 1 pairs of
parentheses. Identify aq with its associated face of Kn+2 and consider the minimal
and amax
of aq . Define the primitive terms of ∆Tn+2 to
and maximal vertices amin
q
q
be
min
max
⊗ Tn+2 + Tn+2 ⊗ Tn+2
.
Tn+2
Let 0 < p < p + q = n and consider distinct p-faces b and b0 of Tn+2 . Say that
b ≤ b0 if there is a path of p-faces from b to b0 along which parentheses shift strictly
min
6= Tn+2
,
to the right. Now given a q-dimensional face aq of Tn+2 such that amin
q
there is a unique path of p-faces b1 ≤ b2 ≤ · · · ≤ br with minimal length such that
min
= bmin
and amin
= bmax
. Up to sign, we define the non-primitive terms of
Tn+2
1
q
r
∆Tn+2 to be
X
± b j ⊗ aq .
To visualize this,
consider the edge d(1,2) (T4 ) ∈ C1 (K4 ) whose minimal vertex
is the point 1, 12 (see Figure 7). The
edges d(0,2) (T4 ) ≤ d(1,1) (T4 ) form a path of
1
.
Consequently,
∆T4 contains the non-primitive
minimal
length
from
(0,
0)
to
1,
2
terms ±d(0,2) ± d(1,1) ⊗ d(1,2) (T4 ⊗ T4 ) .
12
SAMSON SANEBLIDZE AND RONALD UMBLE1
d(2,1) (T4 )
•
N
T4max
•
IN
d(1,2) (T4 )
d(0,1) (T4 )
•
N
T4
d(1,1) (T4 )
•
T4min
I•
d(0,2) (T4 )
Figure 7: Edge paths from T4min to T4max .
Precisely, for T2 ∈ C∗ (K2 ) define ∆T2 = T2 ⊗ T2 ; inductively, assume that the
map ∆ : C∗ (Ki+2 ) → C∗ (Ki+2 ) ⊗ C∗ (Ki+2 ) has been defined for all i < n. For
Tn+2 ∈ Cn (Kn+2 ) define
X
(−1) d(i0q ,`0q ) · · · d(i01 ,`01 ) (Tn+2 ) ⊗ d(ip ,`p ) · · · d(i1 ,`1 ) (Tn+2 )
∆Tn+2 =
0≤p≤p+q=n
where
=
q
X
j=1
i0j (`0j + 1) +
p
X
(ik + k + p + 1)`k ,
k=1
and lower indices (i1 , `1 ) , . . . , (ip , `p ) ; (i01 , `01 ) , . . . , i0q , `0q
of the following system of inequalities:
(4.1)

1 ≤ ij < ij−1 ≤ n + 1







 1 ≤ `j ≤ n + 1 − ij − `(j−1)
n
o

i0r , itk − `0(o0 (tk ))
0 ≤ i0k ≤ 0 min


o (tk )<r<k





1 ≤ `0k = k − i0k − `0(k−1)
range over all solutions

(1) 




(2) 


(3) 







(4) 1≤j≤p
where
{1 < · · · < q } = {1, . . . , n} \ {i1 , . . . , ip } ;
0 = `0 = `00 = ip+1 = i0q+1 = 0;
(4.2)
i0 = i00 = q+1 = `(p+1) = `0(q+1) = n + 1;
Pu
`(u) = j=0 `j for 0 ≤ u ≤ p + 1;
Pu
`0(u) = k=0 `0k for 0 ≤ u ≤ q + 1;
tu = min r | ir + `(r) − `(o(u)) > u > ir ;
o (u) = max {r | ir ≥ u } ; and
o0 (u) = max {r | r ≤ iu } .
1≤k≤q
,
A DIAGONAL ON THE ASSOCIAHEDRA
13
Extend ∆ multiplicatively to all of C∗ (Kn+2 ), using the fact that the cells of Kn+2
are products of cells Ki+2 with i < n.
Note that right-hand and left-hand factors in each component of ∆Tn+2 are
expressed in first and second fundamental form, respectively. In particular, the
terms given by the extremes p = 0 and p = n are the primitive terms of ∆Tn+2 :
d(0,1) · · · d(0,1) ⊗ 1 + 1 ⊗ d(1,1) · · · d(n,1) (Tn+2 ⊗ Tn+2 ) .
P
The sign in ∆Tn+2 = (−1) b ⊗ a is the product of five signs: (−1) = sgn (b) ·
sgn (b0 ) · sgn(b0 , a1 ) · sgn (a1 ) · sgn (a) , where the face b0 is obtained from b by
keeping ε’s but all i0k = 0 (i.e., `0k is replaced by `0k + i0k − i0k−1 ), the face a1 is
obtained from a by keeping ir ’s but all `r = 1 and sgn(b0 , a1 ) is the sign of the
shuffle {ip < · · · < i1 , 1 < 2 < · · · < q }. Geometrically, b0 and a1 lie on orthogonal
faces of the cube I n and are uniquely defined by the property that the canonical
cellural projection Kn+2 → I n maps b0 7→ x1 , ..., x1 −1 , 0, ..., xq −1 , 0, ..., xn and
a1 7→ x1 , ..., xip −1 , 1, ..., xi1 −1 , 1, ..., xn .
Example 5. For T4 ∈ C2 (K4 ) we obtain by direct calculation:
∆T4 = d(0,1) d(0,1) ⊗ 1 + 1 ⊗ d(1,1) d(2,1) + d(0,2) ⊗ d(1,1)
o
+d(0,2) ⊗ d(1,2) + d(1,1) ⊗ d(1,2) − d(0,1) ⊗ d(2,1) (T4 ⊗ T4 ).
A proof of our main result, stated as the following theorem, appears in the appendix:
Theorem 1. For each n ≥ 0, the map ∆ : C∗ (Kn+2 ) −→ C∗ (Kn+2 ) ⊗ C∗ (Kn+2 )
defined above is a chain map.
Identify the sequence of cellular chain complexes {C∗ (Kn )}n≥2 with the A∞ -operad
A∞ [11]. Since ∆ is extended multiplicatively on decomposable faces, we immediately obtain the following interesting fact:
Corollary 1. The sequence of chain maps
{∆ : C∗ (Kn ) −→ C∗ (Kn ) ⊗ C∗ (Kn )}n≥2
induces a morphism of operads
A∞ −→ A∞ ⊗ A∞ .
5. Tensor Products in the A∞ Category
The notion of an A∞ -algebra also appears in Stasheff’s seminal papers [17]. An
A∞ -algebra is a dga in which the associative law holds up to homotopy, the homotopies between the various associations are homotopic, the homotopies between
these homotopies are homotopic, and so on. Although A∞ -algebras first appeared
in topology as the singular chain complex on the loop space of a connected CW complex, they have since assumed their rightful place as fundamental structures
in algebra [10], [15], topology [3], [7], [18], and mathematical physics [4], [5], [12],
[13], [20], [21]. Furthermore, Stasheff’s idea carries over to homotopy versions of
SAMSON SANEBLIDZE AND RONALD UMBLE1
14
coalgebras [14], [16], [19] and Lie algebras [6], and one can deform a classical dga,
dgc or dg Lie algebra to the corresponding homotopy version in a standard way.
An R-module A equipped simultaneously with an A∞ -algebra and an A∞ coalgebra structure is an “A∞ -Hopf algebra” if the two A∞ structures are sufficiently compatible. Our definition of an A∞ -Hopf algebra, which appears in the
sequel, requires an independently interesting tensor product in the A∞ category.
In this concluding section, we apply the diagonal ∆ defined above to obtain this
tensor product in maximal generality. We note that a special case was given by
J. Smith [16] for certain objects with a richer structure than we have here. We
also mention that Lada and Markl [6] defined a A∞ tensor product structure on a
construct different from the tensor product of graded R-modules.
We adopt the following notation and conventions: The symbol R denotes a
commutative ring with unity; all R-modules are assumed to be Z-graded. The
reduced R-module V /V0 of a connected V is denoted by V . All tensor products
and Hom’s are defined over R and all maps are R-module maps unless indicated
otherwise. The symbol 1 : V → V denotes the identity map; the suspension
and desuspension maps, which shift dimension by +1 and −1, are denoted by ↑
⊗n
and ↓, respectively. We
P let V ⊗n = V ⊗ · · · ⊗ V with n > 0 factors and define
⊗0
= R; then T V = n≥0 V . If A is an R-algebra (resp., R-coalgebra) , then
V
T a A (respectively, T c A) denotes the tensor algebra (resp., the tensor coalgebra) of
A. Given a R-modules V1 , . . . , Vn and a permutation σ ∈ Sn , define the permutation
isomorphism σ : V1 ⊗ · · · ⊗ Vn → Vσ−1 (1) ⊗ · · · ⊗ Vσ−1 (n) by σ(x1 · · · xn ) = ±
xσ−1 (1) · · · xσ−1 (n) , where the sign is determined by the Mac Lane commutation rule
⊗m
→ V1⊗m ⊗ V2⊗m is
[9] to which we strictly adhere. In particular, σm : (V1 ⊗ V2 )
the permutation isomorphism induced by σm = (1 3 · · · (2m − 1) 2 4 · · · 2m) . If
f : V ⊗p → V ⊗q is a map, we let fi,n−p−i = 1⊗i ⊗ f ⊗ 1⊗n−p−i : V ⊗n → V ⊗n−p+q ,
where 0 ≤ i ≤ n − p. The abbreviations dgm, dga,and dgc stand for differential
graded R-module, dg R-algebra and dg R-coalgebra, respectively.
We begin with a discussion of A∞ -(co)algebras paying particular attention to
the signs; in structural compatibility considerations, sign issues become critical.
Let A be a connected
graded R-module and consider a sequence of maps {ϕk ∈
k−2
⊗k
A , A }k≥1 . For each k and n ≥ 1, linearly extend ϕk to A⊗n via
Hom
n−k
X
ϕki,n−k−i : A⊗n → A⊗n−k+1 ,
i=0
and consider the induced map of degree −1 given by
n−k
X
↑ ϕk ↓⊗k
i,n−k−i
: ↑A
⊗n
⊗n−k+1
→ ↑A
.
i=0
e → BA
e of degree −1 by
e = T c ↑ A and define a map d e : BA
Let BA
BA
(5.1)
dBA
e =
X
1≤k≤n
0≤i≤n−k
↑ ϕk ↓⊗k
i,n−k−i
.
A DIAGONAL ON THE ASSOCIAHEDRA
[n/2]
The identities (−1)
imply that
(5.2)
↑⊗n ↓⊗n = 1⊗n and [n/2] + [(n + k) /2] ≡ nk + [k/2] (mod 2)
X
dBA
e =
15
[(n−k)/2]+i(k+1)
(−1)
↑⊗n−k+1 ϕki,n−k−i ↓⊗n .
1≤k≤n
0≤i≤n−k
= 0. The maps {ϕn } in an
Definition 5. (A, ϕn )n≥1 is an A∞ -algebra if d2BA
e
A∞ -algebra are called higher homotopies.
Proposition 2. Let (A, ϕn )n≥1 be an A∞ -algebra. The higher homotopies
satisfy the following quadratic relations for n ≥ 1:
X
`(i+1) n−` `+1
(5.3)
(−1)
ϕ
ϕi,n−`−1−i = 0.
ϕk
0≤`≤n−1
0≤i≤n−`−1
Proof: For n ≥ 1,
X
(−1)[(n−k)/2]+i(k+1) ↑ ϕn−k+1 ↓⊗n−k+1 ↑⊗n−k+1 ϕki,n−k−i ↓⊗n
0=
1≤k≤n
0≤i≤n−k
=
X
(−1)n−k+i(k+1) ϕn−k−1 ϕki,n−k−i
1≤k≤n
0≤i≤n−k
n
= − (−1)
X
`(i+1)
(−1)
ϕn−` ϕ`+1
i,n−`−1−i .
0≤`≤n−1
0≤i≤n−`−1
It is easy to prove that
e de
Proposition 3. Given an A∞ -algebra (A, ϕn )n≥1 , BA,
BA is a dgc.
Definition 6. Let (A, ϕn )n≥1 be an A∞ -algebra. The tilde bar construction on A
e de .
is the dgc BA,
BA
Definition 7. Let A and C be A∞ -algebras. A chain map f = f 1 : A → C is a
k
k−1
map of A
∞ -algebras if there exists a family of higher homotopies {f ∈ Hom
⊗k
A , C }k≥2 such that
P
⊗n
P
k ⊗k
e → BC
e
: BA
fe = n≥1
k≥1 ↑ f ↓
is a dgc map.
Dually, consider a sequence of maps {ψ k ∈ Homk−2 A, A⊗k }k≥1 . For each k
and n ≥ 1, linearly extend each ψ k to A⊗n via
n−1
X
i=0
k
ψi,n−1−i
: A⊗n → A⊗n+k−1 ,
SAMSON SANEBLIDZE AND RONALD UMBLE1
16
and consider the induced map of degree −1 given by
n−1
X
↓⊗k ψ k ↑
i,n−1−i
⊗n
⊗n+k−1
: ↓A
→ ↓A
.
i=0
e → ΩA
e of degree −1 by
e = T a ↓ A and define a map d e : ΩA
Let ΩA
ΩA
X
dΩA
↓⊗k ψ k ↑ i,n−1−i ,
e =
n,k≥1
0≤i≤n−1
which can be rewritten as
X
[n/2]+i(k+1)+k(n+1) ⊗n+k−1 k
(5.4)
(−1)
↓
ψi,n−1−i ↑⊗n .
dΩA
e =
n,k≥1
0≤i≤n−1
= 0. The maps {ψ n } in an
Definition 8. (A, ψ n )n≥1 is an A∞ -coalgebra if d2ΩA
e
A∞ -coalgebra are called higher homotopies.
Proposition 4. Let (A, ψ n )n≥1 be an A∞ -coalgebra. The higher homotopies ψ k
satisfy the following quadratic relations for n ≥ 1:
X
`(n+i+1) `+1
(5.5)
(−1)
ψi,n−`−1−i ψ n−` = 0.
0≤`≤n−1
0≤i≤n−`−1
Proof: The proof is similar to the proof of Proposition 2 and is omitted.
Again, it is easy to prove that
e de
Proposition 5. Given an A∞ -coalgebra (A, ψ n )n≥1 , ΩA,
ΩA is a dga.
Definition 9. Let (A, ψ n )n≥1 be an A∞ -coalgebra. The tilde cobar construction
e de .
on A is the dga ΩA,
ΩA
Definition 10. Let A and B be A∞ -coalgebras. A chain map g = g 1 : A → B is a
k
k−1
map of A
∞ -coalgebras if there exists a family of higher homotopies g ∈ Hom
A, B ⊗k }k≥2 such that
⊗n
P
P
⊗k k
e → ΩB,
e
(5.6)
↓
g
↑
: ΩA
ge = n≥1
k≥1
is a dga map.
The essential ingredient in an A∞ -(co)algebra is the given family of higher homotopies. Given such a family, one is free to independently reverse the direction
of each homotopy, i.e., its sign, and subsequently alter the signs in the quadratic
relations. Nevertheless, the signs in the relations relate directly to the initial given
family. The “simplest” signs in the quadratic relations in an A∞ -algebra appear
when the tilde bar differential is the sum of the given higher homotopies, i.e., the
direction of each higher homotopy is unaltered as in (5.1) above. Consequently,
the simplest signs appear in (5.3) above. On the other hand, a straightforward
A DIAGONAL ON THE ASSOCIAHEDRA
17
calculation shows that the “simplest” signs appear in the quadratic relations for an
[(n−1)/2] n
ψ , n ≥ 1, i.e., the direction of
A∞ -coalgebra when ψ n is replaced by (−1)
every third and fourth homotopy is reversed. The choices one makes will depend
on the application; for us the appropriate choices are as in (5.3) and (5.5).
Let (A, ϕn )n≥1 be an A∞ -algebra with structure relations as in (5.3). For each
n ≥ 2, associate Tn ∈ Cn−2 (Kn ) with the operation ϕn via
Tn 7→ (−1)n ϕn
(5.7)
and d(i,`) (Tn ) ∈ Cn−3 (Kn ) with the quadratic composition ϕn−` ϕ`+1
i,n−`−1−i , i.e.,
d(i,`) (Tn ) 7→ ϕn−` ϕ`+1
i,n−`−1−i .
(5.8)
Then (5.7) and (5.8) induce a chain map
(5.9)
ζA : A∞ −→ Hom∗ A⊗n , A n≥2
representing the A∞ -algebra structure on A.
Dually, if (A, ψ n )n≥1 is an A∞ -coalgebra with structure relations as in (5.5), the
associations
Tn 7→ ψ n
and
`+1
ψ n−`
d(i,`) (Tn ) 7→ ψi,n−`−1−i
induce a chain map
ξA : A∞ −→ Hom∗ A, A⊗n n≥2
(5.10)
representing the A∞ -coalgebra structure on A.
Definition 11. Let A and B be A∞ -algebras with respective A∞ -algebra structures
ζA and ζB . The tensor product A ⊗ B is the A∞ -algebra with structure
∗
ζA⊗B (Tn ) = (σn ) (ζA ⊗ ζB ) ∆Tn n≥2 ,
where (σn )∗ = Hom(σn , A ⊗ B). Similarly, if A and B are A∞ -coalgebras with
respective structures ξA and ξB , A ⊗ B is the A∞ -coalgebra with structure
ξA⊗B (Tn ) = σn−1 ∗ (ξA ⊗ ξB ) ∆Tn n≥2 ,
where σn−1 ∗ = Hom A ⊗ B, σn−1 .
Proposition 6. Let A and B be A∞ -algebras with respective A∞ -algebra structures
ζA and ζB ; let A ⊗ B be the A∞ -algebra with structure ζA⊗B . Then the following
diagram commutes for each n ≥ 2 :
C∗ (Kn )
∆
ζA⊗B
−→
↓
C∗ (Kn ) ⊗ C∗ (Kn )
Hom((A ⊗ B)⊗n , A ⊗ B)
↑
−→
ζA ⊗ζ B
(σn )∗
Hom(A⊗n , A) ⊗ Hom (B ⊗n , B)
SAMSON SANEBLIDZE AND RONALD UMBLE1
18
Dually, let A and B be A∞ -coalgebras with respective A∞ -coalgebra structures ξA
and ξB ; let A ⊗ B be the A∞ -coalgebra with structure ξA⊗B . Then the following
diagram commutes for each n ≥ 2 :
ξA⊗B
−→
C∗ (Kn )
∆
Hom(A ⊗ B, (A ⊗ B)⊗n )
↓
↑ (σn−1 )
∗
C∗ (Kn ) ⊗ C∗ (Kn )
−→
ξA ⊗ξ B
Hom(A, A⊗n ) ⊗ Hom (B, B ⊗n )
Proof: This is an immediate consequence of Definition 11 and the canonical isomorphisms Hom(A⊗n , A) ⊗ Hom (B ⊗n , B) ≈ Hom(A⊗n ⊗ B ⊗n , A ⊗ B) and
Hom(A, A⊗n ) ⊗ Hom (B, B ⊗n ) ≈ Hom(A ⊗ B, A⊗n ⊗ B ⊗n ).
Example 6. Given an A∞ -coalgebra A with operations {ψ n }n≥1 , we have the following operations on A ⊗ A:
1
=
ψA⊗A
ψ1 ⊗ 1 + 1 ⊗ ψ1
2
ψA⊗A
=
σ2 ψ 2 ⊗ ψ 2 ,
3
=
ψA⊗A
σ3 ψ02 ψ02 ⊗ ψ 3 + ψ 3 ⊗ ψ12 ψ02 ,
4
=
ψA⊗A
..
.
σ4 ψ02 ψ02 ψ02 ⊗ ψ 4 + ψ 4 ⊗ ψ22 ψ12 ψ02 + ψ03 ψ02 ⊗ ψ12 ψ03
+ψ03 ψ02 ⊗ ψ13 ψ02 + ψ12 ψ03 ⊗ ψ13 ψ02 − ψ02 ψ03 ⊗ ψ22 ψ03
..
.
The stage is now set for the sequel in which we introduce the notion of an A∞ Hopf algebra in full generality. We shall observe that the homology of a loop space
and the double cobar construction admit canonical A∞ -Hopf algebra structures.
6. Appendix
In this appendix we prove that the diagonal ∆ defined in Section 4 above is a
chain map. We begin with some preliminaries. It will be convenient to rewrite face
relation (3) as follows:
q
q+1
q
dq+1
(iq+1 ,`q+1 ) d(iq ,`q ) = d(iq −`q+1 ,`q ) d(iq+1 ,`q+1 ) ;
iq > iq+1 + `q+1 .
(30 )
Definition 12. For 1 ≤ k ≤ m, the k th left-transfer τ`k of a composition d(im ,`m )
· · · d(i2 ,`2 ) d(i1 ,`1 ) in first fundamental form is one of the following compositions:
(a) If ik+1 + `k+1 ≥ ik , apply face relation (2) to dk+1 dk and obtain
· · · dk(ik −ik+1 ,`k ) dk(ik+1 ,`k+1 +`k ) · · · ;
then successively apply face relation (1) to dj+2 dj for j = k, k + 1, . . . , m − 2,
and obtain
k+1
k
1
τ`k = dk(ik −ik+1 ,`k ) dm−1
(im ,`m ) · · · d(ik+2 ,`k+2 ) d(ik+1 ,`k+1 +`k ) · · · d(i1 ,`1 ) .
A DIAGONAL ON THE ASSOCIAHEDRA
19
(b) If ik+1 + `k+1 < ik ≤ iq+1 + `(q+1) − `(k) for some smallest integer q > k, successively apply face relation (30 ) to dj+1 dj for j = k, k + 1, . . . , q, and obtain
q+1
q
k
· · · dq+2
(iq+2 ,`q+2 ) d(ik +`(k) −`(q+1) ,`k ) d(iq+1 ,`q+1 ) · · · d(ik+1 ,`k+1 ) d(ik−1 ,`k−1 ) · · · .
Apply face relation (2) to dq+1 dq ; then successively apply face relation (1) to
dj+2 dj for j = q, q + 1, . . . , m − 2, and obtain
k
τ`k = dq(i,`k ) dm−1
(im ,`m ) · · · d(ik+1 ,`k+1 ) d(ik−1 ,`k−1 ) · · · d(i1 ,`1 ) ,
where i = ik − iq+1 + `(k) − `(q) .
(c) Otherwise, successively apply face relation (30 ) to dj+1 dj for j = k, k+1, . . . , m−
1, and obtain
m−1
k
τ`k = dm
(i,`k ) d(im ,`m ) · · · d(ik+1 ,`k+1 ) d(ik−1 ,`k−1 ) · · · d(i1 ,`1 )
where i = ik + `(k) − `(m) .
Definition 13. For 1 ≤ k ≤ m, the k th right-transfer τrk of a composition d(im ,`m )
· · · d(i2 ,`2 ) d(i1 ,`1 ) in first fundamental form is one of the following compositions:
(a) If ip + `(p) ≤ ik + `(k) < ip−1 + `(p−1) for some greatest integer 1 < p ≤ k,
successively apply face relation (30 ) to dj+1 dj for j = p − 1, . . . , 2, 1; j =
p, . . . , 3, 2; . . . , j = k − 1, k − 2, . . . , k − (p − 1). Then apply face relation
(2) to dj+1 dj for j = k − p, . . . , 2, 1 and obtain
τrk = d(im ,`m ) · · · d(ik+1 ,`k+1 ) dk(ip−1 −`,`p−1 )
k−p
1
1
· · · dk−p+2
(i1 −`,`1 ) d(ik−1 −ik ,`k−1 ) · · · d(ip −ik ,`p ) d(ik ,`) ,
where ` = `(k) − `(p−1) .
(b) Otherwise, successively apply face relation (2) to dj+1 dj for j
k − 2, . . . , 2, 1, and obtain
1
1
τrk = d(im ,`m ) · · · d(ik+1 ,`k+1 ) dk−1
(ik−1 −ik ,`k−1 ) · · · d(i1 −ik ,`1 ) d(i
=
k ,`(k)
k − 1,
).
Note that if I = (i1 , `1 ) , . . . , (im , `m ) is a type I sequence and a = dI (Tn+2 ) ∈
Cn−m (Kn+2 ) , then for each k, the k th right transfer τrk (Tn+2 ) expresses a in
first fundamental form as a face of some (n − 1)-face b of Tn+2 . The expressions
τr1 (Tn+2 ) , . . . , τrm (Tn+2 ) determine the m distinct (n − 1)-faces b containing a.
There are analogous left and right transfers for compositions in second fundamental form.
Definition 14. For 1 ≤ k ≤ m, the k th left-transfer τ`k of a composition d(im ,`m )
· · · d(i2 ,`2 ) d(i1 ,`1 ) in second fundamental form is one of the following compositions:
(a) If ik+1 ≤ ik , apply face relation (2) to dk+1 dk , then successively apply face
relation (1) to dj+2 dj for j = k, k + 1, . . . , m − 2, and obtain
k+1
k
1
τ`k = dk(ik −ik+1 ,`k ) dm−1
(im ,`m ) · · · d(ik+2 ,`k+2 ) d(ik+1 ,`k+1 +`k ) · · · d(i1 ,`1 ) .
20
SAMSON SANEBLIDZE AND RONALD UMBLE1
(b) If ik+1 > ik ≥ iq+1 for some smallest integer q > k, successively apply face
relation (3) to dj+1 dj for j = k, k + 1, . . . , q. Apply face relation (2) to
dq+1 dq ; then successively apply face relation (1) to dj+2 dj for j = q, q+1, . . . , m−
2, and obtain
q+1
q
q−1
τ`k = dq(ik −iq+1 ,`k ) dm−1
(im ,`m ) · · · d(iq+2 ,`q+2 ) d(iq+1 ,`q+1 +`k ) d(iq +`k ,`q )
· · · dk(ik+1 +`k ,`k+1 ) d(ik−1 ,`k−1 ) · · · d(i1 ,`1 ) .
(c) Otherwise, successively apply face relation (3) to dj+1 dj for j = k, k + 1,
. . . , m − 1, and obtain
m−1
k+1
k
τ`k = dm
(ik ,`k ) d(im +`k ,`m ) · · · d(ik+2 +`k ,`k+2 ) d(ik+1 +`k ,`k+1 ) d(ik−1 ,`k−1 ) · · · d(i1 ,`1 ) .
Definition 15. For 1 ≤ k ≤ m, the k th right-transfer τrk of a composition d(im ,`m )
· · · d(i2 ,`2 ) d(i1 ,`1 ) in second fundamental form is one of the following compositions:
(a) If ip−1 < ik ≤ ip for some greatest integer 1 < p ≤ k, successively apply
face relation (3) to dj+1 dj for j = p − 1, . . . , 2, 1; j = p, . . . , 3, 2; . . . ; j =
k − 1, k − 2, . . . , k − (p − 1). Then apply face relation (2) to dj+1 dj for j =
k − p, . . . , 2, 1 and obtain
k−p
τrk = d(im ,`m ) · · · d(ik+1 ,`k+1 ) dk(ip−1 ,`p−1 ) · · · dk−p+2
(i1 ,`1 ) d(ik−1 −ik ,`k−1 )
· · · d1(ip −ik ,`p ) d1(ik +`(p−1) ,`(k) −`(p−1) ) .
(b) Otherwise, successively apply face relation (2) to dj+1 dj for j = k − 1, . . . ,
2, 1, and obtain:
τrk = d(im ,`m ) · · · d(ik+1 ,`k+1 ) dk(ik−1 −ik ,`k−1 ) · · · d1(i1 −ik ,`1 ) d1(ik ,`(k) ) .
Once again, if I = (i1 , `1 ) , . . . , (im , `m ) is a type II sequence and a = dI (Tn+2 ) ∈
Cn−m (Kn+2 ) , then for each k, the k th right-transfer τrk (Tn+2 ) expresses a in second
fundamental form as a face of some (n − 1)-face b; the expressions τr1 (Tn+2 ) , . . . ,
τrm (Tn+2 ) determine the m distinct (n − 1)-faces b containing a. Thus two (n − m)faces a1 and a2 expressed in first and second fundamental form, respectively, are
k
contained in the same (n − 1)-face b if and only if there exist right transfers τr j
kj
such that aj = τr (Tn+2 ) , j = 1, 2, and b = d(i,`) (Tn+2 ), where d(i,`) is the rightmost face operator common to τrk1 and τrk2 . We state this formally in the following
lemma:
Lemma 1. Let 0 ≤ m1 , m2 ≤ n and assume that a1 = d(im1 ,`m1 ) · · · d(i1 ,`1 ) (Tn+2 )
and a2 = d(i0m ,`0m ) · · · d(i01 ,`01 ) (Tn+2 ) are expressed in first and second fundamen2
2
tal form, respectively. Then a1 and a2 are contained in the same (n − 1)-face
d(i,`) (Tn+2 ) if and only if there exist integers kj ≤ mj and greatest integers 1 ≤
pj < kj for j = 1, 2, such that
ik1
i0p2 < i0k2 ,
+ `(k1 ) − `(p1 ) < ip1
A DIAGONAL ON THE ASSOCIAHEDRA
21
and
i = ik1 = i0k2 + `0(p2 ) ,
` = `(k1 ) − `(p1 ) = `0(k2 ) − `0(p2 ) .
Consider a solution ((i1 , `1 ), . . . , (ip , `p ); (i01 , `01 ), . . . , (i0q , `0q )) of system (4.1) and
its related (p, q)-shuffle {ip < ·· · < i1 , 1 < 2 < · · · < q }. Given 1 ≤ k ≤ q + 1,
let k1 = k − 1, kj+1 = o0 tkj for j ≥ 1 and note that kj+1 < kj for all j. For
0 ≤ m < `0k , consider the following selection algorithm:
if k = 1 then z = p + 1 − i01 − m
else z = n + 2; j = 0
repeat
j ←j+1
if i0kj < i0k + m then iz = kj − i0kj + i0k + m
if itkj − `0(kj+1 ) = i0k + m then z = tkj
until z < n + 2
endif
It will be clear from the proof of Lemma 2 below that the selection algorithm
eventually terminates.
Example 7. Let p = q = 4 and consider the following solution of system (4.1):
((7, 1) , (6, 1) , (4, 2) , (2, 3) ; (0, 1) , (1, 1) , (1, 2) , (0, 4)) .
Then t1 = 5, t2 = 4, t3 = 3, t4 = 4 and the selection algorithm produces the
following:
k
m
z
1
0
5
2 3 3
0 0 1
5 4 3
4 4
0 1
5 4
4 4
2 3
2 1
The key to our proof that ∆ is a chain map is given by our next lemma.
Lemma 2. Given 1 ≤ k ≤ q + 1 and 0 ≤ m < `0k , let z be the integer given by the
selection algorithm.
(a) z > o(k)
(b) iz + `(z) − `(o(k)) ≥ k .
(c) i0k + m = iz − `0(o0 (z)) ;
(d) If o0 (z) < r < k, then i0k + m ≤ i0r ;
n
o
i0r , itk − `0(o0 (tk )) , then
(e) If i0k + m > 0 min
o (tk )<r<k
(e.1) z < tk and iz + `(z) − `(o(k)) = k ;
(e.2) o (k) = max r | ir > iz + `(z) − `(r) ;
r<z
SAMSON SANEBLIDZE AND RONALD UMBLE1
22
(e.3) o0 (z) = max {r | i0r < i0k + m} .
r<k
Proof:
(a) When k = 1, z = (p + 1 − 1 ) + (`01 − m) > p + 1 − 1 = o (1) . For k > 1,
first note that itkj < kj < k ≤ io(k) for all j ≥ 1, by the definition of tkj in (4.2).
So the result follows whenever iz ≤ itkj for some j ≥ 1. If i0k−1 < i0k + m, then
iz = k−1 + i0k + m − i0k−1 < k−1 + i0k + `0k − i0k−1 = k . If i0kj < i0k + m for some j ≥ 2
and i0k +m ≤ i0kr ≤ itkr −`0(kr+1 ) for all r < j, where the later follows from inequality
(3) of system (4.1), then iz = kj + i0k + m − i0kj < kj + itkj−1 − `0(kj ) − i0kj = itkj−1 .
(b) When k = 1, indices 0 = ip+1 < ip < · · · < ip+2−1 are consecutive; hence
ir + `(r) = ir−1 + `(r−1) + `r − 1 ≥ ir−1 + `(r−1) for p + 3 − 1 ≤ r ≤ p + 1.
Since z ≥ p + 2 − 1 we have iz + `(z) − `(o(1)) ≥ ip+2−1 + `(p+2−1 ) − `(p+1−1 ) =
1 −1+`p+2−1 ≥ 1 . So consider k > 1. If i0k−1 < i0k +m, then k−1 < iz < k < io(k)
so that k = iz + z − o (k) ≤ iz + `(z) − `(o(k)) . If i0k−1 ≥ i0k + m and z = tk−1 , then
o (k − 1) − o (k) = k − k−1 − 1 so that iz + `(z) − `(o(k)) = iz + `(z) − `(o(k−1)) +
`(o(k−1)) − `(o(k)) > k−1 + k − k−1 − 1 = k − 1 by the definition of tk−1 in (4.2). If
i0k2 < i0k +m then k2 < iz < itk1 < k1 so that `(z) −`(tk ) ≥ z−tk1 = itk1 −iz , where
1
equality holds iff `tk1 +1 = · · · = `z = 1; hence iz + `(z) − `(o(k)) = iz + `(z) − `(tk ) +
1
`(tk ) − `(o(k)) ≥ iz + itk1 − iz + `(tk ) − `(o(k)) > k − 1 by the previous calculation.
1
1
If i0k2 ≥ i0k + m and z = tk2 , then iz + `(z) − `(o(k1 )) = iz + `(z) − `(o(k2 )) + `(o(k2 )) −
`(o(k1 )) > k2 + `(o(k2 )) − `(o(k1 )) ≥ itk1 + `(tk ) − `(o(k2 )) + `(o(k2 )) − `(o(k1 )) > k1
1
since tk1 < tk2 implies itk1 + `(tk ) − `(o(k2 )) ≤ k2 by the definition of tk2 . Thus
1
iz + `(z) − `(o(k)) = iz + `(z) − `(o(k1 )) + `(o(k1 )) − `(o(k)) > k − 1. Note that in
general, itkj + `(tk ) − `(o(k1 )) > k1 so that itkj + `(tk ) − `(o(k)) ≥ k for all j ≥ 1.
j
j
So if necessary, continue in like manner until the desired z is found at which point
the result follows.
(c) This result follows immediately from the choice of z.
(d) Note that k > 1. If i0k1 < i0k + m, then o0 (z) = k − 1 and the case is vacuous.
So assume that i0k + m ≤ i0k1 . If either i0k + m = itk1 − `0(k2 ) or i0k2 < i0k + m, then
o0 (z) = o0 (tk1 ) and i0k1 ≤ i0r for o0 (z) < r < k. Otherwise, i0k + m ≤ min i0k1 , i0k2 .
If either i0k + m = itk2 − `0(k3 ) or i0k3 < i0k + m, then o0 (z) = o0 (tk2 ) and i0k2 ≤ i0r
for o0 (z) < r ≤ k2 . But i0k1 ≤ i0r for k2 = o0 (tk1 ) < r < k, so the desired inequality
n
o
holds for o0 (z) < r < k. Continue in this manner until i0k + m ≤ min i0k1 , . . . , i0kj
for some j, at which point the conclusion follows.
(e) If k = 1, o0 (z) = 0 so that i01 + m = iz and iz + `(z) ≥ 1 . Now i01 + m > it1 by
assumption, hence z < t1 . But t1 is the smallest integer r such that ir + `(r) > 1 ;
therefore iz + `(z) = 1 , by (b). Let k > 1 and suppose that z ≥ tk . Then iz ≤ itk ,
in which case o0 (z) = o0 (tk ) . But by (c), i0k + m = iz − `0(o0 (z)) ≤ itk − `0(o0 (tk )) and
by (d), i0k + m ≤ i0r for o0 (tk ) < r < k, contradicting the hypothesis. Hence z < tk .
But tk is the smallest possible integer such that itk + `(tk ) − `(o(k)) > k , so (e.1)
follows from by (b). Results (e.2) and (e.3) are obvious.
Proof of Theorem 1:
A DIAGONAL ON THE ASSOCIAHEDRA
23
Let n ≥ 0. We show that if u ⊗ v is a component of d⊗2 ∆(Tn+2 ) or ∆d(Tn+2 ),
there is a corresponding component that cancels it, in which case (d⊗2 ∆−∆d)(Tn+2 )
= 0. Let a0 ⊗ a be a component of ∆Tn+2 . We consider various cases.
Case I. Consider a component dr(i,`) ⊗1 of d⊗1, where i+` ≤ `0r , i ≥ 0, 1 ≤ ` < `0r
and 1 ≤ r ≤ q + 1. Reduce b0 = dr(i,`) (a0 ) to an expression in second fundamental
form, i.e., if 1 ≤ r ≤ q, successively apply relation (1) to dj+1 dj for j = q, ..., r + 1;
r
apply (2) to replace dr(i,`) dr(i0 ,`0 ) by dr+1
(i0r ,`0r −`) d(i0r +i,`) ; then successively apply (3) to
r r
dj+1 dj for either j = r − 1, ..., β if i + ` < `0r or for j = q, ..., β if r = q + 1, where
β is the greatest integer such that 1 ≤ β ≤ r and i + ` ≥ i0β−1 . Then for i + ` < `0r ,
1 ≤ r ≤ q + 1 we have
r+2
b0 = dq+1
(i0 ,`0 ) · · · d(i0
q
q
0
r+1 ,`r+1 )
r
dr+1
(i0 ,`0 −`) d(i0r−1 −`,`0r−1 )
r
r
β
β−1
· · · dβ+1
(i0β −`,`0β ) d(i0r +i,`) d(i0β−1 ,`0β−1 )
· · · d1(i01 ,`01 ) (Tn+2 ),
and for i + ` = `0r , 1 ≤ r ≤ q + 1 we have
r+2
b0 = dq+1
(i0 ,`0 ) · · · d(i0
q
q
0
r+1 ,`r+1 )
r−1
r
dr+1
(i0 ,`0 −`) d(i0r +i,`) d(i0
r
r
0
r−1 ,`r−1 )
· · · d1(i01 ,`01 ) (Tn+2 ).
Case Ia. Let i + ` < `0r and i0r + i + ` > i0β−1 . As in the case of a0 above,
the inequalities for lower indices in an expression of b0 as a face of Tn+2 in first
fundamental form are strict, but whose number now is increased by one. Obviously
we will have that
β = i0r + i + `0(β−1) + ` = ik
for certain 1 ≤ k ≤ p. Apply the k th left-transfer to obtain
a = τ`k (Tn+2 ) = dk̃(ı̃,`)
˜ (b),
where k̃ = α − 1, ı̃ = ik − iα − `(α−1) − `(k) , `˜ = `k and α is the smallest integer
k ≤ α ≤ p + 1 with iα + `(α) − `(k) ≥ ik . Then b0 ⊗ b is a component of ∆Tn+2 as
well.
Case Ib. Let i + ` < `0r and i0r + i + ` = i0β−1 . Apply the (β − 1)th left-transfer to
obtain
0
b0 = τ`β−1 (Tn+2 ) = dβ−1
˜ (c ),
(ı̃,`)
where ı̃ = ` and `˜ = `0 β−1 . Then c0 ⊗ a is also a component of ∆Tn+2 .
Case Ic. Let i + ` = `0r ; there are two subcases:
Subcase i. If 1 ≤ r ≤ q and i0r + i ≤ min(itr − `0(o0 (tr )) , i0j ), o0 (tr ) < j < r, apply
the (r + 1)th left-transfer to obtain
0
b0 = τ`r+1 (Tn+2 ) = dr̃(ı̃,`)
˜ (c ),
where r̃ = α, ı̃ = i0r − i0α , `˜ = `0r − ` and α is the smallest integer r ≤ α ≤ p + 1
with i0α ≤ i0r . Although certain i0 ’s are increased by i, while `0r is reduced by i to
obtain `, it is straightforward to check that c0 ⊗ a is also a component of ∆Tn+2 .
Subcase ii. Suppose that 1 ≤ r ≤ q +1 and i0r +i > min(itr −`0(o0 (tr )) , i0j ), o0 (tr ) <
j < r. In view of Lemma 2 (with k = r and m = i) we have
r = iz + `(z) − `(o(r)) and i0r + i = iz − `0(o0 (z)) ,
from which we also establish the equality
`(z) − `(o(r)) = `0(r−1) + ` − `0(o0 (z)) .
24
SAMSON SANEBLIDZE AND RONALD UMBLE1
The hypotheses of Lemma 1 are satisfied by setting k1 = z, p1 = o(r) and k2 =
r, p2 = o0 (z). Hence, b0 ⊗ a is a component of ∆d(ı̃,`)
˜ (Tn+2 ) with ı̃ = iz and
˜
` = `(z) − `(o(r)) .
Case II. Consider a component 1 ⊗ dr(i,`) of 1 ⊗ d, where i + ` ≤ `r , i ≥ 0,
1 ≤ ` < `r , and 1 ≤ r ≤ p + 1. Reduce b = dr(i,`) (a) to the first fundamental form,
i.e., if 1 ≤ r ≤ p, successively apply relation (1) to dj+1 dj for j = p, ..., r + 1, and
r
apply (2) to replace dr(i,`) dr(ir ,`r ) by dr+1
(ir ,`r −`) d(ir +i,`) . Then if i > 0, successively
apply (3) to dj+1 dj , for j = r − 1, ..., β; or if r = p + 1, successively apply (3)
to dj+1 dj for j = p, ..., β, where β is the greatest integer with 1 ≤ β ≤ r and
ir + i + `r−1 + ... + `β ≤ iβ−1 . Then for β = r, we have ir + i ≤ ir−1 ; and for i > 0,
1 ≤ r ≤ p + 1,
r+2
r+1
r
b = dp+1
(ip ,`p ) · · · d(ir+1 ,`r+1 ) d(ir ,`r −`) d(ir−1 ,`r−1 )
β
β−1
1
· · · dβ+1
(iβ ,`β ) d( ˜ı,`) d(iβ−1 ,`β−1 ) · · · d(i1 ,`1 ) (Tn+2 ),
where ı̃ = ir + i + `(r−1) − `(β−1) , and
r+1
r−1
r
1
b = dp(ip−1 ,`p−1 ) · · · dr+2
(ir+1 ,`r+1 ) d(ir ,`r −`) d(ir ,`) d(i r−1 ,`r−1 ) · · · d(i1 ,`1 ) (Tn+2 ),
for i = 0, 1 ≤ r ≤ p + 1.
Case IIa. Let i > 0 and ir +i+`(r−1) −`(β−1) < iβ−1 . Once again, the inequalities
for the lower indices in the expression of b in first fundamental are strict inequalities
as they were for a but whose number now is increased by one. Obviously we will
have that
iβ = ir + i + `(r−1) − `(β−1) = k
for certain 1 ≤ k ≤ q. Apply the k th left-transfer to obtain
0
a0 = τ`k (Tn+2 ) = dk̃(ı̃,`)
˜ (b ),
where k̃ = α − 1, ı̃ = i0k − i0α , `˜ = `0k and α is the smallest integer k ≤ α ≤ q + 1
with i0α ≤ i0k . Then b0 ⊗ b is a component of ∆Tn+2 as well.
Case IIb. Let i > 0 and ir + i + `(r−1) − `(β−1) = iβ−1 . Apply the (β − 1)th
left-transfer to obtain
b = τ`β−1 (Tn+2 ) = dβ−1
˜ (c),
(ı̃,`)
where ı̃ = 0 and `˜ = `β−1 . Then a0 ⊗ c is also a component of ∆Tn+2 .
Case IIc. Let i = 0; there are two subcases:
Subcase i. If 1 ≤ r ≤ p and no integer 1 ≤ k ≤ q exists with r = tk , then
k = ir + ` + `(r−1) − `(o(k)) and i0k = ir − `0(o0 (r)) .
Apply the (r + 1)th left-transfer to obtain
b = τ`r+1 (Tn+2 ) = dr̃(ı̃,`)
˜ (c),
where r̃ = α, ı̃ = ir − iα + `(α−1) − `(r) , `˜ = `r − ` and α is the smallest integer
r < α ≤ p + 1 such that iα + `(α) − `(r) ≥ ir ; namely, α = r + 1 or α = to0 (r) .
Now the required inequality for the i0k ’s could conceivably be violated for k > o0 (r),
but this is not so since it is easy to see that: (a) if r is not realized as tk for some
k > o0 (r), then each z with α < z < r is so; and (b) if r = tk for some k > o0 (r),
while k = ir + ` + `(r−1) − `(o(k)) , then α (and not r) serves as tk for indices of face
A DIAGONAL ON THE ASSOCIAHEDRA
25
0
operators in expressions of a and c; moreover, for
n either α = r +o1 or α = to (r) (in
0
0
0
0
0
0
ij , iα − `(o0 (α)) so that a ⊗ c is
which case ik ≤ io0 (r) ), one has ik ≤ 0 min
o (α)<j<k
also a component of ∆Tn+2 .
Subcase ii. If 1 ≤ r ≤ p + 1 and r = tk for some 1 ≤ k ≤ q, then
k = ir + ` + `(r−1) − `(o(k)) and i0k = ir − `0(o0 (r)) ,
from which we establish the equality
` + `(r−1) − `(o(k)) = `0(k) − `0(o(r)) .
The hypotheses of Lemma 1 are satisfied by putting k1 = r = tk , p1 = o(k) and
k2 = k, p2 = o0 (r) so that a0 ⊗ c is a component of ∆d(ı̃,`)
˜ (Tn+2 ) with ı̃ = ir and
`˜ = ` + `(r−1) − `(o(k)) .
Case III. Let c0 ⊗ c be a component of ∆d(i,`) (Tn+2 ). Reduce c and c0 to the first
and second fundamental forms, respectively. According to Lemma 1, we have either
i = ir1 or i = i0r2 + `0r2 + · · · + `k2 = i0r2 +1 + `0r2 +1 for certain integers r1 , r2 , k2 , i.e.,
ir1 +1 = ir1 or i0r2 +1 +`0r2 +1 = i0r2 . Note that the shuffles under consideration prevent
both cases from occurring simultaneously, and we obtain the situation dual to either
Subcase ii of Case IIc, or to Subcase ii of Case Ic. Thus we obtain components c0 ⊗a
1
2
(a) = c or dr(i,`)
(a0 ) = c0 , respectively.
or a0 ⊗ c of ∆Tn+2 with dr(i,`)
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A. Razmadze Mathematical Institute, Georgian Academy of Sciences, M. Aleksidze
st., 1, 380093 Tbilisi, Georgia
E-mail address: [email protected] <mailto:[email protected]>
Department of Mathematics, Millersville University of Pennsylvania, Millersville,
PA. 17551
E-mail address: [email protected] <mailto:[email protected]>