(Co)homology of triassociative algebras

We study homology and cohomology of triassociative algebras with non-trivial coefficients.

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 certain cosimplicial spectrum), modeled after the algebraic (co)simplicial object that defines (co)homology.
In the classical case of an associative algebra R, the coefficients in Hochschild homology 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 cohomology are not the same. The natural coefficient for triassociative cohomology is a representation, which can be defined using the 11 triassociative algebra axioms. This leads to an expected algebraic deformation theory. To define triassociative homology with coefficients, we will construct the universal enveloping algebra (UA), which is a unital associative algebra of a triassociative algebra A. A left UA-module is exactly an A-representation. We then define an A-corepresentation to be a right UA-module. These right UA-modules are the coefficients in triassociative homology. In the classical Hochschild theory, the universal 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 cohomology with nontrivial coefficients, in which we begin with a brief discussion of representations over a triassociative algebra. The construction of triassociative algebra cohomology 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-representation M (Theorem 2.2) and give descriptions of the low-dimensional cohomology modules H * Trias for * ≤ 2. It is also shown that H n Trias (A,−) is trivial for all n ≥ 2 when A is a free triassociative algebra (Theorem 2.3).
In Section 3, we construct triassociative algebra homology with nontrivial coefficients by using the universal enveloping algebra (UA) of a triassociative algebra A. The A-representations 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 Poincaré-Birkhoff-Witt theorem (Corollary 3.3). We then use the right UA-modules as the coefficients for triassociative algebra homology, analogous 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 Trias 0 and H Trias 1 . Section 4 is 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 A is called rigid if every deformation of A is equivalent to the trivial deformation. It is observed that the cohomology module H 2 Trias (A,A) can be thought of as the obstruction to the rigidity of A. Namely, we observe that A is rigid, provided that the module H 2 Trias (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 2 Trias (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 cohomology 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 3 Trias (A,A) is trivial (Corollary 4.7). We remark that the work of Balavoine [1] gives another approach to study deformations of triassociative algebras.

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].

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: 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 space M 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 via ϕ, namely, 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.

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: Trees in T n are said to have degree n, 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., rightmost) 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 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 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 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, if ψ = , 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 maps d i and degeneracy maps s i .
Proof. Recall that the simplicial relations are (2.6) All of them are immediate from the definitions. To define the coboundary maps, we need the following operations. For 0 ≤ i ≤ n + 1, define a function according to the following rules. Let ψ be a tree in T n+1 , which is written uniquely as (2.10) Finally, • ψ n+1 is given by For example, for the 11 trees ψ in T 3 (from left to right), we have Now define the map to be the alternating sum where for f ∈ C n Trias (A,M), ψ ∈ T n+1 , and a 1 ,...,a n+1 ∈ A. Theorem 2.2. The maps δ n i satisfy the cosimplicial identities, in Loday and Ronco [8], that is, δ n i = Hom(d i ,M). It is shown there that these d i satisfy the simplicial identities. Therefore, to prove the cosimplicial identities, it suffices to consider the following three cases: (i) i = 0, j = n + 2, (ii) i = 0 < j < n + 2, and (iii) 0 < i < j = n + 2.
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 (2.24) Here we used the simplicial identity On the other hand, we have In the cases 2 ≤ j ≤ n + 1, we have that In the case j = 1, we need to show the identity We break it into three cases.
0 . Therefore, it follows from axioms (2.1d), (2.1e), and (2.1j) that (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 n l 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. (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 The theorem now follows from the Koszulness of Trias.

Abelian extensions and H 2
Trias . Define an abelian triassociative algebra to be a triassociative 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 element e ∈ E such that π(e) = a. Now consider an A-representation M. By an abelian extension of A by M, we mean a short exact sequence Here ψ ∈ T 2 is given by (2.43) Note that this is basically (2.36). We will always identify the 3 trees in T 2 with 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 2 Trias (A,M) being a 2-cocycle. For instance, we have 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 2 Trias (A,M) is a 2-cocycle. Then one can define a triassociative 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, in which i and π are, respectively, the inclusion into the first factor and the projection onto the second factor. 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.

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 coefficients, we first need to discuss the universal enveloping algebra. The dialgebra analogue of the results in this section is worked out in [4].

Universal enveloping algebra. Fix a triassociative algebra A.
We would like to identify 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 actions. Therefore, we make the following definition. For a K-vector space V , T(V ) denotes the tensor algebra of V , which is the free unital associative K-algebra generated by V .

Filtration on UA.
Each homogeneous element in UA has a length: elements in K have length 0. The elements γ * (a), where γ = α,β, * = l,r,m, and a ∈ A, have length 1. Inductively, the homogeneous elements in UA of length at most k + 1 (k ≥ 1) are the Klinear 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: These submodules form an increasing and exhaustive filtration of UA, so that F 0 UA = K, F k UA ⊆ F k+1 UA, and UA = ∪ 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, where Gr k UA = F k UA/F k−1 UA. It is clear that Gr 0 UA = K. The following result identifies the other associated quotients.
Theorem 3.2. In the associated graded algebra Gr * UA, one always has Gr n UA = 0 for all n ≥ 3. Moreover, there are isomorphisms In particular, if A is of finite-dimension d over K, then Gr * UA has dimension 1 + 6d + 7d 2 over K.
Proof. The first isomorphism is clear, since Gr 0 UA = F 0 UA = K and the elements of length 1 are linearly generated by the α * (a) and β * (a). For the second isomorphism, observe that any other generator of length ≤ 2 is identified with an element of length 1 by one of the 33 conditions, leaving only the 7 displayed generators. For example, β l (b) · β l (a) ∈ β l β l A ⊗2 is identified with β l (a b) ∈ β l A by condition (1) in the definition of the universal enveloping algebra (UA). Finally, to show that Gr n UA = 0 for n ≥ 3, it suffices to show that F 2 UA = F 3 UA. It is straightforward to check that multiplying any one of the 7 generators of length 2 with a generator of length 1 always yields an element of length ≤ 2. For example, α l (a) · α l (b) · β l (c) = α l (a) · α l (b) · β l (c) = α l (a b) · β l (c), (3.5) which lies in α l β l A ⊗2 , by condition (7) in the definition of UA.
An immediate consequence of this result is the following.

Corepresentation. By a corepresentation of A, or an
A-corepresentation, we mean a right UA-module. This definition can be made more explicit as follows. Let N be an A-corepresentation. Set x < a : = x · β l (a), x > a := x · β r (a), for x ∈ N and a ∈ A. This gives rise to three left actions <,>,∧ : A ⊗ N → N and three right actions <,>,∧ : N ⊗ A → N. With these notations, the condition that N is an Acorepresentation is equivalent to the following 33 axioms (for x ∈ N and a,b ∈ A), corresponding to the axioms for UA in Section 3.1: , Here are some examples of corepresentations (1) UA is an A-corepresentation via the right action of UA on itself.

Deformations of triassociative algebras
In this final section, we describe algebraic deformations of a triassociative algebra A = (A, , ,⊥), using the cohomology theory constructed in Section 2. The theory of triassociative algebra deformations is what one would expect from the existing literature on deformations. The arguments in this section are rather straightforward and are very similar to the cases of associative algebras [5] and dialgebras [9], whose arguments are provided in detail and can be easily adapted to the present case. Also, Balavoine [1] has another way of doing algebraic deformations that can be applied to triassociative algebras. Therefore, we will safely omit most of the proofs in this section. Thinking of A also as an A-representation via the identity self-map, we consider the cochain complex C * Trias (A,A). A 2-cochain θ ∈ C 2 Trias (A,A) will be identified with the triple (λ,ρ,μ), where each component is a binary operation on A, via for x, y ∈ A.

Deformation and equivalence.
By a deformation of A, we mean a power series Θ t = ∞ i=0 θ i t i , in which each θ i = (λ i ,ρ i ,μ i ) is a 2-cochain in C 2 Trias (A,A) and θ 0 = ( , ,⊥), satisfying the following 11 conditions (same as (2.1)) for all x, y,z ∈ A, where L t = ∞ i=0 λ i t i , R t = ∞ i=0 ρ i t i , and M t = ∞ i=0 μ i t i : (1) L t (L t (x, y),z) = L t (x,L t (y,z)), (2) L t (L t (x, y),z) = L t (x,R t (y,z)), and so forth.
Extending linearly, this gives a triassociative algebra structure on the power series A [[t]]. We will also denote a deformation Θ t by the triple (L t ,R t ,M t ) of power series.
Trias (A,A) ∼ = Hom K (A,A) and φ 0 = Id A . Let Θ t = (L t ,R t ,M t ) be another deformation of A. Then Θ t and Θ t are said to be equivalent if there exists a formal isomorphism Φ t such for O = L,R,M, and for all x, y,z ∈ A. In this case, we write Θ t = Φ t Θ t Φ −1 t . Conversely, given a deformation Θ t and a formal isomorphism Φ t , one can define a new deformation as Θ t := Φ t Θ t Φ −1 t .

Infinitesimal.
Given a deformation Θ t = ∞ i=0 θ i t i of A, the linear coefficient θ 1 is called the infinitesimal of Θ t .
Theorem 4.1. Let Θ t = ∞ i=0 θ i t i be a deformation of A. Then the infinitesimal θ 1 is a 2cocycle in C 2 Trias (A,A), whose cohomology class is well defined by the equivalence class of Θ t . Moreover, if θ i = 0 for 1 ≤ i ≤ l, then θ l+1 is a 2-cocycle in C 2 Trias (A,A). This theorem allows one to think of an infinitesimal as a cohomology class, instead of just a cochain.

Rigidity.
By the trivial deformation of A, we mean the deformation Θ t = ( , ,⊥)t 0 . We say that A is rigid if every deformation of A is equivalent to the trivial deformation.
The following result will lead to a cohomological criterion for rigidity.
Proposition 4.2. Let Θ t = θ 0 + θ l t l + θ l+1 t l+1 + ··· , with θ i = (λ i ,ρ i ,μ i ), be a deformation of A for some l ≥ 1 in which θ l is a 2-coboundary in C 2 Trias (A,A). Then there exists a formal isomorphism of the form Φ t = Id A +φt l such that the deformation defined by Θ t = ∞ i=0 θ i t i := Φ t Θ t Φ −1 t satisfies θ i = 0 for 1 ≤ i ≤ l. Combining Theorem 4.1 and Proposition 4.2, we obtain the following cohomological criterion for rigidity.  Next we want to obtain a cohomological criterion for the existence of a deformation with a prescribed 2-cocycle as its infinitesimal.
If θ 1 ∈ C 2 Trias (A,A) is a 2-cocycle, then θ 0 + θ 1 t is a deformation of order 1. Thus, given a 2-cocycle θ 1 , in order to determine the existence of a deformation with θ 1 as its infinitesimal, it suffices to determine the obstruction to extend a deformation of order N to one of order N + 1 for N ≥ 1.

4.5.
Obstructions. Fix a deformation Θ t = N i=0 θ i t i , with θ i = (λ i ,ρ i ,μ i ), of order N < ∞ as above. Define a 3-cochain Ob Θ ∈ C 3 Trias (A,A) as follows. Let x, y,z be elements of A Lemma 4.5. The element Ob Θ ∈ C 3 Trias (A,A) is a 3-cocycle. The proof of the lemma is a long but elementary computation. The following result is independent of Lemma 4.5.
Theorem 4.6. Let Θ t = N i=0 θ i t i , with θ i = (λ i ,ρ i ,μ i ), be a deformation of order N < ∞ of A and let θ N+1 be an element of C 2 Trias (A,A). Define the polynomial Θ t := Θ t + θ N+1 t N+1 . Then Θ t is a deformation of order N + 1 if and only if Ob Θ = δ 2 θ N+1 .
Combining Theorem 4.6 with Lemma 4.5, one concludes that the obstruction to extend a deformation Θ t of order N to one of order N + 1 is the cohomology class of Ob Θ , which lies in H 3 Trias (A,A). Therefore, all these obstructions vanish if the cohomology module H 3 Trias (A,A) is trivial.