DONALD YAU

*Received 24 October 2005; Revised 28 March 2006; Accepted 6 April 2006*

We study homology and cohomology of triassociative algebras with nontrivial coeﬃ- cients. The cohomology theory is applied to study algebraic deformations of triassociative algebras.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

**1. Introduction**

Triassociative algebras, introduced by Loday and Ronco [8], are analogues of associative
algebras in which there are three binary operations, satisfying 11 relations. These algebras
are closely related to some very interesting combinatorial objects. In fact, the operad*Trias*
for triassociative algebras is modeled by the standard simplices. This operad is Koszul, in
the sense of Ginzburg and Kapranov [6]. Its Koszul dual operad*TriDend*is modeled by
the Stasheﬀpolytopes [11,12], or equivalently, planar trees. The algebras for the latter
operad are called tridendriform algebras, which have three binary operations whose sum
is an associative operation.

The purpose of this paper is to advance the study of triassociative algebras in the fol- lowing ways: (1) algebraic foundation: when studying (co)homology, the first task is to figure out the correct coeﬃcients. We introduce homology and cohomology with non- trivial coeﬃcients for triassociative algebras. In fact, Loday and Ronco [8] already defined homology with trivial coeﬃcients for these algebras. We will build upon their construc- tion. (2) Deformation: once (co)homology is in place, we use it to study algebraic de- formations of triassociative algebras, following the pattern established by Gerstenhaber [5].

We should point out that deformations of dialgebras [7], the analogues of triasso-
ciative algebras with only two binary operations, have been studied by Majumdar and
Mukherjee [9]. One diﬀerence between triassociative algebras and dialgebras is the co-
boundary operator in cohomology. Specifically, in both cases, the coboundary involves
certain products*◦*^{ψ}* _{j}*. The definition of this product is more delicate in the triassociative al-
gebra case than in the dialgebra case. This is due to the fact that planar trees, which model

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 69248, Pages1–21

DOI10.1155/IJMMS/2006/69248

*triassociative algebra cohomology, are more complicated than planar binary trees, which*
model dialgebra cohomology. A more fundamental diﬀerence is that the sets of planar
*trees form a simplicial set, whereas the sets of binary trees form only an almost simplicial*
set [7, Section 3.10]. This is important, since the author intends to adapt the algebraic
works in the present paper to study topological triassociative (co)homology, in analogy
with topological Hochschild (co)homology (see, e.g., [2,3,10]). Such objects should be
defined as the geometric realization of a certain simplicial spectrum (or the*Tot*of a cer-
tain cosimplicial spectrum), modeled after the algebraic (co)simplicial object that defines
(co)homology.

In the classical case of an associative algebra*R, the coe*ﬃcients in Hochschild homol-
ogy and cohomology are the same, namely, the*R-representations. In contrast, in the case*
of triassociative algebras (and also dialgebras), the coeﬃcients in homology and coho-
mology are not the same. The natural coeﬃcient for triassociative cohomology is a repre-
sentation, which can be defined using the 11 triassociative algebra axioms. This leads to
an expected algebraic deformation theory. To define triassociative homology with coeﬃ-
cients, we will construct the universal enveloping algebra (UA), which is a unital associa-
tive algebra of a triassociative algebra*A. A leftUA-module is exactly anA-representation.*

We then define an*A-corepresentation to be a rightUA-module. These rightUA-modules*
are the coeﬃcients in triassociative homology. In the classical Hochschild theory, the uni-
versal enveloping algebra is*R*^{e}*=**R**⊗**R** ^{op}*.

We remark that all of the results in this paper can be reproduced for tridendriform and tricubical algebras in [8] with minimal modifications.

*Organization. The next section is devoted to defining triassociative algebra cohomol-*
ogy with nontrivial coeﬃcients, in which we begin with a brief discussion of represen-
tations over a triassociative algebra. The construction of triassociative algebra cohomol-
ogy requires a discussion of planar trees. We observe that the sets of planar trees form
a simplicial set (Proposition 2.1). After that we define the cochain complex*C*^{∗}_{Trias}(A,M)
for a triassociative algebra*A*with coeﬃcients in an*A-representationM* (Theorem 2.2)
and give descriptions of the low-dimensional cohomology modules*H*_{Trias}* ^{∗}* for

*∗ ≤*2. It is also shown that

*H*

_{Trias}

*(A,*

^{n}*−*) is trivial for all

*n*

*≥*2 when

*A*is a free triassociative algebra (Theorem 2.3).

InSection 3, we construct triassociative algebra homology with nontrivial coeﬃcients
*by using the universal enveloping algebra (UA) of a triassociative algebraA. TheA-repre-*
sentations are identified with the left*UA-modules (Proposition 3.1). We then compute*
the associated graded algebra of*UA*under the length filtration (Theorem 3.2). This leads
to a triassociative version of the Poincar´e-Birkhoﬀ-Witt theorem (Corollary 3.3). We then
*use the rightUA-modules as the coeﬃcients for triassociative algebra homology, analo-*
gous to the dialgebra case [4]. To show that the purported homology complex is actually a
*chain complex, we relate it to the cotangent complex (Proposition 3.4), which can be used*
to define both the homology and the cohomology complexes. The cotangent complex is
the analogue of the bar complex in Hochschild homology. When the coeﬃcient module is
taken to be the ground field (i.e., trivial coeﬃcients), our triassociative homology agrees
with the one constructed in Loday and Ronco [8]. That section ends with descriptions of
*H*0^{Trias}and*H*1^{Trias}.

Section 4is devoted to studying algebraic deformations of triassociative algebras. We
define algebraic deformations and their infinitesimals for a triassociative algebra*A. It is*
observed that an infinitesimal is always a 2-cocycle in*C*^{2}_{Trias}(A,A) whose cohomology class
is determined by the equivalence class of the deformation (Theorem 4.1). A triassociative
algebra*Ais called rigid if every deformation ofA*is equivalent to the trivial deformation.

It is observed that the cohomology module*H*_{Trias}^{2} (A,A) can be thought of as the obstruc-
tion to the rigidity of*A. Namely, we observe thatA*is rigid, provided that the module
*H*_{Trias}^{2} (A,A) is trivial (Corollary 4.3). As examples, free triassociative algebras are rigid
(Corollary 4.4). Finally, we identify the obstructions to extend 2-cocycles in*C*_{Trias}^{2} (A,A) to
deformations. Given a 2-cocycle, there is a sequence of obstruction classes in*C*^{3}_{Trias}(A,*A),*
which are shown to be 3-cocycles (Lemma 4.5). The simultaneous vanishing of their co-
homology classes is equivalent to the existence of a deformation whose infinitesimal is
the given 2-cocycle (Theorem 4.6). In particular, these obstructions always vanish if the
cohomology module*H*_{Trias}^{3} (A,*A) is trivial (Corollary 4.7). We remark that the work of*
Balavoine [1] gives another approach to study deformations of triassociative algebras.

**2. Cohomology of triassociative algebras**

For the rest of this paper, we work over a fixed ground field*K*. Tensor products are taken
over*K*. We begin by recalling some relevant definitions about triassociative algebras and
planar trees from [8].

**2.1. Triassociative algebras and representations. A triassociative algebra is a vector space***A*that comes equipped with three binary operations,(left),(right), and*⊥*(middle),
satisfying the following 11 triassociative axioms for all*x,y,z**∈**A:*

(x*y)**z**=**x*(y*z),* (2.1a)

(x*y)**z**=**x*(y*z),* (2.1b)

(x*y)**z**=**x*(y*z),* (2.1c)

(x*y)**z**=**x*(y*z),* (2.1d)

(x*y)**z**=**x*(y*z),* (2.1e)

(x*y)**z**=**x*(y*⊥**z),* (2.1f)

(x*⊥**y)**z**=**x**⊥*(y*z),* (2.1g)

(x*y)**⊥**z**=**x**⊥*(y*z),* (2.1h)

(x*y)**⊥**z**=**x*(y*⊥**z),* (2.1i)

(x*⊥**y)**z**=**x*(y*z),* (2.1j)

(x*⊥**y)**⊥**z**=**x**⊥*(y*⊥**z).* (2.1k)

Note that the first 5 axioms state that (A,,) is a dialgebra [7], and they do not involve
the middle product*⊥*. From now on,*A* will always denote an arbitrary triassociative
algebra, unless stated otherwise.

*A morphism of triassociative algebras is a vector space map that respects the three*
products.

An*A-representation is a vector spaceM*together with (i) 3 left operations,,*⊥*:*A**⊗*
*M**→**M, (ii) 3 right operations* ,,*⊥*:*M**⊗**A**→**M* satisfying (2.1a)–(2.1k) whenever
exactly one of*x,* *y,z* is from*M*and the other two are from*A. Thus, there are 33 total*
axioms. From now on,*M*will always denote an arbitrary*A-representation, unless stated*
otherwise.

For example, if*ϕ*:*A**→**B*is a morphism of triassociative algebras, then*B*becomes an
*A-representation viaϕ, namely,xb**=**ϕ(x)b*and*bx**=**bϕ(x) forx**∈**A,b**∈**B,*
and*∈ {*,,*⊥}*. In particular,*A*is naturally an*A-representation via the identity map.*

In order to construct triassociative algebra cohomology, we also need to use planar trees.

**2.2. Planar trees. For integers***n**≥*0, let*T**n*denote the set 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 first four sets*T** _{n}*are listed below:

*T*0*=* , *T*1*=*

, *T*2*=*

, , ^{},

*T*3*=*

, , , , , , , , , , ^{}*.*

(2.2)

Trees in*T*_{n}*are said to have degreen, denoted by**|**ψ**| =**n*for*ψ**∈**T** _{n}*. The

*n*+ 1 leaves of a tree in

*T*

*n*are labeled

*{*0, 1,. . .,

*n*

*}*from left to right.

*A leaf is said to be left oriented (resp., right oriented) if it is the leftmost (resp., 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*ψ*0,. . .,ψ_{k}*, one can form a new tree by the operation of grafting. Namely,*
the grafting of these*k*+ 1 trees is the tree*ψ*0*∨ ··· ∨**ψ**k*obtained by arranging*ψ*0,. . .,ψ*k*

from left to right and joining the*k*+ 1 roots to form a new (lowest) internal vertex, which
is connected to a new root. The degree of*ψ*0*∨ ··· ∨**ψ** _{k}*is

*ψ*0+ 1^{}+*···*+^{}*ψ**k*+ 1^{}*−*1*=**k*+

*k*
*i**=*0

*ψ**i**.* (2.3)

Conversely, every tree*ψ*can be written uniquely as the grafting of*k*+ 1 trees,*ψ*0*∨ ··· ∨*
*ψ** _{k}*, where the valence of the lowest internal vertex of

*ψ*is

*k*+ 1.

Before defining cohomology, we make the following observation, which is not used in the rest of the paper but maybe useful in future studies of triassociative cohomology.

For 0*≤**i**≤**n*+ 1, define a function

*d** _{i}*:

*T*

_{n+1}*−→*

*T*

*, (2.4)*

_{n}which sends a tree*ψ**∈**T**n+1*to the tree*d**i**ψ**∈**T**n*obtained from*ψ*by deleting the*ith leaf.*

For 0*≤**i**≤**n, define another function*

*s**i*:*T**n**−→**T**n+1* (2.5)

as follows: for*ψ**∈**T**n*,*s**i**ψ**∈**T**n+1*is the tree obtained from*ψ*by adding a new leaf to the
internal vertex connecting to leaf*i, and this new leaf is placed immediately to the left of*
the original leaf*i. For example, ifψ**=* , then*s*0(ψ)*=* and*s*1(ψ)*=**s*2(ψ)*=* .
*Proposition 2.1. The sets**{**T*_{n}*}**n**≥*0*form a simplicial set with face mapsd*_{i}*and degeneracy*
*mapss**i**.*

*Proof. Recall that the simplicial relations are*
*d*_{i}*d*_{j}*=**d*_{j}* _{−}*1

*d*

*if*

_{i}*i < j,*

*d**i**s**j**=*

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

*s**j**−*1*d**i* if*i < j,*
Id if*i**=**j,j*+ 1,
*s*_{j}*d*_{i}* _{−}*1 if

*i > j*+ 1,

*s*

_{i}*s*

_{j}*=*

*s*

_{j+1}*s*

*if*

_{i}*i*

*≤*

*j.*

(2.6)

All of them are immediate from the definitions.

**2.3. Cohomology. For integers***n**≥**0, define the module ofn-cochains ofAwith coeﬃ-*
*cients inM*to be

*C*_{Trias}* ^{n}* (A,M) :

*=*Hom

*K*

*K*^{}*T*_{n}^{}*⊗**A*^{⊗}* ^{n}*,M

^{}

*.*(2.7) To define the coboundary maps, we need the following operations. For 0

*≤*

*i*

_{≤}*n*+ 1, define a function

*◦**i*:*T*_{n+1}*−→ {*,,*⊥}* (2.8)

according to the following rules. Let*ψ* be a tree in*T** _{n+1}*, which is written uniquely as

*ψ*

*=*

*ψ*0

*∨ ··· ∨*

*ψ*

*k*in which the valence of the lowest internal vertex of

*ψ*is

*k*+ 1. Also, write

*◦*

^{ψ}*i*for

*◦*

*i*(ψ). Then

*◦*

*0 is given by*

^{ψ}*◦** ^{ψ}*0

*=*

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

if^{}*ψ*0*=*0,*k**=*1,
if^{}*ψ*0*>*0,

*⊥* if^{}*ψ*0*=*0,*k >*1.

(2.9)

For 1*≤**i**≤**n,**◦*^{ψ}*i* is given by

*◦*^{ψ}_{i}*=*

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

if the*ith leaf ofψ*is left oriented,
if the*ith leaf ofψ*is right oriented,

*⊥* if the*ith leaf ofψ*is a middle leaf.

(2.10)

Finally,*◦*^{ψ}*n+1*is given by

*◦*^{ψ}*n+1**=*

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

if^{}*ψ**k**>*0,
if*k**=*1,^{}*ψ*1*=*0,

*⊥* if*k >*1,^{}*ψ*_{k}^{}*=*0.

(2.11)

For example, for the 11 trees*ψ*in*T*3(from left to right), we have

*◦** ^{ψ}*0

*=*,,,,,,

*⊥*,

*⊥*,,,

*⊥*,

*◦*^{ψ}*n+1**=*,,,,,,,*⊥*,*⊥*,,*⊥**.* (2.12)
Now define the map

*δ** ^{n}*:

*C*

_{Trias}

*(A,M)*

^{n}*−→*

*C*

_{Trias}

*(A,M) (2.13) to be the alternating sum*

^{n+1}*δ*^{n}*=*

*n+1*
*i**=*0

(*−*1)^{i}*δ*_{i}* ^{n}*, (2.14)

where

*δ*_{i}^{n}*f*^{}*ψ;a*1,*. . .,a*_{n+1}^{}*=*

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

*a*1*◦** ^{ψ}*0

*f*

^{}

*d*0

*ψ;a*2,

*. . .,a*

_{n+1}^{}if

*i*

*=*0,

*f*

^{}

*d*

*i*

*ψ*;

*a*1,. . .,a

*i*

*◦*

^{ψ}*i*

*a*

*i+1*,. . .,a

*n*

if 1*≤**i**≤**n,*
*f*^{}*d**n+1**ψ;a*1,. . .,a*n*

*◦*^{ψ}*n+1**a**n+1* if*i**=**n*+ 1,

(2.15)

for *f* *∈**C*^{n}_{Trias}(A,*M),ψ**∈**T**n+1*, and*a*1,*. . .,a**n+1**∈**A.*

*Theorem 2.2. The mapsδ*_{i}^{n}*satisfy the cosimplicial identities,*

*δ*^{n+1}_{j}*δ*_{i}^{n}*=**δ*_{i}^{n+1}*δ*^{n}_{j}_{−}_{1}, (2.16)
*for 0**≤**i < j**≤**n+ 2. In particular, (C*^{∗}* _{Trias}*(A,M),

*δ) is a cochain complex.*

*Proof. First, note that for 1**≤**i**≤**n, the mapδ*^{n}* _{i}* is dual to the map

*d*

*i*:

*C*

_{n+1}^{Trias}(A)

*=*

*K*

^{}

*T*

*n+1*

*⊗**A*^{⊗}^{n+1}*−→**K*^{}*T**n*

*⊗**A*^{⊗}^{n}*=**C*_{n}^{Trias}(A) (2.17)
in Loday and Ronco [8], that is,*δ*^{n}_{i}*=*Hom(d* _{i}*,M). It is shown there that these

*d*

*satisfy the simplicial identities. Therefore, to prove the cosimplicial identities, it suﬃces to consider the following three cases: (i)*

_{i}*i*

*=*0,

*j*

*=*

*n*+ 2, (ii)

*i*

*=*0

*< j < n*+ 2, and (iii) 0

*< i < j*

*=*

*n*+ 2.

Consider case (i). Suppose that*ψ**∈**T**n+2*,*a*1,. . .,*a**n+2**∈**A, and* *f* *∈**C*^{n}_{Trias}(A,M). Let
*α*be the element (ψ;*a*1,. . .,*a**n+2*) in*K*[T*n+2*]*⊗**A*^{⊗}* ^{n+2}*. Write

*ψ*as

*ψ*0

*∨ ··· ∨*

*ψ*

*k*uniquely

with*k*+ 1 being the valence of the lowest internal vertex of*ψ. We have*
*δ*_{n+2}^{n+1}*δ*_{0}^{n}*f*^{}(α)*=*

*δ*_{0}^{n}*f*^{}*d**n+2**ψ;a*1,. . .,a*n+1*

*◦*^{ψ}*n+2**a**n+2*

*=*

*a*1*◦** ^{d}*0

^{n+2}

^{ψ}*f*

^{}

*d*0

*d*

*n+2*

*ψ;a*2,. . .,a

*n+1*

*◦*^{ψ}*n+2**a**n+2*

*=*

*a*1*◦** ^{d}*0

^{n+2}

^{ψ}*y*

^{}

*◦*

^{ψ}*n+2*

*a*

*n+2*,

(2.18)

where*y**=* *f*(d_{n+1}*d*0*ψ*;*a*2,. . .,a* _{n+1}*). Here we used the simplicial identity

*d*0

*d*

_{n+2}*=*

*d*

_{n+1}*d*0. On the other hand, we have

*δ*_{0}^{n+1}*δ*_{n+1}^{n}*f*^{}(α)*=**a*1*◦** ^{ψ}*0

*δ*^{n}_{n+1}*f*^{}*d*0*ψ;a*2,. . .,a*n+2*

*=**a*1*◦** ^{ψ}*0

*y**◦*^{d}*n+1*^{0}^{ψ}*a**n+2*

*.* (2.19)

In order to show that

*a*1*◦** ^{d}*0

^{n+2}

^{ψ}*y*

^{}

*◦*

^{ψ}*n+2*

*a*

*n+2*

*=*

*a*1

*◦*

*0*

^{ψ}*y**◦*^{d}*n+1*^{0}^{ψ}*a**n+2*

, (2.20)

we need to consider 7 diﬀerent cases.

(1) If*k**=*1 and*|**ψ*0*| =*0, then*◦** ^{ψ}*0

*== ◦*

*0*

^{d}

^{n+2}*. In this case, it follows from the triasso- ciative algebra axioms (2.1a), (2.1b), and (2.1f) that*

^{ψ}*a*1*y*^{}*∗**a*_{n+2}*=**a*1

*y**∗*^{}*a*_{n+2}^{} (2.21)

regardless of what*∗*,*∗*^{}*∈ {*,,*⊥}*are.

(2) If *k**=*1 and*|**ψ*1*| =*0, then*◦*^{ψ}*n+2**== ◦*^{d}*n+1*^{0}* ^{ψ}*. It then follows from axioms (2.1d),
(2.1e), and (2.1j) that

*a*1*∗**y*^{}*a*_{n+2}*=**a*1*∗*^{}

*y**a*_{n+2}^{} (2.22)

regardless of what*∗*,*∗*^{}*∈ {*,,*⊥}*are.

(3) For*k**≥*1, if both*|**ψ*0*|*and*|**ψ**k**|*are positive, then*◦** ^{d}*0

^{n+2}

^{ψ}*== ◦*

*0 and*

^{ψ}*◦*

^{ψ}*n+2*

*==*

*◦*^{d}*n+1*^{0}* ^{ψ}*. Therefore, (2.20) holds by axiom (2.1c).

(4) If*k**=*2 and*|**ψ*0*| =*0*= |**ψ*2*|*, then*◦** ^{d}*0

^{n+2}

^{ψ}*=*,

*◦*

^{ψ}*n+2*

*= ⊥*,

*◦*

*0*

^{ψ}*= ⊥*, and

*◦*

^{d}*n+1*

^{0}

^{ψ}*=*. It follows from axiom (2.1h) that (2.20) holds.

(5) If*k >*2 and*|**ψ*0*| =*0*= |**ψ**k**|*, then*◦** ^{d}*0

^{n+2}

^{ψ}*= ◦*

^{ψ}

_{n+2}*= ◦*

*0*

^{ψ}*= ◦*

^{d}

_{n+1}^{0}

^{ψ}*= ⊥*. Thus, (2.20) fol- lows from axiom (2.1k).

(6) If*k**≥*2,*|**ψ*0*| =*0, and*|**ψ**k**|**>*0, then*◦** ^{d}*0

^{n+2}

^{ψ}*= ⊥ = ◦*

*0 and*

^{ψ}*◦*

^{ψ}

_{n+2}*== ◦*

^{d}

_{n+1}^{0}

*. There- fore, it follows from axiom (2.1g) that (2.20) holds.*

^{ψ}(7) If*k**≥*2,*|**ψ*0*|**>*0, and*|**ψ*_{k}*| =*0, then*◦** ^{d}*0

^{n+2}

^{ψ}*== ◦*

*0 and*

^{ψ}*◦*

^{ψ}

_{n+2}*= ⊥ = ◦*

^{d}

_{n+1}^{0}

*. There- fore, (2.20) follows from axiom (2.1i).*

^{ψ}Therefore, the cosimplicial identity holds in the case*i**=*0, *j**=**n*+ 2.

Next, we consider case (ii), where*i**=*0,*j**=*1,*. . .,n*+ 1. We have
*δ*^{n+1}_{j}*δ*_{0}^{n}*f*^{}(α)*=*

*δ*_{0}^{n}*f*^{}*d*_{j}*ψ;a*1,. . .,*a*_{j}*◦*^{ψ}_{j}*a** _{j+1}*,

*. . .,a*

_{n+2}^{}

*=*

⎧⎨

⎩

*a*1*◦** ^{ψ}*1

*a*2

*◦** ^{d}*0

^{1}

^{ψ}*z*if

*j*

*=*1,

*a*1

*◦*

*0*

^{d}

^{j}

^{ψ}*x*if 2

*≤*

*j*

*≤*

*n*+ 1,

(2.23)

where

*z**=* *f*^{}*d*0*d*0*ψ;a*3,. . .,a*n+2*
,

*x**=* *f*^{}*d**j**−*1*d*0*ψ;a*2,. . .,*a**j**◦*^{ψ}_{j}*a**j+1*,*. . .,a**n+2*

*.* (2.24)

Here we used the simplicial identity*d*0*d*_{j}_{=}*d*_{j}* _{−}*1

*d*0for 0

*< j. On the other hand, we have*

*δ*

_{0}

^{n+1}*δ*

^{n}

_{j}

_{−}_{1}

*f*

^{}(α)

*=*

*a*1

*◦*

*0*

^{ψ}*δ*^{n}_{j}_{−}_{1}*f*^{}*d*0*ψ;a*2,. . .,*a*_{n+2}^{}

*=*

⎧⎨

⎩
*a*1*◦** ^{ψ}*0

*a*2*◦** ^{d}*0

^{0}

^{ψ}*z*

^{}if

*j*

*=*1,

*a*1

*◦*

*0*

^{ψ}*w*if 2

*≤*

*j*

*≤*

*n*+ 1,

(2.25)

where

*w**=**f*^{}*d**j**−*1*d*0*ψ;a*2,. . .,a*j**◦*^{d}_{j}^{0}_{−}* ^{ψ}*1

*a*

*j+1*,. . .,a

*n+2*

*.* (2.26)

In the cases 2*≤**j**≤**n*+ 1, we have that*◦*^{ψ}*j* *= ◦*^{d}*j*^{0}*−** ^{ψ}*1 and

*◦*

*0*

^{d}

^{j}

^{ψ}*= ◦*

*0, and hence*

^{ψ}*a*1

*◦*

*0*

^{d}

^{j}

^{ψ}*x*

*=*

*a*1

*◦*

*0*

^{ψ}*w. In the casej*

*=*1, we need to show the identity

*a*1*◦** ^{ψ}*1

*a*2

*◦** ^{d}*0

^{1}

^{ψ}*z*

_{=}*a*1

*◦*

*0*

^{ψ}*a*2*◦** ^{d}*0

^{0}

^{ψ}*z*

^{}

*.*(2.27) We break it into three cases.

(1) If*|**ψ*0*| ≥*2, then*◦** ^{d}*0

^{1}

^{ψ}*== ◦*

*0*

^{d}^{0}

*. Therefore, it follows from axioms (2.1d), (2.1e), and (2.1j) that*

^{ψ}*a*1*∗**a*2

*z**=**a*1*∗*^{}

*a*2*z*^{} (2.28)

regardless of what*∗*,*∗*^{}*∈ {*,,*⊥}*are.

(2) If*|**ψ*0*| =*1, then*◦** ^{ψ}*1

*== ◦*

*0 and*

^{ψ}*◦** ^{d}*0

^{0}

^{ψ}*= ◦*

*0*

^{d}^{1}

^{ψ}*=*

⎧⎨

⎩

if*k**=*1,

*⊥* if*k**≥*2. (2.29)

Therefore, (2.27) holds by axiom (2.1c) if*k**=*1 and by axiom (2.1i) if*k**≥*2.

(3) Now suppose that*|**ψ*0*| =*0. If*k**=*1, then*◦** ^{ψ}*0

*== ◦*

*1. It follows that (2.27) holds by axioms (2.1a), (2.1b), and (2.1f). If*

^{ψ}*k*

*≥*2, then

*◦*

*0*

^{ψ}*= ⊥*. To figure out what the other three operations are, we need to consider two subcases.

(i) If *|**ψ*1*|**>*0, then*◦** ^{d}*0

^{0}

^{ψ}*=*,

*◦*

*1*

^{ψ}*=*, and

*◦*

*0*

^{d}^{1}

^{ψ}*= ⊥*. Thus, (2.27) holds by axiom (2.1h).

(ii) If*|**ψ*1*| =*0, then*◦** ^{ψ}*1

*= ⊥*and

*◦** ^{d}*0

^{1}

^{ψ}*= ◦*

*0*

^{d}^{0}

^{ψ}*=*

⎧⎨

⎩

if*k**=*2,

*⊥* if*k >*2. (2.30)

It follows that (2.27) holds by axiom (2.1g) when *k**=*2 and by axiom (2.1k)
when*k >*2.

This proves the cosimplicial identities for the*δ*_{l}* ^{n}*when

*i*

*=*0 and 1

*≤*

*j*

*≤*

*n*+ 1. The proof for the case 1

*≤*

*i*

*≤*

*n*+ 1,

*j*

*=*

*n*+ 2 is similar to the argument that was just given.

In view ofTheorem 2.2, we define the*nth cohomology ofAwith coeﬃcients in the rep-*
*resentationM*to be

*H*_{Trias}* ^{n}* (A,M) :

*=*

*H*

^{n}^{}

*C*

^{∗}_{Trias}(A,M),δ

^{}(2.31) for

*n*

*≥*0. We describe the first three cohomology modules below.

**2.4.***H*_{Trias}^{0} **and***H*_{Trias}^{1} **. A linear map***ϕ*:*A**→**Mis called a derivation ofAwith values inM*
if it satisfies the condition

*ϕ(a*_{∗}*b)*_{=}*ϕ(a)*_{∗}*b*+*a*_{∗}*ϕ(b),* (2.32)
for all*a,b**∈**A*and*∗ ∈ {*,,*⊥}*. Denote by Der(A,M) the submodule of Hom* _{K}*(A,M)
consisting of all the derivations of

*A*with values in

*M. For an elementm*

*∈*

*M, define the*map

*ad*

*:*

_{m}*A*

*→*

*M*, where

*ad**m*(a) :*=**a**m**−**m**a* (2.33)

for *a**∈**A. Such a map is called an inner derivation of* *A* *with values inM. It follows*
immediately from the triassociative algebra axioms (2.1) that each map*ad** _{m}* belongs to
Der(A,M). Let Inn(A,M) denote the submodule of Der(A,M) consisting of all the inner
derivations of

*A*with values in

*M.*

Identify*C*_{Trias}^{0} (A,*M) withM*and*C*_{Trias}^{1} (A,M) with Hom* _{K}*(A,M). Under such identifi-
cations, the coboundary map

*δ*

^{0}:

*M*

*→*Hom

*K*(A,M) is given by

*δ*

^{0}(m)

*=*

*ad*

*m*. Therefore, the cohomology module

*H*

_{Trias}

^{0}(A,M), which is the kernel of

*δ*

^{0}, is the following submod- ule of

*M:*

*H*_{Trias}^{0} (A,*M)*^{∼}*=**M** ^{A}*:

*=*

*m**∈**M**|**a**m**=**m**a**∀**a**∈**A*^{}*.* (2.34)
The image of*δ*^{0}is the module Inn(A,M). Now if *f* *∈**C*_{Trias}^{1} (A,M), then

*δ*^{1}*f*^{}(ψ;*a,b)**=**a**∗**f*(b)*−**f*(a*∗**b) +f*(a)*∗**b,* (2.35)
where

*∗ =*

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

if*ψ**=* ,
if*ψ**=* ,

*⊥* if*ψ**=* *.*

(2.36)

In particular, the kernel of*δ*^{1}is exactly Der(A,M). Therefore, as in the cases of associative
algebras and dialgebras [4], we have

*H*_{Trias}^{1} (A,M)^{∼}* _{=}*Der(A,M)

Inn(A,M)*.* (2.37)

**2.5.***H*_{Trias}* ^{i}* (A,

*−*

**) when**

*A*

**is free. It is shown in [8, Theorem 4.1] that the operad**

*Trias*for triassociative algebras is Koszul. In other words,

*H*

_{n}^{Trias}(A,K)

*=*0 for

*n*

*≥*2 if

*A*is a free triassociative algebra. Here

*H*

_{n}^{Trias}(A,K) denotes the triassociative homology of

*A*with trivial coeﬃcients defined by Loday and Ronco [8]. It agrees with our triassociative ho- mology defined below. The Koszulness of

*Trias*implies that free triassociative algebras have trivial higher cohomology modules.

*Theorem 2.3. LetAbe a free triassociative algebra. Then*

*H*_{Trias}* ^{n}* (A,M)

*=*0 (2.38)

*for alln**≥**2 and anyA-representationM.*

*Proof. The first line in the proof of*Theorem 2.2implies that for any triassociative algebra
*A,*

*H*_{Trias}* ^{n}* (A,M)

^{∼}*=*Hom

*K*

*H*_{n}^{Trias}(A,K),*M*^{}*.* (2.39)

The theorem now follows from the Koszulness of*Trias.*

**2.6. Abelian extensions and***H*_{Trias}^{2} * . Define an abelian triassociative algebra to be a trias-*
sociative algebra

*P*in which all three products,,,

*⊥*, are equal to 0. In this case, we will just say that

*P*is abelian. Any vector space becomes an abelian triassociative algebra when equipped with the trivial products. Suppose that

*ξ*: 0

*→*

*P*

*−*

*→*

^{i}*E*

*−*

*→*

^{π}*A*

*→*0 is a short exact sequence of triassociative algebras in which

*P*is abelian. Then

*P*has an induced

*A-representation structure via*

*a*

*∗*

*p*

*=*

*e*

*∗*

*i(p) and*

*p*

*∗*

*a*

*=*

*i(p)*

*∗*

*e*for

*∗ ∈ {*,,

*⊥}*,

*p*

*∈*

*P,a*

*∈*

*A, and any elemente*

*∈*

*E*such that

*π(e)*

*=*

*a.*

Now consider an*A-representationM. By an abelian extension ofAbyM, we mean a*
short exact sequence

*ξ*: 0*−→**M**−−→*^{i}*E**−−→*^{π}*A**−→*0 (2.40)

of triassociative algebras in which*M*is abelian and such that the induced*A-representa-*
tion structure on*M* coincides with the original one. An abelian extension is said to be
*trivial if it splits triassociative algebras. Given another abelian extensionξ*^{}*= {*0*→**M**→*
*E*^{}*→**A**→*0*}*of*A*by*M, we say thatξ*and*ξ*^{}*are equivalent if there exists a mapϕ*:*E**→**E** ^{}*
of triassociative algebras making the obvious ladder diagram commutative. (Note that
such a map

*ϕ*must be an isomorphism.) Denote by [ξ] the equivalence class of an abelian extension

*ξ*and denote by Ext(A,M) the set of equivalence classes of abelian extensions of

*A*by

*M.*

Suppose that*ξ* is an abelian extension of*A*by*M* as in (2.40). By choosing a vector
space splitting*σ*:*A**→**E, one can identify the underlying vector space ofE*with*M**⊕**A.*

As usual, there exists a map

*f**ξ*:*K*^{}*T*2

*⊗**A*^{⊗}^{2}*−→**M* (2.41)

such that the products in*E*become
(m,a)*∗*(n,b)*=*

*m**∗**b*+*a**∗**n*+*f** _{ξ}*(ψ;

*a,b),a*

*∗*

*b*

^{}(2.42) for

*∗ ∈ {*,,

*⊥}*,

*m,n*

*∈*

*M, anda,b*

*∈*

*A. Hereψ*

*∈*

*T*2is given by

*ψ**=*

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

if*∗ =*,
if*∗ =*,
if*∗ = ⊥**.*

(2.43)

Note that this is basically (2.36). We will always identify the 3 trees in*T*2with the products
*{*,,*⊥}*like this. It is easy to check that the 11 triassociative algebra axioms (2.1) in*E*
are equivalent to *f**ξ**∈**C*_{Trias}^{2} (A,M) being a 2-cocycle. For instance, we have

(0,x)(0,*y)*^{}(0,z)*=*
*f**ξ*

;*x,y*^{},*x**y*^{}(0,z)

*=*
*f**x*

;*x,y*^{}*z*+*f**ξ*

;*x**y,z*^{}, (x*y)**z*^{}*.*
(2.44)
On the other hand, a similar calculation yields

(0,x)

(0,*y)**⊥*(0,z)^{}*=*
*x**f**ξ*

;*y,z*^{}+*f**ξ*

;*x,y**⊥**z*^{},x(y*⊥**z)*^{}*.* (2.45)
These two expressions are equal by axiom (2.1f). By equating the first factors, we obtain

*δ*^{2}*f**ξ*

;*x,y,z*^{}*=*0. (2.46)

Similar arguments, using the other 10 triassociative algebra axioms, show that (δ^{2}*f**ξ*)(ψ;

*x,y,z)**=*0 for the other 10 trees*ψ**∈**T*3.

Conversely, suppose that*g**∈**C*_{Trias}^{2} (A,M) is a 2-cocycle. Then one can define a trias-
sociative algebra structure on the vector space*M**⊕**A*using (2.42) with*g* in place of *f**ξ*.
Again, the triassociative algebra axioms are verified because of the cocycle condition on
*g*. This yields an abelian extension of*A*by*M,*

*ζ**g*: 0*−→**M**−−→*^{i}*M**⊕**A**−−→*^{π}*A**−→*0 (2.47)
in which*i*and*π* are, respectively, the inclusion into the first factor and the projection
onto the second factor.

*Theorem 2.4. The above constructions induce well-defined maps*
Ext(A,M)*−→**H*_{Trias}^{2} (A,M), [ξ]*−→*

*f*_{ξ}^{},

*H*_{Trias}^{2} (A,M)*−→*Ext(A,M), [g]*−→*

*ζ*_{g}^{}, (2.48)

*that are inverse to each other. In particular, there is a canonical bijectionH*_{Trias}^{2} (A,*M)*^{∼}* _{=}*
Ext(A,M

*).*

The proof is basically identical to that of the classical case for associative algebras (see, e.g., [13, pages 311-312] or [4, Sections 2.8-2.9]). Therefore, we omit the details. Note that one only needs to see that the maps are well defined, since they are clearly inverse to each other.

**3. Universal enveloping algebra and triassociative homology**

The purpose of this section is to construct triassociative algebra homology with nontrivial
coeﬃcients. In [8], Loday and Ronco already constructed triassociative homology with
trivial coeﬃcients (i.e.,*K*). Our homology agrees with the one in Loday and Ronco [8]

by taking coeﬃcients in*K. In order to obtain the nontrivial coe*ﬃcients, we first need
to discuss the universal enveloping algebra. The dialgebra analogue of the results in this
section is worked out in [4].

**3.1. Universal enveloping algebra. Fix a triassociative algebra***A. We would like to iden-*
tify the*A-representations as the left modules of a certain associative algebra. This requires*
6 copies of*A, since 3 copies are needed for the left actions and another 3 for the right ac-*
tions. Therefore, we make the following definition. For a*K-vector spaceV,T*(V) denotes
the tensor algebra of*V*, which is the free unital associative*K-algebra generated byV*.

*Define the universal enveloping algebra ofA*to be the unital associative*K-algebraUA*
obtained from the tensor algebra*T*(α*l**A**⊕**α**r**A**⊕**α**m**A**⊕**β**l**A**⊕**β**r**A**⊕**β**m**A) on 6 copies*
of*A*by imposing the following 33 relations for*a,b**∈**A*(3 relations for each of the 11
triassociative axioms (2.1)):

*β** _{l}*(b)

*·*

*β*

*(a)*

_{l}^{(1)}

*=*

*β*

*(a*

_{l}*b)*

^{(2)}

*=*

*β*

*(a*

_{l}*⊥*

*b)*

^{(3)}

*=*

*β*

*(a*

_{l}*b),*

*β*

*(b)*

_{l}*·*

*α*

*(a)*

_{l}^{(4)}

*=*

*α*

*(a)*

_{l}*·*

*β*

*(b)*

_{l}^{(5)}

*=*

*α*

*(a)*

_{l}*·*

*β*

*(b)*

_{m}^{(6)}

*=*

*α*

*(a)*

_{l}*·*

*β*

*(b),*

_{r}*α** _{l}*(a

*b)*

^{(7)}

*=*

*α*

*(a)*

_{l}*·*

*α*

*(b)*

_{l}^{(8)}

*=*

*α*

*(a)*

_{l}*·*

*α*

*(b)*

_{m}^{(9)}

*=*

*α*

*(a)*

_{l}*·*

*α*

*(b),*

_{r}*β*

*r*(a

*b)*

^{(10)}

*=*

*β*

*r*(b)

*·*

*β*

*r*(a)

^{(11)}

*=*

*β*

*r*(b)

*·*

*β*

*l*(a)

^{(12)}

*=*

*β*

*r*(b)

*·*

*β*

*m*(a),

*α*

*r*(a)

*·*

*β*

*r*(b)

^{(13)}

*=*

*β*

*r*(b)

*·*

*α*

*r*(a)

^{(14)}

*=*

*β*

*r*(b)

*·*

*α*

*l*(a)

^{(15)}

*=*

*β*

*r*(b)

*·*

*α*

*m*(a),

*α**r*(a)*·**α**r*(b)^{(16)}*=* *α**r*(a*b)*^{(17)}*=* *α**r*(a*b)*^{(18)}*=* *α**r*(a*⊥**b),*
*β**l*(b)*·**β**r*(a)^{(19)}*=* *β**r*(a*b),* *β**l*(b)*·**α**r*(a)^{(20)}*=* *α**r*(a)*·**β**l*(b),

*α**l*(a*b)*^{(21)}*=* *α**r*(a)*·**α**l*(b),

*β** _{l}*(b)

*·*

*β*

*(a)*

_{m}^{(22)}

*=*

*β*

*(a*

_{m}*b),*

*β*

*(b)*

_{l}*·*

*α*

*(a)*

_{m}^{(23)}

*=*

*α*

*(a)*

_{m}*·*

*β*

*(b),*

_{l}*α** _{l}*(a

*⊥*

*b)*

^{(24)}

*=*

*α*

*(a)*

_{m}*·*

*α*

*(b),*

_{l}*β** _{m}*(b)

*·*

*β*

*(a)*

_{l}^{(25)}

*=*

*β*

*(a*

_{m}*b),*

*β*

*(b)*

_{m}*·*

*α*

*(a)*

_{l}^{(26)}

*=*

*α*

*(a)*

_{m}*·*

*β*

*(b),*

_{r}*α*

*(a*

_{m}*b)*

^{(27)}

*=*

*α*

*(a)*

_{m}*·*

*α*

*(b),*

_{r}*β**m*(b)*·**β**r*(a)^{(28)}*=* *β**r*(a*⊥**b),* *β**m*(b)*·**α**r*(a)^{(29)}*=* *α**r*(a)*·**β**m*(b),
*α**m*(a*b)*^{(30)}*=* *α**r*(a)*·**α**m*(b),

*β**m*(b)*·**β**m*(a)^{(31)}*=* *β**m*(a*⊥**b),* *β**m*(b)*·**α**m*(a)^{(32)}*=* *α**m*(a)*·**β**m*(b),
*α**m*(a*⊥**b)*^{(33)}*=* *α**m*(a)*·**α**m*(b).

(3.1)
*Proposition 3.1. LetMbe aK-vector space. Then anA-representation structure onMis*
*equivalent to a leftUA-module structure onM.*

*Proof. The correspondence is given by (a**∈**A,x**∈**M)*

*α**l*(a)*·**x**=**a**x,* *α**r*(a)*·**x**=**a**x,* *α**m*(a)*·**x**=**a**⊥**x,*

*β**l*(a)*·**x**=**x**a,* *β**r*(a)*·**x**=**x**a,* *β**m*(a)*·**x**=**x**⊥**a.* (3.2)
So*α** _{∗}*and

*β*

*correspond to left and right*

_{∗}*A-actions, respectively, and the subscript indi-*cates where the product points to. In particular, the conditions (1)–(9) above correspond to axioms (2.1a), (2.1b), and (2.1f). Conditions (10)–(18) correspond to axioms (2.1d), (2.1e), and (2.1j). The other 5 groups of conditions correspond to the remaining 5 axioms

in (2.1).

**3.2. Filtration on***UA. Each homogeneous element inUA*has a length: elements in*K*
have length 0. The elements*γ** _{∗}*(a), where

*γ*

*=*

*α,β,*

*∗ =*

*l,r,m, anda*

*∈*

*A, have length 1.*

Inductively, the homogeneous elements in*UA*of length at most*k*+ 1 (k*≥*1) are the*K-*
linear combinations of the elements*γ** _{∗}*(a)

*·*

*x*and

*x*

*·*

*γ*

*(a), where*

_{∗}*x*has length at most

*k.*

For*k**≥*0, consider the following submodule of*UA:*

*F*_{k}*UA**= {**x**∈**UA*:*x*has length *≤**k**}**.* (3.3)
These submodules form an increasing and exhaustive filtration of*UA, so thatF*0*UA**=**K,*
*F**k**UA**⊆**F**k+1**UA, andUA**= ∪**k**≥*0*F**k**UA. Moreover, it is multiplicative, in the sense that*
(F_{k}*UA)**·*(F_{l}*UA)**⊆**F*_{k+l}*UA. Therefore, it makes sense to consider the associated graded*
algebra*Gr*_{∗}*UA**= ⊕**k**≥*0*Gr**k**UA, whereGr**k**UA**=**F**k**UA/F**k**−*1*UA. It is clear thatGr*0*UA**=*
*K. The following result identifies the other associated quotients.*