GERSTENHABER STRUCTURE AND DELIGNE'S
CONJECTURE FOR LODAY ALGEBRAS
DONALD YAU
Abstract. A method for establishing a Gerstenhaber algebra structure
on the cohomology of Loday-type algebras is presented. This method
is then applied to dendriform dialgebras and three types of trialgebras
introduced by Loday and Ronco. Along the way, our results are com-
bined with a result of McClure-Smith to prove an analogue of Deligne's
conjecture for Loday algebras.
1. Introduction and statements of the main results
The Hochschild cohomology HH*(A, A) of an associative algebra A with
coefficients in itself has a rich structure. Indeed, the classical results of G*
*er-
stenhaber [4] shows that HH*(A, A) has a graded Lie bracket and a graded
commutative cup product of which the Lie bracket is a graded derivation.
This structure, which is a graded version of a Poisson algebra, is now called
a Gerstenhaber algebra, or a G-algebra for short. Several other types of al-
gebras, including coalgebras and graded associative algebras, have also been
shown to admit a G-algebra structure in cohomology. On the other hand,
commutative algebra (Harrison) cohomology and Lie algebra (Chevalley-
Eilenberg) cohomology are graded Lie algebras but not G-algebras [7]. In
general, whenever a new kind of algebra arises, it is an interesting and impor-
tant problem to determine if it admits a G-algebra structure in cohomology.
Meanwhile, Loday's program [9] of studying periodicity phenomenon in
algebraic K-theory has generated a number of new algebras. In that pro-
gram, a Lie algebra is replaced by a Leibniz algebra, which has a bracket
that satisfies a version of the Jacobi identity. If the bracket happens to be
anti-symmetric, then it is a Lie algebra. The role of an associative algebra is
played by a dialgebra, which has two associative operations satisfying three
more associative-style axioms. A dialgebra gives rise to a Leibniz algebra
the same way an associative algebra gives rise to a Lie algebra. With this
analogy in mind, it is reasonable to expect that dialgebra cohomology also
admits a G-algebra structure. The work of Majumdar and Mukherjee [11]
shows that this is indeed the case.
In view of the result of [11], it is natural to ask the question:
____________
Key words and phrases. Loday algebras, operads, G-algebras, brace algebras, *
*homotopy
G-algebras, Deligns's conjecture.
1
2 DONALD YAU
Do other Loday algebras admit a G-algebra structure in co-
homology?
The main purpose of this note is to extract the method used in [11], putting it
in context in such a way that it allows easy applications to other cases. This
gives a positive answer (more or less) to the question above. This method
will then be illustrated with several examples of Loday algebras. Moreover,
we combine our results with those of McClure-Smith [14] to establish a
positive answer to a variation of Deligne's conjecture for Loday algebras.
We also hope that the method and examples here will make it easier to
construct G-algebra structures on the cohomology of other Loday algebras
and related algebras that may come up in the future.
A more detailed descriptions of our results follow.
1.1. Gerstenhaber structure. The method used in [11] can be summa-
rized as follows. The main objectives are (i) to establish a non- operad
structure [12, 13] on the cochain modules and (ii) to show the existence of
a multiplication on this operad. The operad structure arises naturally from
a certain collection of functions, satisfying four conditions. These functions
are defined on certain sets that parametrize the cochain modules. The re-
sults of Gerstenhaber and Voronov [6] about operads, brace algebras, and
(homotopy) G-algebras are then used to obtain the desired G-algebra struc-
ture on cohomology.
For many of the Loday algebras (e.g. (dendriform) dialgebras, associa-
tive/dendriform/cubical trialgebras [10]), there is a sequence of non-empty
sets U = {Un :n 1} such that the cochain modules of an algebra A of that
type are given by
C(n) = Cn(A, A) = Hom K (K[Un] A n , A) (1.1.1)
for n 1, where K is the ground field. For example, in the case of dialge-
bras, Un = Yn is the set of binary trees with n + 1 leaves. For dendriform
dialgebras, Un = Cn = {1, . .,.n} is the finite set with n elements.
Definition 1.2. Given a sequence of non-empty sets U = {Un :n 1},
define a pre-operadic system on U to be a collection of functions
R = {R0(k; n1, . .,.nk), Ri(k; n1, . .,.nk): k, n1, . .,.nk 1, 1 i k},
where
R0(k; n1, . .,.nk): Un1+...+nk! Uk
and
Ri(k; n1, . .,.nk): Un1+...+nk! Uni.
These functions are required to satisfy the following conditions: Let
m1, . .,.mN be positive integers, where N = n1 + . .+.nk. Write Ni =
G-STRUCTURE AND DELIGNE'S CONJECTURE FOR LODAY ALGEBRAS 3
n1+. .+.ni(N0 0), Mi= m1+. .+.mi(M0 0), and Ti= MNi-MNi-1.
Then the functions are required to satisfy:
(1) Identity: R0(k; 1, . .,.1) (k occurrences of 1's) is the identity func-
tion on Uk for each k 1.
(2) Idempotency:
R0(k; n1, . .,.nk)R0(N; m1, . .,.mN ) = R0(k; T1, . .,.Tk).
(3) Commutativity: For each i 2 {1, . .,.k},
Ri(k; n1, . .,.nk)R0(N; m1, . .,.mN ) =
R0(ni; mNi-1+1, . .,.mNi)Ri(k; T1, . .,.Tk).
(4) Closure: For each i 2 {1, . .,.k} and each j 2 {1, . .,.ni},
RNi-1+j(N; m1, . .,.mN ) = Rj(ni; mNi-1+1, . .,.mNi)Ri(k; T1, . .,.Tk).
Now let A be a type of Loday algebras (e.g. (dendriform) dialgebras,
associative/dendriform/cubical trialgebras) so that its cochain modules are
given by (1.1.1)for some sequence of non-empty sets U. Suppose that R is
a pre-operadic system on U. (It will be shown below that such an R does
exist for Loday algebras.) Then for k, n1, . .,.nk 1, define maps
fl :C(k) C(n1) . . .C(nk) ! C(N)
by:
fl(f; g1 . .g.k)(r; x1, . .,.xN )
= f(R0(r); g1(R1(r); x1, . .,.xN1) . . .
(1.2.1)
gi(Ri(r); xNi-1+1, . .,.xNi) . . .
gk(Rk(r); xNk-1+1, . .,.xN )).
Here N and the Niare as before, xi2 A, r 2 UN , and Ri= Ri(k; n1, . .,.nk)
for 0 i k. Denote by IdA 2 C(1) = Hom K (K[U1] A, A) the canonical
1-cochain given by
IdA(r; x) = x
for r 2 U1 and x 2 A.
Theorem 1.3. With the maps fl and the 1-cochain IdA 2 C(1), the collection
of vector spaces C = {C(n): n 1} becomes a non- operad.
In this case, we say that this operad is generated by the pre-operadic
system R.
A 2-cochain ss 2 C(2) is called a multiplication on C if it satisfies
fl(ss; ss, IdA) = fl(ss; IdA, ss). (1.3.1)
4 DONALD YAU
Using the results and arguments of Gerstenhaber and Voronov [6], such
a multiplication generates a homotopy G-algebra structure on (the brace
algebra generated by) C. Passing to cohomology, we obtain the following
result:
Corollary 1.4. If ss 2 C(2) is a multiplication on the operad C, then the
corresponding cohomology H*(C, d) has the structure of a G-algebra, where
d is the differential generated by ss.
Both Theorem 1.3 and Corollary 1.4 apply easily to the Loday algebras
mentioned above. We record it as follows.
Theorem 1.5. Let A be one of the following types of Loday algebras: dialge-
bras, dendriform dialgebras, associative trialgebras, dendriform trialgebras,
or cubical trialgebras (so that the cochains of A has the form (1.1.1)). Then
there exists a pre-operadic system R on U. Moreover, the resulting operad
structure on C = {Cn(A, A)} admits a multiplication.
Combining Theorem 1.3, Corollary 1.4, and Theorem 1.5, we obtain the
following result in cohomology.
Corollary 1.6. Let A be a Loday algebra as in Theorem 1.5. Then the
cohomology H*(A, A) of A has the structure of a G-algebra.
For dialgebras, this Corollary simply recovers the results of [11]. The
other examples will be proved below in Section 4.
We now discuss a variation of Deligne's conjecture for Loday algebras.
1.7. Deligne's conjecture for Loday algebras. Deligne's conjecture [2]
states that:
Deligne's Conjecture. The Hochschild cochain complex C*(A, A) of an
associative algebra A is an algebra over the singular chain operad Ssingof
the little squares operad C2.
In topology, the operad C2 is used to recognize double loop spaces and is
closely related to the geometry of configuration spaces. Deligne's conjecture,
therefore, expresses a deep connection between algebra and topology.
An affirmative solution to Deligne's conjecture was given by McClure-
Smith [14], which can be summarized as follows. There is an operad H whose
algebras are brace algebras with multiplication, which includes C*(A, A)
when A is an associative algebra. McClure and Smith showed that H is
quasi-isomorphic as a chain operad to Ssing. This gives a positive answer to
Deligne's conjecture. There are also other solutions to Deligne's conjecture
(see the references in [14]).
G-STRUCTURE AND DELIGNE'S CONJECTURE FOR LODAY ALGEBRAS 5
Now let A be one of the types of Loday algebras in Theorem 1.5, so that
C = {Cn(A, A)} is an operad with multiplication. This induces the structure
of a brace algebra with multiplication on C (Corollary 2.4 and Theorem 1.5).
In particular, C is an algebra over the operad H. Combined with the result
of McClure-Smith [14] mentioned above, this gives the following variation
of Deligne's conjecture.
Corollary 1.8 (Deligne's conjecture for Loday algebras). Let A be a Lo-
day algebra as in Theorem 1.5. Then the cochains C = C*(A, A) verify
Deligne's conjecture: Namely, C is an algebra over an operad H that is
quasi-isomorphic to the singular chain operad Ssingof the little squares op-
erad.
It should be noted that for this Corollary to hold, we do not exactly need
a Loday algebra. More precisely, it suffices to assume that:
(1) A is a type of algebras whose cochain modules are in the form (1.1.1)
for some non-empty sets Un.
(2) There exists a pre-operadic system on U = {Un}.
(3) There exists a multiplication on the resulting operad structure on
C = {Cn(A, A)}.
This might come in handy for other algebras that may come up in the future.
1.9. Organization. The rest of this paper is organized as follows. The
following section begins with the definition of an operad, followed by the
proof of Theorem 1.3. It also discusses brace algebras and multiplications.
Section 3 discusses (homotopy) G-algebras, leading to Corollary 1.4. Our
discussion on brace algebras, (homotopy) G-algebras, and multiplication
follow Gerstenhaber and Voronov [6]. Section 4 contains a proof of Theorem
1.5, and hence Corollary 1.6, for the new examples, i.e. dendriform dialgebras
and the three types of trialgebras. In the final section, we show that the
differential d induced by the multiplication ss agrees with the differential ff*
*i,
up to a sign, for that particular type of algebras (Theorem 5.4). This ensures
that the cohomology modules in Corollary 1.6 are the intended ones.
1.10. Acknowledgment. The author would like to thank Mark Behrens
for a discussion about this project and Jim McClure for reading an earlier
version of this paper. The author also thanks the referee for his/her helpful
suggestions.
2.Operads, brace algebras and multiplications
We work over a fixed field K. In this section, suppose that A is a type
of Loday algebras whose cochain modules are given by (1.1.1)for some
6 DONALD YAU
sequence of non-empty sets U = {Un :n 1}. Also, suppose that R is a
pre-operadic system on U (see Definition 1.2).
2.1. Algebraic operads. A non- operad [12, 13] is a collection O =
{O(n), n 1} of vector spaces together with structure maps
fl :O(k) O(n1) . . .O(nk) ! O(n1 + . .+.nk),
for k, n1, . .,.nk 1. The structure maps are required to satisfy the asso-
ciativity condition:
fl(fl(f; g1, . .,.gk); h1, . .,.hN )
= fl(f; fl(g1; h1, . .,.hN1), . .,. (2.1.1)
fl(gi; hNi-1+1, . .,.hNi), . .,.fl(gk; hNk-1+1, . .,.hNk )).
Here f 2 O(k), gi 2 O(ni), hj 2 O(mj), N = n1 + . .+.nk, and Ni =
n1 + . .+.ni. It is also required that there be an identity element Id 2 O(1)
such that
fl(-; Id, . .,.Id): O(k) ! O(k) (2.1.2)
is the identity map.
From now on, whenever we write operad, we mean a non- operad.
2.2. Proof of Theorem 1.3. Using property (1) in Definition (1.2), it is
clear that the 1-cochain IdA satisfies (2.1.2).
For associativity (2.1.1), we use the notations above. Suppose that hj 2
C(mj) for 1 j N. Set N0 = M0 = 0, Ni = n1 + . .+.ni, Mi =
m1 + . .+.mi, M = m1 + . .+.mN , and Ti= MNi - MNi-1. Let x1, . .,.xM
be elements of A, and let r 2 UM . We will write ys,tfor the sequence
ys, . .,.yt when s t, where y can be g, h, n, m, T , or x. Then, on the one
hand, we have
fl(fl(f; g1,k); h1,N)(r; x1,M)
= f(R0(k; n1,k)R0(N; m1,N)(r); . . .
gi(Ri(k; n1,k)R0(N; m1,N)(r); . . .
hNi-1+j(RNi-1+j(N; m1,N)(r); xMNi-1+j-1+1, MNi-1+j)
. .).
. .)..
Here 1 i k and, for each i, 1 j ni. In particular, not counting the
U components, the expression above displays the ith typical input gi(. .).in
G-STRUCTURE AND DELIGNE'S CONJECTURE FOR LODAY ALGEBRAS 7
f and the jth typical input hNi-1+j(. .).in gi. On the other hand, we have
fl(f; . .,.fl(gi; hNi-1+1, Ni), . .).(r; x1,M)
= f(R0(k; T1, k)(r); . . .
gi(R0(ni; mNi-1+1, Ni)Ri(k; T1, k)(r); . . .
hNi-1+j(Rj(ni; mNi-1+1, Ni)Ri(k; T1, k)(r); xMNi-1+j-1+1, MNi-1+j)
. .).
. .)..
Comparing the first inputs (the U components) in f, gi, and hNi-1+j, the
associativity of fl now follows from properties (2), (3), and (4) of Definition
1.2.
This proves that C = {C(n) = Hom K(K[Un] A n , A): n 1} is an
operad.
2.3. Brace algebras. For a graded vector space O = O(n) and an element
x 2 O(n), set deg x = n and |x| = n - 1.
Recall from [6, Definition 1] that a brace algebra is a graded vector space
O = O(n) together with a collection of braces x{x1, . .,.xn} of degree -n,
satisfying
x{x1, . .,.xm }{y1, . .,.yn}
X
= (-1)"x{y1, . .y.i1, x1{yi1+1, . .,.yj1}, yj1+1, . .,.yim,
0 i1 ... im n
xm {yim +1, . .,.yjm}, yjm +1, . .,.yn}.
P m P ip
Here " = p=1(|xp| q=1|yq|).
According to [6, Proposition 1], an operad C gives rise to a brace algebra
via the braces:
X
x{x1, . .,.xn} := (-1)"fl(x; Id, . .,.Id, x1, Id, . .,.Id, xn, Id, . .,.I*
*d).
(2.3.1)
Here the sum runs over all possible substitutions of x1, . .,.xn into fl(x; . .*
*).
P n
in the given order and " = p=1|xp|ip, where ip is the total number of inputs
P n
in front of xp. The degree of x{x1, . .,.xn} is ( p=1 degxp) + degx - n, so
this operation is of degree -n. This leads to the following consequence of
Theorem 1.3.
Corollary 2.4. With the braces (2.3.1), the graded vector space C(n) =
Cn(A, A) admits the structure of a brace algebra.
In such a brace algebra, define a "comp" operation and a bracket:
x O y:= x{y},
(2.4.1)
[x, y]:= x O y - (-1)|x||y|y O x.
8 DONALD YAU
By convention, {} is the identity operation, i.e., x{} = x.
2.5. Multiplications. In an operad or a brace algebra O, a multiplication
is an element m 2 O(2) such that
m O m = 0. (2.5.1)
Since deg(m) = 2, this is equivalent to
fl(m; m, Id) = fl(m; Id, m).
Given such a multiplication m, one defines a dot product,
x . y := (-1)degxm{x, y}, (2.5.2)
of degree 0 and a degree 1 map d,
dx := [m, x]
(2.5.3)
= m O x - (-1)|x|x O m.
According to [6, Proposition 2], the map d is a differential (i.e., d2 = 0). We
say that d is generated by m. Moreover, the dot product is associative for
which d is a derivation. In particular, the dot product induces an operation
on the cohomology modules defined by the differential d.
If ss 2 C(2) = Hom K (K[U2] A 2, A) is a multiplication, then we denote
the corresponding cohomology modules by
Hn (A, A) := Hn (C, d).
We will decorate this notation with a subscript for a given type of algebras.
For a given type of Loday algebras, there is usually a canonical choice of a
multiplication. The condition (2.5.1)often amounts to either the defining
axioms of that type of Loday algebras or the associativity of ss, as we will
discuss in the examples in Section 4.
The relationships between the dot product, the bracket, and the differen-
tial are discussed next.
3. Homotopy G-algebras
We keep the same assumptions as in the previous section. Also, suppose
that ss 2 C(2) = Hom K (K[U2] A 2, A) is a multiplication.
3.1. Homotopy G-algebras. Recall from [6, Definition 2] that a homotopy
G-algebra is a brace algebra V = V n with a degree 1 differential d and
a degree 0 dot product . that make V into a differential graded associative
algebra. The dot product is required to satisfy the identity:
Xn
(x1 . x2){y1, . .,.yn} = (-1)"x1{y1, . .,.yk} . x2{yk+1, . .,.yn},
k=0
G-STRUCTURE AND DELIGNE'S CONJECTURE FOR LODAY ALGEBRAS 9
P k
where " = |x2| p=1|yp|. The differential is required to satisfy the identity:
d(x{x1, . .,.xn+1}) - (dx){x1, . .,.xn+1}
n+1X
- (-1)|x| (-1)|x1|+...+|xi-1|x{x1, . .,.dxi, . .,.xn+1}
i=1
= (-1)|x||x1|+1x1 . x{x2, . .,.xn+1}
Xn
+ (-1)|x| (-1)|x1|+...+|xi-1|x{x1, . .,.xi. xi+1, . .,.xn+1}
i=1
-x{x1, . .,.xn} . xn+1.
A multiplication m on an operad O = {O(n)} gives rise to a homotopy
G-algebra structure on the brace algebra O(n), where the dot product and
the differential are defined as in, respectively, (2.5.2)and (2.5.3). This is
Theorem 3 in [6]. In particular, this applies to the operad C and multipli-
cation ss.
Corollary 3.2. With the multiplication ss, the brace algebra Cn(A, A) in
Corollary 2.4 admits the structure of a homotopy G-algebra.
3.3. G-algebras. A G-algebra [6, 2.2] is a graded vector space H = Hn
with a degree 0 dot product
- . -: Hm Hn ! Hm+n
and a degree -1 graded Lie bracket
[-, -]: Hm Hn ! Hm+n-1 ,
satisfying the following conditions:
(1) The dot product is graded commutative,
x . y = (-1)degx degyy . x.
(2) The Lie bracket is a graded derivation for the dot product, in the
sense that
[x, y . z] = [x, y] . z + (-1)|x| degyy . [x, z].
Corollary 1.4 in the Introduction now follows from [6, Corollary 5] and its
argument. The dot product and the Lie bracket are induced by the ones de-
fined in, respectively, (2.4.1)and (2.5.2). In particular, that the dot product
is graded commutative and that the bracket is a graded derivation for the
dot product are both consequences of the homotopy G-algebra structure in
Corollary 3.2 [6, (8) and (9)].
10 DONALD YAU
4. Proof of Theorem 1.5
In this section, we give a proof of Theorem 1.5, and hence Corollary 1.6.
We will not discuss dialgebras, since this was done in [11]. In each case,
we first recall some relevant definitions and the constructions of the cochain
modules.
Fix a ground field K.
4.1. Dendriform dialgebras. A dendriform dialgebra E over K [9, Sec-
tion 5] is a K-vector space equipped with two binary operations,
: E E ! E,
: E E ! E,
such that
(x y) z = x (y z + y z), (4.1.1a)
(x y) z = x (y z), (4.1.1b)
(x y + x y) z = x (y z) (4.1.1c)
for all x, y, z 2 E. Dendriform dialgebras are the operadic duals [8] of
dialgebras.
Given a dendriform dialgebra, define a single binary operation * by adding
the two given operations:
x * y := x y + x y.
The sum of the three axioms, (4.1.1a), (4.1.1b), and (4.1.1c), states that *
is associative. In particular, a dendriform dialgebra can be thought of as an
associative algebra whose binary operation splits into two operations and
whose associative condition splits into three conditions.
Fix a dendriform dialgebra E. Denote by Cn the n-element set {1, . .,.n}.
Define the module of n-cochains as:
Cndidend(E, E) := Hom K (K[Cn] E n , E).
Notice that an n-cochain can be interpreted as an n-tuple of n-ary operations
on E.
Using the notations in Theorem 1.3, define
R0(k; n1,k): CN ! Ck
by
R0(k; n1,k)(r) = i if Ni-1+ 1 r Ni
for r 2 CN . For each j 2 {1, . .,.k}, define
Rj(k; n1,k): CN ! Cnj
G-STRUCTURE AND DELIGNE'S CONJECTURE FOR LODAY ALGEBRAS 11
by 8
>><1 if 1 r Nj-1,
Rj(k; n1,k)(r) = r - Nj-1 if 1 + Nj-1 r Nj,
>>:
nj if 1 + Nj r N.
We claim that
R = {R0(k; n1,k), Rj(k; n1,k): k, ni 1, 1 j k}
is a pre-operadic system on {Cn :n 1}. To see this, first note that
R0(k; 1, . .,.1) (k occurrences of 1's) is the identity function. To prove idem-
potency, note that the left-hand side of condition (2) in Definition 1.2 gives
R0(k; n1,k)R0(N; m1,N)(r) = l 2 {1, . .,.k}
if and only if
Nl-1+ 1 R0(N; m1,N)(r) Nl,
which is equivalent to
T1 + . .+.Tl-1+ 1 r T1 + . .+.Tl.
This is exactly when R0(k; T1,k)(r) is equal to l, thereby proving idempo-
tency.
With a similar reasoning, one observes that both Ri(k; n1,k)R0(N; m1,N)(r)
and R0(ni; mNi-1+1, Ni)Ri(k; T1,k)(r) in Cni are equal to:
8
>><1 if 1 r MNi-1,
>>:l 2 {1, . .,.ni}ifMNi-1+l-1+ 1 r MNi-1+l,
ni if 1 + MNi r M.
This proves commutativity (condition (3) in Definition 1.2).
For closure (condition (4) in Definition 1.2), one observes just as above
that both Rj(ni; mNi-1+1,Ni)Ri(k; T1,k)(r) and RNi-1+j(r) in CmNi-1+j are
equal to:
8
>><1 if 1 r MNi-1+j-1,
>>:l 2 {1, . .,.mNi-1+j}ifr = l + MNi-1+j-1,
mNi-1+j if 1 + MNi-1+j r M.
This proves closure.
Therefore, R is a pre-operadic system on {Cn :n 1}, as claimed. It
follows from Theorem 1.3 that R generates an operad structure on
Cdidend(E) = {Cndidend(E, E): n 1}.
Moreover, the 2-cochain ss 2 C2didend(E, E) given by
ss(r; x y) = x * y = x y + x y, (4.1.2)
12 DONALD YAU
for r 2 C2 and x, y 2 E, is a multiplication on Cdidend(E). In fact, the
condition ss O ss = 0 is equivalent to the associativity of *. Corollary 1.4 now
implies that the corresponding cohomology,
H*didend(E, E) = H*(Cdidend(E), d),
has a G-algebra structure.
4.2. Associative trialgebras. An associative trialgebra [10] is a vector
space A that comes equipped with three binary operations, a (left), ` (right),
and ? (middle), satisfying the following 11 relations for all x, y, z 2 A:
(x a y) a z= x a (y a z), (4.2.1a)
(x a y) a z= x a (y ` z), (4.2.1b)
(x ` y) a z= x ` (y a z), (4.2.1c)
(x a y) ` z= x ` (y ` z), (4.2.1d)
(x ` y) ` z= x ` (y ` z), (4.2.1e)
(x a y) a z= x a (y ? z), (4.2.1f)
(x ? y) a z= x ? (y a z), (4.2.1g)
(x a y) ? z= x ? (y ` z), (4.2.1h)
(x ` y) ? z= x ` (y ? z), (4.2.1i)
(x ? y) ` z= x ` (y ` z), (4.2.1j)
(x ? y) ? z= x ? (y ? z). (4.2.1k)
To define the cochain modules, consider the set Tn of planar trees with
n + 1 leaves and one root in which each internal vertex has valence at least
2. We will call them trees from now on. The leaves of a tree _ 2 Tn are
labelled 0, 1, . .,.n, from left to right. Here are the first three sets Tn:
T1 = { },
T2 = { , , },
T3 = { , , , , , , , , , , }.
Then the cochain modules of an associative trialgebra A are defined as
Cntrias(A, A) := Hom K (K[Tn] A n , A).
To define the functions R, first define the maps
di:Tn ! Tn-1 (0 i n), (4.2.2)
where di_ is the tree obtained from _ by deleting the ith leaf. These maps
satisfy the simplicial relations
didj = dj-1di (4.2.3)
G-STRUCTURE AND DELIGNE'S CONJECTURE FOR LODAY ALGEBRAS 13
for i < j. Using the same kind of abbreviations and notations as in the
proof of Theorem 1.3, we define
R0(k; n1,k):= d1, N1-1dN1+1, N2-1. .d.Nk-1+1, Nk-1:TN ! Tk,
Rj(k; n1,k):= d0, Nj-1-1dNj+1, N:TN ! Tnj
for k, n1, . .,.nk 1, 1 j k. In other words, the function R0 leaves the
0th, N1th, . .,.Nkth leaves alone and deletes the other leaves from right to
left. The function Rj leaves the Nj-1th, (Nj-1+1)st, . .,.Njth leaves alone
and deletes the other leaves from right to left. These functions R admit the
same formulas as those in [11, Definition 4.2], where they are denoted by
and are defined on the sets of binary trees.
It is clear that R0(k; 1, . .,.1) is the identity function. Properties (2) -
(4) of Definition 1.2 are proved by the exact same argument used in [11,
Lemma 4.5]. In fact, they all follow from the simplicial relations (4.2.3).
Therefore, R = {R0(k; n1,k), Rj(k; n1,k): k, ni 1, 1 j k} is a pre-
operadic system on T = {Tn :n 1}. It follows from Theorem 1.3 that
Ctrias(A) = {Cntrias(A, A): n 1} is an operad, which is generated by R.
To obtain a G-algebra structure, we need a multiplication. Let ss 2
C2trias(A, A) be the 2-cochain:
8
>>>:
x ` y if_ = .
Then, for _ 2 T3, it is easy to see that the condition
(ss O ss)(_; x, y, z) = 0
is equivalent to the trialgebra axioms (4.2.1). In fact, the 11 possibilities of
_ correspond to the 11 trialgebra axioms. Therefore, ss is a multiplication
on the operad Ctrias(A) = {Cntrias(A, A): n 1}. It follows from Corollary
1.4 that the corresponding cohomology,
H*trias(A, A) = H*(Ctrias(A), d),
admits a G-algebra structure.
4.3. Dendriform trialgebras. Recall from [10] that a dendriform trialge-
bra is a vector space D together with three binary operations, (left),
(right), and . (middle), satisfying the following 7 conditions for x, y, z 2 D:
(x y) z= x (y * z), (4.3.1a)
(x y) z= x (y z), (4.3.1b)
(x * y) z= x (y z), (4.3.1c)
(x y) . z= x (y . z), (4.3.1d)
14 DONALD YAU
(x y) . z= x . (y z), (4.3.1e)
(x . y) z= x . (y z), (4.3.1f)
(x . y) .=zx . (y . z). (4.3.1g)
Here x * y = x y + x . y + x y. The operation * is associative, which one
can see by adding the seven axioms above [10]. Dendriform trialgebras are
the operadic duals of associative trialgebras.
To define the cochain modules, let Pn be the set of non-empty subsets
of [n] := {1, . .,.n}. (Note: The notation in [10] is slightly different from
ours.) Then the cochain modules of D are defined as
Cntridend(A, A) := Hom K (K[Pn] A n , A).
With the notations of the proof of Theorem 1.3, we can define the func-
tions R. Let X be an element of PN . Then
R0(k; n1,k): PN ! Pk
is defined such that i 2 R0(k; n1,k)X if and only if r 2 X for some r such
that Ni-1+ 1 r Ni. For 1 j k, the function
Rj(k; n1,k): PN ! Pnj
is defined by the condition: i 2 Rj(k; n1,k)X if and only if
8
>>>:i + Nj-1 2 X if2 i nj - 1,
r 2 X for some r such thatNj r N ifi = nj.
We claim that R = {R0(k; n1,k), Rj(k; n1,k): k, ni 1, 1 j k} is a
pre-operadic system on P = {Pn :n 1}. To see this, let X be an element of
PM , where M = m1+. .+.mN . Properties (1), (2), and (4) in Definition 1.2
follow easily by direct inspection. For property (3), suppose that 1 i k
and 1 l ni. Then one observes that l 2 Ri(k; n1,k)R0(N; m1,N)X if and
only if
8
>>>:r 2 X for some r with MNi-1+l-1+ 1 r MNi-1+l if2 l ni- 1,
r 2 X for some r with MNi-1 + 1 r M ifl = ni.
Similarly, one observes that the above condition is equivalent to l 2
R0(ni; mNi-1+1, Ni)Ri(k; T1,k)X. This proves property (3). Therefore, R
is a pre-operadic system on P. By Theorem 1.3, R generates an operad
structure on Ctridend(A) = {Cntridend(A, A): n 1}.
G-STRUCTURE AND DELIGNE'S CONJECTURE FOR LODAY ALGEBRAS 15
To obtain the desired G-algebra structure, let ss 2 C2tridend(A, A) be the
2-cochain defined by:
ss(X; a, b)= a * b
= a b + a . b + a b.
The condition ss O ss = 0 is equivalent to the associativity of *. Therefore,
ss is a multiplication on the operad Ctridend(A). Corollary 1.4 now implies
that the corresponding cohomology,
H*tridend(A, A) = H*(Ctridend(A), d),
admits the structure of a G-algebra.
4.4. Cubical trialgebras. Recall from [10] that a cubical trialgebra is a
vector space A together with three binary operations, a (left), ` (right),
and ? (middle), such that
(x O1 y) O2 z = x O1 (y O2 z), (4.4.1)
for O1, O2 2 {a, `, ?}. There are 9 axioms in (4.4.1), the sum of which states
that the operation x * y := x a y + x ? y + x ` y is associative. Cubical
trialgebras are operadically self-dual.
Let Qn be the set {-1, 0, +1}n. The ith component of an element X 2 Qn
is denoted by Xi. The cochain module of A is defined as
Cntricub(A, A) := Hom K (K[Qn] A n , A).
To define the functions R, let X be an element of QN . Then the function
R0(k; n1,k): QN ! Qk
is given by the formula
niY
(R0(k; n1,k)X)i = XNi-1+t.
t=1
for 1 i k. For 1 j k, the function
Rj(k; n1,k): QN ! Qnj
is defined by the formula
(Rj(k; n1,k)X)l= XNj-1+l
for 1 l nj. We claim that R = {R0(k; n1,k), Rj(k; n1,k): k, ni 1, 1
j k} is a pre-operadic system on Q = {Qn :n 1}. Indeed, it is clear that
16 DONALD YAU
R0(k; 1, . .,.1) is the identity function. Properties (2) and (4) in Definition
(1.2)follow by direct inspection. For (3), one observes that
MNi-1+jY
(Ri(k; n1,k)R0(N; m1,N)X)j = Xt
t=MNi-1+j-1
= (R0(ni; mNi-1+1, Ni)Ri(k; T1,k)X)j
for 1 i k and 1 j ni. Therefore, by Theorem 1.3, R generates an
operad structure on Ctricub(A) = {Cntricub(A, A): n 1}.
Let ss 2 C2tricub(A, A) be the 2-cochain given by:
ss(X; a, b)= a * b
= a a b + a ? b + a ` b.
As in the example of dendriform trialgebras, the condition ss Oss = 0 is equiv-
alent to the associativity of *. Therefore, the corresponding cohomology,
H*tricub(A, A) = H*(Ctricub(A), d),
has the structure of a G-algebra.
5. Comparison of cohomology
The purpose of this section is to show that the cohomology modules in
Corollary 1.6, which arise from the differential d induced by the multiplica-
tion ss 2 C(2), are the actual ones for that particular type of algebras. Since
the various cases are rather similar, we will only work out the details in
the case of associative trialgebras, which can be easily adapted to the other
cases.
So let A be an associative trialgebra. The differential ffi in C*trias(A, A) *
*can
be figured out by considering formal deformations of associative trialgebras,
along the lines of Gerstenhaber [5]. Deformations in the more general setting
of algebras over a quadratic operad were worked out by Balavoine [1]. We
make the differential ffi explicit in the associative trialgebra case to compare
it with d. In order to do that, we need to define a few functions on the set
of planar trees.
5.1. Functions on trees. For an element _ 2 Tn, write |_| = n. A leaf
in _ is said to be left oriented (respectively, right oriented) if it is the le*
*ft
most (respectively, right most) leaf of the vertex underneath it. Leaves that
are neither left nor right oriented are called middle leaves. For example, in
the tree , leaves 0 and 2 are left oriented, while leaf 3 is right oriented.
Leaf 1 is a middle leaf.
Given trees _0, . .,._k, their grafting is the tree _0 _ . ._._k obtained by
arranging _0, . .,._k from left to right and joining the k + 1 roots to form a
G-STRUCTURE AND DELIGNE'S CONJECTURE FOR LODAY ALGEBRAS 17
new (lowest) internal vertex, which is connected to a new root. Conversely,
every tree _ can be written uniquely as the grafting of k+1 trees, _0_. ._._k,
where the valence of the lowest internal vertex of _ is k + 1.
For 0 i n + 1, define a function Oi:Tn+1 ! {a, `, ?} according to
the following rules. Let _ be a tree in Tn+1, which is written uniquely as
_ = _0 _ . ._._k as in the previous paragraph. Also, write O_ifor Oi(_).
Then set:
8
>> 0,
>>:
? if|_0| = 0 and k > 1,
8
>>>:
? if theith leaf of_ is a middle leaf,
8
>> 0,
O_n+1= ` ifk = 1 and |_1| = 0,
>>:
? ifk > 1 and |_k| = 0.
5.2. The differential ffi. Now the differential ffi in C*trias(A, A) is given by
n+1X
ffin = (-1)iffini:Cntrias(A, A) ! Cn+1trias(A, A),
i=0
where
8
>>>: i
f(dn+1_; a1, . .,.an) O_n+1an+1 ifi = n + 1,
for f 2 Cntrias(A, A), _ 2 Tn+1, and a1, . .,.an+1 2 A. Here the di are as in
(4.2.2). This differential ffi is similar to the one in dialgebras [3, 2.3].
5.3. The differential d. From the construction in x4.2, there is another
differential d in C*trias(A, A) given by
df = ss O f - (-1)|f|f O ss,
where ss 2 C2trias(A, A) is defined in (4.2.4). Now consider dn and
ffin :Cntrias(A, A) ! Cn+1trias(A, A).
Theorem 5.4. For each n, we have dn = (-1)n+1ffin. In particular, the
cohomology modules defined by (C*trias(A, A), d) and (C*trias(A, A), ffi) are t*
*he
same.
18 DONALD YAU
Proof.Pick an element f 2 Cntrias(A, A). Using the notations from earlier
sections, we have
dnf = ss O f + (-1)nf O ss
= ss{f} + (-1)nf{ss}
= (-1)n-1fl(ss; Id f) + fl(ss; f Id) (5.4.1)
Xn i j
+ (-1)n+i-1fl f; Id (i-1) ss Id (n-i).
i=1
We will show that these n + 2 terms are exactly the (-1)iffini, 0 i n + 1,
up to the sign (-1)n+1.
Consider an element x = _ a 2 K[Tn+1] A (n+1), where _ 2 Tn+1
and a = a1 . . .an+1 with each ai2 A. Then
fl(ss; Id f)(x) = ss(R0(2; 1, n)(_); a1 f(R2(2; 1, n)(_); a2, n+1)).
Using the descriptions in x4.2, R2(2; 1, n)(_) 2 Tn is obtained from _ by
(1) leaving leaves 1, 2, . .,.n + 1 alone, and (2) deleting the 0th leaf. That
is, R2(2; 1, n)(_) = d0_. Likewise, the tree R0(2; 1, n)(_) 2 T2 is obtained
from _ by (1) leaving leaves 0, 1, and n + 1 alone, and (2) deleting leaves n,
n - 1, . .,.2, in this order. Therefore, we have
8
>>< if|_0| = 0 and k = 1,
R0(2; 1, n)(_) = if|_0| > 0,
>>:
if|_0| = 0 and k > 1.
This shows that
fl(ss; Id f) = ffin0f. (5.4.2)
A similar argument shows that
fl(ss; f Id) = ffinn+1f. (5.4.3)
To finish the proof, note that for 1 i n, we have
i j
fl f; Id (i-1) ss Id (n-i)(x)
= f (R0(_); a1, i-1 ss(Ri(_); ai, ai+1) ai+2,.n+1)
Denoting a k-tuple of 1 by 1k, we have R0(_) = R0(n; 1i-1, 2, 1n-i)(_). This
is simply _ with its ith leaf deleted. That is, R0(_) = di_. Likewise, for
1 i n, Ri(_) = Ri(n; 1i-1, 2, 1n-i)(_) is the tree in T2 obtained from
_ by (1) leaving leaves i - 1, i, and i + 1 alone, and (2) deleting the other
leaves from right to left. Therefore, we have
8
>>< if the ith leaf of _ is left oriented,
Ri(n; 1i-1, 2, 1n-i)(_) = if the ith leaf of _ is right oriented,
>>:
if the ith leaf of _ is a middle.leaf
G-STRUCTURE AND DELIGNE'S CONJECTURE FOR LODAY ALGEBRAS 19
This shows that
i j
fl f; Id (i-1) ss Id (n-i)= ffinif. (5.4.4)
The required identity, dn = (-1)n+1ffin, is now an immediate consequence of
(5.4.1), (5.4.2), (5.4.3), and (5.4.4). This finishes the proof.
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E-mail address: dyau@math.ohio-state.edu
Department of Mathematics, The Ohio State University Newark, 1179 Uni-
versity Drive, Newark, OH 43055, USA