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Journal of Generalized Lie Theory and Applications
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Notes on cohomologies of ternary algebras of associative type

H. ATAGUEMA and A. MAKHLOUF

Laboratoire de Mathématiques, Informatique et Applications, Université de Haute Alsace, 4 rue des Fréres Lumiére F-68093 Mulhouse, France E-mails: [email protected], [email protected]

Received Date: January 15, 2009

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Abstract

The aim of this paper is to investigate the cohomologies for ternary algebras of associative type.We study in particular the cases of partially associative ternary algebras and weak totally associative ternary algebras. Also, we consider the Takhtajan's construction, which was used to construct a cohomology of ternary Nambu-Lie algebras using Chevalley-Eilenberg cohomology of Lie algebras, and discuss it in the case of ternary algebras of associative type. One of the main results of this paper states that a usual deformation cohomology does not exist for partially associative ternary algebras which implies that their operad is not a Koszul operad.

Introduction

The paper is dedicated to studying cohomologies adapted to deformation theory of ternary algebraic structures appearing more or less naturally in various domains of theoretical and mathematical physics and data processing. Indeed, theoretical physics progress of quantum mechanics and the discovery in 1973 of the Nambu mechanics (see [48]), as well as a work of Okubo on Yang-Baxter equation (see [49]), gave impulse to a signi cant development on ternary algebras and more generally n-ary algebras. The ternary operations, in particular cubic matrices, were already introduced in the nineteenth century by Cayley. The cubic matrices were considered again and generalized by Kapranov, Gelfand, and Zelevinskii in 1994 (see [30]) and Sokolov in 1972 (see [54]). Another recent motivation to study ternary operation comes from string theory and M-branes where appeared naturally a so-called Bagger-Lambert algebra (see [3]). For other physical applications, see [1,31-35].

The concept of ternary algebras was introduced rst by Jacobson (see [29]). In connection with problems from Jordan theory and quantum mechanics, he de ned the Lie triple systems. A Lie triple system consists of a space of linear operators on vector space V that is closed under the ternary bracket [x, y, z]T = [[x, y], z], where [x, y] = xyyx. Equivalently, a Lie triple system may be viewed as a subspace of the Lie algebra closed relative to the ternary product. A Lie triple system arose also in the study of symmetric spaces (see [43]). We distinguish two kinds of generalization of binary Lie algebras: ternary Lie algebras (resp., n- ary Lie algebras) in which the Jacobi identity is generalized by considering a cyclic summation over S5 (resp., S2n–1) instead of S3 (see [25,47]), and ternary Nambu algebras (resp., n-ary Nambu algebras) in which the fundamental identity generalizes the fact that the adjoint maps are derivations. The fundamental identity appeared rst in Nambu mechanics (see [48]), see also [55] for the algebraic formulation of the Nambu mechanics. The abstract de nitions of ternary and more generally n-ary Nambu algebras or n-ary Nambu-Lie algebras (when the bracket is skew-symmetric) were given by Fillipov in 1985 (see [13] in Russian). While the n-ary Leibniz algebras were introduced and studied in [8]. For deformation theory and cohomologies of ternary algebras of Lie type, we refer to [14,15,26,38,56,57].

In another hand, ternary algebras or more generally n-ary algebras of associative type were studied by Carlsson, Lister, and Loos (see [6,40,42]). The typical and founding example of totally associative ternary algebra was introduced by Hestenes (see [27]) de ned on the linear space of rectangular matrices Equation with complex entries by AB*C where B* is the conjugate transpose matrix of B. This operation is strictly speaking not a ternary algebra product on Mm,n as it is linear on the rst and the third arguments but conjugate-linear on the second argument. The ternary operation of associative type leads to two principal classes: totally associative ternary algebras and partially associative ternary algebras. Also they admit some variants. The totally associative ternary algebras are also called associative triple systems. The operads of n-ary algebras were studied by Gnedbaye (see [20,21]), see also [22,28]. The cohomology of totally associative ternary algebras was studied by Carlsson through the embedding (see [7]). In [2], we extended to ternary algebras of associative type, the 1-parameter formal deformations introduced by Gerstenhaber [16-18], see [44] for a review. We built a 1-cohomology and 2-cohomology of partially associative ternary algebras tting with the deformation theory.

The generalized Poisson structures and n-ary Poisson brackets were discussed in [9,10,24,47]. While the quantization problem was considered in [11,12]. Further generalizations and related works could be found in [4,5,23,50-52].

In this paper, we summarize in Section 2 the de nitions of ternary algebras of associative type and Lie type with examples, and recall the basic settings of homological algebra. Section 3 is devoted to study the cohomology of partially associative ternary algebras with values in the algebra. We provide the rst and the second coboundary operators and show that their extension to a usual 3-coboundary does not exist. This shows that the operad of partially ternary algebras is not a Koszul operad. In Section 4, we consider weak totally associative ternary algebras for which we construct a p-coboundary operator extending, to any p, the 2-coboundary operators already de ned by Takhtajan (see [56]). In Section 5, we discuss Takhtajan's construction for ternary algebras of associative type. The process was introduced by Takhtajan to construct a cohomology for ternary algebras of Lie type starting from a cohomology of binary algebras. It was used to derive the cohomology of ternary Nambu-Lie algebras from the Chevalley-Eilenberg cohomology of Lie algebras. We show that a usual cohomology of ternary algebras of partially associative type cannot be constructed from binary algebras of associative type. We also show in Section 6, that the skew-associative binary algebras do not carry a usual cohomology tting with deformation theory and therefore their operad is not Koszul as well.

Generalities

In this section, we summarize the de nitions of di erent ternary algebra structures of associative type and Lie type and provide some examples, and then give general settings for cohomology theories.

Ternary algebra structures

Let Equation be an algebraically closed eld of characteristic zero and let V be a Equation-vector space. A ternary operation on V is a linear map Equation or a trilinear map m :Equation. If V is n-dimensional vector space and Equation is a basis of V , the ternary operation m is completely determined by its structure constants Equation, where

Equation

A ternary operation is said to be symmetric (resp., skew-symmetric) if

Equation

and, respectively,

Equation

where Sgn(σ) denotes the signature of the permutation Equation.

We have the following type of \associative" ternary operations.

Definition 2.1. A totally associative ternary algebra is given by a Equation-vector space V and a ternary operation m satisfying, for every Equation,

Equation

Example 2.2. Let Equation be a basis of a 2-dimensional space V =Equation2, ternary operation on V given by

Equation

de nes a totally associative ternary algebra.

Definition 2.3. A weak totally associative ternary algebra is given by a Equation-vector space V and a ternary operation m satisfying, for every Equation,

Equation

Naturally, every totally associative ternary algebra is a weak totally associative ternary algebra.

Definition 2.4. A partially associative ternary algebra is given by a Equation-vector space V and a ternary operation m satisfying, for every Equation,

Equation

Example 2.5. Let Equation be a basis of a 2-dimensional space V =Equation2, ternary operation on V given by Equation de nes a partially associative ternary algebra.

We introduce in the following some variants of partial total associativity of ternary operations.

Definition 2.6. An alternate partially associative ternary algebra of rst kind is given by a Equation-vector space V and a ternary operation m satisfying, for every Equation,

Equation

The alternate partially associative ternary algebra is of second kind it satis es

Equation

Remark 2.7. Let (V,·) be a bilinear associative algebra. Then, the ternary operation, de- ned by Equation, determines on the vector space V a structure of totally associative ternary algebra which is not partially associative.

Definition 2.8. A ternary operation m is said to be commutative if

Equation

Remark 2.9. A symmetric ternary operation is commutative.

In the following, we recall the de nitions of ternary algebras of Lie type.

Definition 2.10. A ternary Lie algebras is a skew-symmetric ternary operation [ , , ] on a Equation-vector space V satisfying Equation the following generalized Jacobi condition:

Equation

As in the binary case, there is a functor which makes correspondence to any partially associative ternary algebra a ternary Lie algebra (see [20,21]).

Proposition 2.11. To any partially associative ternary algebra on a vector space V with ternary operation m, one associates a ternary Lie algebra on V defined Equation by the bracket

Equation

There is another kind of ternary algebras of Lie type, they are called ternary Nambu algebra. They appeared naturally in Nambu mechanics which is a generalization of classical mechanics.

Definition 2.12. A ternary Nambu algebra is a ternary bracket on a Equation-vector space V satisfying a so-called fundamental or Filippov identity:

Equation

When the bracket is skew-symmetric, the ternary algebra is called ternary Nambu-Lie algebra.

The Lie triple system is de ned as a vector space V equipped with a ternary bracket that satis es the same fundamental identity as a Nambu-Lie bracket but instead of skewsymmetry, it satis es the condition

Equation

Example 2.13. The polynomial algebra of 3 variables x1, x2, x3, endowed with a ternary operation de ned by the functional Jacobian:

Equation

is a ternary Nambu-Lie algebra.

We have also the following fundamental example.

Example 2.14. Let V = Equation4 be the 4-dimensional oriented Euclidian space over Equation. The bracket of 3 vectors Equation is given by

Equation

where Equation are the coordinates of Equation with respect to orthonormal basisEquation.

Then, (V, [·,·,·]) is a ternary Nambu-Lie algebra.

Remark 2.15. Every ternary Nambu-Lie algebra on Equation4 could be deduced from the previous example (see [14]).

Homological algebra of ternary algebras

The basic concepts of homological algebra are those of a complex and homomorphisms of complexes, de ning the category of complexes (see, e.g., [58]). A chain complex C: is a sequence C = {Cp}p of abelian groups or more generally objects of an abelian category and an indexed set Equation of homomorphismsEquation such that Equation for all p. A chain complex can be considered as a cochain complex by reversing the enumeration Cp = C–p and Equation. A cochain complex C is a sequence of abelian groups and homomorphisms Equation with the property Equationfor all p.

The homomorphisms Equation are called coboundary operators or codifferentials. A cohomology of a cochain complex C is given by the groups Equation.

The elements of Cp are p-cochains, the elements of Equation are p-cocycles, and the elements of Equation are p-coboundaries. Because Equationfor all p, we have Equation for all p. The pth cohomology group is the quotient Equation.

We introduce in the following the p-cochains for a ternary algebra of associative type A = (V, m).

Definition 2.16. We call p-cochain of a ternary algebra A = (V, m) a linear map Equation : Equation. The p-cochains set on V is

Equation

Remark 2.17. The set Cp(A,A) is an abelian group.

We de ne a circle operation on cochains as usual, that is, a map

Equation

such that

Equation

One has a cochain complex for ternary algebras A with values in A if there exists a sequence of abelian groups and homomorphisms Equation such that for all p, Equation.

Cohomology of partially associative ternary algebras

We have studied in [2] deformations of partially associative ternary algebras which are intimately linked to cohomology groups. We have introduced the operators Equation1 and Equation2 which should correspond to a complex of partially associative ternary algebra de ning a deformation cohomology. In the following, we recall the de nitions of Equation1 and Equation2 and show that it is impossible to extend these operators to a usual operator Equation3. As a consequence, we deduce that the operad of partially associative ternary algebras is not Koszul, see [45,19] about Koszulity.

Let A = (V,m) be a partially associative ternary algebra on a Equation-vector space V.

Definition 3.1. We call ternary 1-coboundary operator the map

Equation

de ned by

Equation

Definition 3.2. We call ternary 2-coboundary operator the map

Equation

de ned by

Equation

Remark 3.3. The operator Equation2 can also be de ned by Equation.

Proposition 3.4. We have Equation.

Proof. Let f be a 0-cochain. We compute Equation.

We have for all Equation,

Equation

The cohomology spaces relative to these coboundary operators are as follows.

Definition 3.5. The 1-cocycles space of A is

Equation

The 2-coboundaries space of A is

Equation

The 2-cocycles space of A is

Equation

Remark 3.6. One has Equation, because Equation. Note also that Z1(A, A) corresponds to the derivations space, denoted also by Der(A), of the partially associative ternary algebra A.

Definition 3.7. We call the pth cohomology group (p = 0, 1) of the partially associative ternary algebra A the quotient

Equation

The following proposition shows that we cannot extend the 1-cohomology and 2-cohomology corresponding to the operators Equation1 and Equation2 to a usual 3-cohomology.

Proposition 3.8. Let A = (V, m) be a partially associative ternary algebra. There is no usual 3-cohomology extending the 2-cohomology corresponding to the coboundary operator

Equation

defined for all Equation by

Equation

Proof. We consider a 3-cochain f, that is, a map Equation, and set

Equation

Equation

Equation

Then, we obtain

Equation

Equation

The equation is satis ed for all Equation if and only if a1, . . . , a8 are all equal to 0.

Corollary 3.9. A usual deformation cohomology of partially associative ternary algebras does not exist. Then, the operad of the partially associative ternary algebras pAss(3) is not a Koszul operad.

Remark 3.10. In [28], it is shown that the operad of totally associative ternary algebras is Koszul because it has a Poincare-Birkho -Witt basis. Moreover, its dual, the operad of partially associative ternary algebras, is also Koszul when the operations are in degree one. See also [22] for constructions in this case and the recent preprint [53]. The corollary claims that the operad is not a Koszul operad when the operations are in degree zero.

Remark 3.11. Using the same approach, we can show that the alternate partially associative ternary algebras of rst and second kind do not carry a usual deformation cohomology as well, then their operads are not koszul operads.

Cohomology of weak totally associative ternary algebras

In this section, we generalize to p-cohomology, for all p, the 1-cohomology and 2-cohomology of weak totally associative ternary algebra de ned by Takhtajan (see [56]). Let A = (V, m) be a weak totally associative ternary algebras on a Equation-vector space V.

The 1-coboundary and 2-coboundary operators for weak totally associative ternary algebras A are de ned as follows

Definition 4.1. A 1-coboundary operator of a weak totally associative ternary algebra A = (V, m) is the map

Equation

defined for all Equation by

Equation

A 2-coboundary operator of a weak totally associative ternary algebra A is the map

Equation

defined for all Equation by

Equation

Remark 4.2. One can easily show that Equation. Indeed,

Equation

We introduce for weak associative ternary algebras the following generalized coboundary map.

Definition 4.3. Let f be a (p–1)-cochain of a weak associative ternary algebra A = (V, m) and Equation. We set

Equation

In particular, we have

Equation

Proposition 4.4. We have Equation for all p ≥ 1.

Proof. We have Equation. Assume Equation. We have to show that Equation.

Let Equation be a p-cochain and Equation,

Equation

Then Equation vanishes. Really,

Equation

Equation

Takhtajan's construction

In this section, we aim to extend to ternary algebras of associative type a process introduced by Takhtajan to construct a complex of ternary algebras starting from a complex of binary algebras (see [56]). Let (V, m) be a ternary algebra of a given type. We associate to it a binary algebra on Equation and a map Δ. Assume that Equation is a complex for the ternary algebras and (M, d) is a complex for the binary algebras.

We de ne a map Δ such that Δp associates to any p-cochain on V a p-cochain on W. It is defined by

Equation

such that, for example, Equation One extends this operation to

 

defined, for example, using the remark that Equation, by

Equation

Let us assume that one has a complex (M, d):

Equation

i.e., for all p, Equation,

Consider for any p > 0, the linear maps Equation satisfying

Equation

The equality is well de ned.

Indeed, one has for p ≥ 1,

Equation

Lemma 5.1. Let p > 1. If Equation

Equation

As a consequence of the previous lemma, one may obtain a complex of ternary algebras starting from a complex of binary algebras and a map Δ. This process was used by Takhtajan to construct a cohomology of ternary Nambu algebras using the Chevalley-Eilenberg cohomology of Lie algebras. The binary multiplication used to that end is de ned as follows:

Let (V, [ , , ]) be a ternary Nambu algebra. Set Equation. The multiplication on W is de ned for Equation by

Equation

Takhtajan's construction and ternary algebras of associative type

In the sequel, we show that we cannot derive a cohomology of a partially associative ternary algebra from a cohomology of binary algebras of associative type. A construction is possible in the case of totally associative ternary algebras but the cohomology obtained is the cohomology of weak totally associative ternary algebras described above.

A binary algebra is called of associative type if it is given by a vector space V and a multiplication μ satisfying an identity of the form

Equation

where λ is a scalar element different from zero. In particular, we have associative algebras for λ = –1 and skew-associative algebras for λ = 1. In the last section, we show that the skew-associative algebras cannot carry a usual cohomology adapted to deformation theory.

In the following, we try to adapt the Takhatjan's procedure to ternary algebras of associative type. We set

Equation

In order to check whether μ is a binary operation of associative type, we compute

Equation

and

Equation

The multiplication μ is of associative type if A1 + λA2 = 0, that is,

Equation

If m is a ternary operation which de nes a partially associative ternary algebra of a given type, then A1 + λA2 = 0 if α(1 + λ) = 0 and the coefficients (1,α,λ), (α2, λα, λα2) are proportional.

The rst condition is satis ed when α = 0 or λ = –1. The case α = 0 is impossible.

If λ = –1, the coefficients Equation should be proportional. This is possible only over Equation with Equation. The associativity condition needed must be of one of the following forms:

Equation

In the both cases, one may construct a cohomology of ternary algebras according to Takhtajan's construction and using the Hochschild complex of associative binary multiplication.

Therefore, we have the following proposition.

Proposition 5.2. It is impossible to construct, using Takhtajan's construction, a cohomology of ternary algebras (V, m) which are partially associative (resp., alternate partially associa- tive) starting from a complex of binary algebra of associative type.

Remark 5.3. If the ternary algebra m is totally associative, then the corresponding binary algebra is of associative type if

Equation

which implies that α = 0 and λ = –1

Therefore, using Takhtajan procedure, we can construct a cohomology of totally associative ternary algebras ((V, m) with a binary multiplication μ defined on Equation by

Equation

Let Equation be a p-cochain of the totally associative ternary algebra (V, m). We set Equation. It turns out that in this case, we recover the coboundary map of the weak totally associative ternary algebras discussed in Section 4.

Really, using the Hochschild coboundary of the binary associative algebra (W, μ), we have

Equation

Equation

Then, we set

Equation

and recover the cohomology of weak totally associative ternary algebras de ned above.

On deformation cohomology of skew-associative algebras

In this section, we show that the 1-cohomology and 2-cohomology guided by 1-parameter formal deformations cannot be extended to a usual 3-cohomology. Therefore, the operad of skew-associative binary algebras is not Koszul.

Definition 6.1. A skew-associative binary algebra is given by a Equation-vector space V and a bilinear multiplication μ satisfying, for every Equation,

Equation

The formal deformation theory leads to the following 1-coboundary and 2-coboundary operators for a cohomology of skew-associative binary algebra A = (V, μ) adapted to formal deformation theory. The 1-coboundary operator of A is the map

Equation

de ned by

Equation

The 2-coboundary operator of A the map

Equation

defined by

Equation

One may characterize the operator Equation2 using the following skew-associator map:

Equation

defined by

Equation

We have Equation. Note also thatEquation.

Proposition 6.2. A usual 3-coboundary operator extending the maps Equation1 and Equation2 to a complex of for skew-associative binary algebras does not exist.

Proof. We set the following general form of a usual 3-coboundary operator:

Equation

We consider a 3-cochain f, that is, a map Equation, and a 2-cochain g, that is, a map Equation. We compute Equation and substitute Equation byEquation. Then, we obtain

Equation

Corollary 6.3. A usual deformation cohomology for skew-associative binary algebras does not exist. Then the operad of skew-associative binary algebras is not Koszul.

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