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**Hammimi ATAGUEMA ^{*} and Abdenacer MAKHLOUF**

University of Haute Alsace, Laboratoire de Math´ematiques, Informatique et Applications, 4 rue des Fr`eres Lumi`ere, F-68093 Mulhouse, France

- *Corresponding Author:
- Hammimi ATAGUEMA
[email protected] and [email protected]

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**Received date:** December 5, 2006; **Revised date:** January 26, 2007

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

The aim of this paper is to extend to ternary algebras the classical theory of formal deformations of algebras introduced by Gerstenhaber. The associativity of ternary algebras is available in two forms, totally associative case or partially associative case. To any partially associative algebra corresponds by anti-commutation a ternary Lie algebra. In this work, we summarize the principal definitions and properties as well as classification in dimension 2 of these algebras. Then we focuss ourselves on the partially associative ternary algebras, we construct the first groups of a cohomolgy adapted to formal deformations and then we work out a theory of formal deformation in a way similar to the binary algebras.

We are concerned in this work by certain ternary algebraic structures which appear more or less naturally in various domains of theoretical and mathematical physics. Indeed, theoretical physics progress of quantum mechanics and the discovery of the mechanics of Nambu, as well as works of S. Okubo gave impulse to a significant development involving ternary structures. The quark model proposed by Y. Nambu in 1973 [26,35] represents a particular case of ternary system of algebras and was known since under the name of “Nambu Mechanics”. The cubic matrices and a generalization of the determinant, called the “hyperdeterminant”, also illustrates the ternary algebras. It was first introduced by Cayley in 1840, then found again and generalized by Kapranov, Gelfand and Zelevinskii in 1990 [26] . Other ternary and cubic algebras have been studied by Lawrence, Dabrowski, Nesti and Siniscalco, Plyushchay, Rausch de Traubenberg, and other authors. The ternary operation gives rise to partially associative, totally associative or Lie ternary algebras, one direction of our work is devoted to classification up to isomorphism of ternary algebras in small dimensions. However, in the second part we are interested in the deformation and degeneration of ternary algebras. These two fundamental concepts are useful in order to have more information about the ternary algebra or to construct new algebras starting from a given algebra. The study of deformations of the ternary algebras leads to the cohomology study of these algebras.

The main purpose of our work is to build and study a cohomology for the partially associative ternary algebras, totally associative ternary algebras and the ternary Lie algebras. In this paper we focuss on partially associative ternary algebras and express the first cohomology groups adapted to formal deformation.

Let V be a vector space over an algebraically closed field of characteristic zero. A ternary operation on V is a linear map Assume that V is n-dimensional with n finite and let be a basis of V , the ternary operation m is completely determined by its structure constants defined by

(1.1)

**Definition 1.1. **A totally associative ternary algebra is given by a -vector space V and a
ternary operation m satisfying

for all

**Definition 1.2. **A partially associative ternary algebra is given by a -vector space V and a
ternary operation m satisfying

If there is no ambiguity on the ternary operation and in order to simplify the writing, one will denote

**Remark 1.1. **Let (V, ·) be a bilinear associative algebra. Then, the ternary operation, defined
by

(1.4)

determines on the vector space V a structure of totally associative ternary algebra which is not partially associative.

**Free associative ternary algebras.**

In this paragraph, we give the construction of the free totally associative ternary algebra and the free partially associative ternary algebra on a finite-dimensional vector space V . This construction is a particular case of the k-ary algebras studied by Gnedbaye [18]. The free totally associative ternary algebra is given by the following

**Proposition 1.1. ***Let V be a finite-dimensional vector space. We set*

(1.5)

The space T<2>(V ) provided with the ternary operation induced by triple concatenation given by

(1.6)

where defines the free totally associative ternary algebra on the vector space V . This ternary algebra is denoted by tAss<2>(V ).

The free partially associative ternary algebra results from the previous proposition. One denotes the space of the symmetric group According to the previous proposition, one may consider the symmetric and free totally associative ternary algebra which we denote by stAss<2>(V ) on a vector space V , described by the space

(1.7)

provided with a ternary operation induced by triple concatenation.

**Proposition 1.2. ***The construction of the free partially associative ternary algebra is represented
by a sequence of vector spaces defined by the relation*

*We denote the partially associative free ternary algebra on V by pAss<2>(V )*

**Symmetric ternary algebras and ternary Lie algebras**

In what follows the ternary operation is denoted by [x, y, z] for all x, y, z in V.

**Definition 1.3.** A ternary algebra on a vector space V is said symmetric if

(1.9)

**Definition 1.4. **A ternary operation is said commutative if

(1.10)

**Remark 1.2.** A symmetric ternary operation is commutative.

**Definition 1.5.** An antisymmetric ternary algebra is characterized by the relation

(1.11)

**Definition 1.6.** A ternary Lie algebra is an antisymmetric ternary operation satisfying the
generalized Jacobi condition:

(1.12)

The concept of Lie algebra was generalized to Lie n-ary algebras by V. Fillipov in 1985 [13] (in Russian) and in addition by Ph. Hanlon and M. Wachs in 1990 [22], see also [33].

**Remark 1.3.** The generalized Jacobi condition is still written

**Remark 1.4.** One can easily check that for a ternary Lie algebra

As in the binary case, there is a functor which makes correspond to any partially associative ternary algebra a ternary Lie algebra.

**Proposition 1.3. ***To any partially associative ternary algebra on a vector space V with ternary
operation m, one associates a ternary Lie algebra on V defined by the bracket*

(1.14)

**Proof. **It is clear that the bracket is antisymmetric and direct calculation shows that the generalized
Jacobi condition is satisfied.

Let V be n-dimensional -vector space with n finite and be a basis of V . Let m be
a ternary operation on V . The multilinearity of m implies that for all *e _{i}, e_{j} , e_{k}* one has

(2.1)

Consequently, the ternary operation is completely determined by the set of structure constants:

(2.2)

**Algebraic varieties of totally associative ternary algebras**

The set of n-dimensional totally associative ternary algebras is determined by the following system of algebraic polynomial equations

(2.3)

where This set forms a quadratic algebraic variety embedded in and is
denoted by tAss_{n}.

**Algebraic varieties of partially associative ternary algebras**

The set of n-dimensional partially associative ternary algebras is denoted by *pAss _{n}*., it forms an
algebraic variety embedded in and is determined by the following polynomial system:

(2.4)

where

**Algebraic varieties of ternary Lie algebras**

The set of n-dimensional Lie ternary algebras denoted by is an algebraic variety embedded in The variety of ternary Lie algebras is determined by the following polynomial system :

where i_{1}, . . . , i_{6} take values 1 to n and

**Action of on varieties of ternary algebras**

The action of the group on an algebraic variety of ternary algebras , where indicates

Thus

(2.5)

Let the orbit of m is denoted by (m) and defined by

(2.6)

In other words,

(2.7)

The orbits are in correspondence with the isomorphism classes of n-dimensional ternary algebras. The stabilizer subgroup of m,

is *Aut(m)*, the automorphisms group of m.

The orbit (m) is identified with the homogeneous space Thus,

The orbit (m) is provided, when (a complex field), with the structure of a differentiable manifold. In fact, (m) is image through the action of the Lie group of the point m, considered as a point of Hom

The Zariski tangent space to at the point m corresponds to Z^{2}(m,m) and the tangent
space to the orbit corresponds to B^{2}(m,m), the cohomology group Z^{2}(m,m) and B^{2}(m,m) are
described in section (4.2).

An algebras whose orbit is open for the topology of Zariski is called rigid and constitute an interesting class for the geometrical study of algebraic varieties [21]. Indeed, the Zariski closure of an open orbit constitutes an irreducible component of the algebraic variety.

The last section of this paper is devoted to classifications up to isomorphism of 2-dimensional partially associative ternary algebras, totally associative ternary algebras and ternary Lie algebras.

An operad is an algebraic tool which provides a modeling of operations with n variables on a certain type of algebras, such that (Lie algebras, commutative algebras, associative algebras, partially associative ternary algebras and totally associative ternary algebras etc).

The operads of the binary algebras were studied by many authors [14, 28] and those of the ternary algebras by Gnedbaye [19].

An operad (resp. unital operad) on a sequence of -vector spaces denoted is an associative
algebra (resp. unital associative algebra) in the monoidal category More
explicitly, - linear operad is a collection of n-dimensional vector spaces provided with actions of the symmetric groups *S _{n}* and a distinguished element I for (the unit) and
compositions

such that for

satisfying the following axioms.

• Equivariance: compatibility of the symmetric group action with compositions. By setting

the permutations and The equivariance is can be expressed by

• Associativity of the compositions: for all we have

(3.1)

(3.2)

• The unit I is defined by

(3.3)

for all and i = 1, . . . , k. In this case the operad is unital. For the examples see [9], [28].

• pAss^{(3)}: operad of partially associative ternary algebras,

• tAss^{(3)}: operad of totally associative ternary algebras,

• Lie^{(3)}: operad of ternary Lie algebras,

• tAssSym^{(3)}: operad of totally associative symmetric ternary algebras.

The concept of Koszul duality for the associative algebras is an algebraic theory developed in the seventies by S. Priddy. Later in 1994, V. Ginzburg et M. M. Kapranov in their article [17] generalized this concept for the algebraic operads.

Let be an operad, one denotes by P! the dual operad within the meaning of Koszul duality. For the ternary algebras there are the following results [19].

**Theorem 3.1.** One has

**Formal deformations of ternary algebras**

In this section we extend to ternary algebras the formal deformation theory introduced in 1964 by Gerstenhaber [15] for associative algebras, and in 1967 by Nijenhuis and Richardson for Lie algebras [36]. The formal deformations of mathematical objects is one of the oldest technics used by mathematicians. The deformations give more information about the structure of the object, for example one can try to see which properties are stable under deformation.

Let V be a vector space over a field and m_{0} be a ternary operation on V . Let be
the power series ring in one variable t and coefficients in and be the extension of V by
extending the coefficients domain from to . Then is a -module and when V is
finite-dimensional and we have

(4.1)

One notes that V is a submodule of V [[t]]. The extension of ternary operations to V [[t]],

may be considered by using the -linearity as a map

**Definition 4.1.** A formal deformation of ternary operation m_{0} on V is given by a ternary
operation m_{t} defined by

**Deformations of partially associative ternary algebras**

The deformation of a partially associative ternary algebra is determined by the deformation of the ternary operation.

**Definition 4.2. **Let V be a -vector space and be a partially associative ternary
algebra. A deformation of on V is given by a linear map

defined by

satisfying the following condition

We call the condition (4.4) the deformation equation of partially associative ternary algebra .

**Deformation equation**

In the following, we study the equation (4.4) and thus characterize the deformations of partially associative ternary algebras. The equation may be written

or

**Definition 4.3. **We call ternary partial associator the map

defined for all by

**Remark 4.1.** This associator can be generalized in the following way. Let

By using the ternary partial associator, the deformation equation may be written as follows

(4.8)

This equation is equivalent to the following infinite system:

In particular,

• this corresponds to the partial associativity of m0,

**Equivalent and trivial deformations**

In this paragraph, we characterize the equivalent and trivial deformations of a partially associative ternary algebras.

**Definition 4.4.** Given two deformations of a partially associative ternary algebra and of we say that they are equivalent if there is a formal isomorphism which is a- linear map that may be written in the form

such that

(4.9)

A deformation mt of m0 is said to be trivial if and only if mt is equivalent to m0.

The condition (4.9) may be written.

(4.10)

which is equivalent to

or

By identification of coefficients, one obtains that the constant coefficients are identical

and for coefficients of t one has

from which it follows

Consequently,

**Cohomological approach of a partially associative ternary algebras**

The study of the deformation equation leads us to certain elements of the cohomology of partially associative ternary algebras. The existence of this cohomology is ensured by the operadic structure of the ternary algebras.

Let be a partially associative ternary algebra on a -vector space V.

**Definition 4.5.** We call ternary p-cochain a linear map The set of p-cochains
on V is

The 1-coboundary and 2-coboundary operators for partially associative ternary algebras are defined as follows

Definition 4.6. We call ternary 1-coboundary of partially associative ternary algebra the τ map

defined by

**Definition 4.7. **We call ternary 2-coboundary operator of partially associative ternary algebra τ
the map

defined by

Remark 4.2. The operator δ^{2} can also be defined by

where • is the operation defined in the paragraph (4.1.1). Note that

The cohomology spaces relative to these coboundary operators are

**Definition 4.8. **The space of 1-cocycles of is

The space of 2-coboundaries of τ is

The space of 2-cocycles of τ is

**Remark 4.3.** One has because Note also that gives the
space of derivations of a ternary algebra

**Definition 4.9. **We call the p^{th} cohomology group of the partially associative ternary algebra τ the quotient

We characterize now deformations in terms of cohomology. Let mt be a deformation of a partially associative ternary algebra

By using the definition of 2-coboundaries and by gathering the first and the last term, the deformation equation (4.4) may be written

**Lemma 4.1. **The first term m_{1} in the deformation mt is a 2-cocycle of the cohomology of the
partially associative ternary algebra τ_{0}.

**Proof. **Take k = 1 in the deformation equation (4.4).

**Definition 4.10. **Let be a partially associative ternary algebra and m1 be an
ePlement of The cocycle m_{1} is said integrab/le if there exists a deformation mt =

**Proposition 4.1.** *Let mt be a deformation of a partially associative ternary algebra. The
integrability of m1 depends only on its cohomology class.*

**Proof. **We saw in the previous section that if two deformations m and are equivalent then Recall that two elements are cohomologous if there difference is a coboundary.
Thus,

which end the proof.

**Proposition 4.2.** There is, over a one to one correspondence between the elements of and the infinitesimal deformation

(4.12)

**Proof.** The proof follows from a direct calculation.

Assume now that

such that

The deformation equation implies which means If further with then using a formal morphism we obtain that the deformation m_{t} is equivalent to the deformation given for all by

and again

Thus, we have the following theorem.

**Theorem 4.1.** *Let be a partially associative ternary algebra and mt be a one
parameter family of deformations of m0. Then mt is equivalent to*

The previous theorem leads to the following fundamental corollary.

**Corollary 4.1.** *If then all deformations of τ0 are equivalent to a trivial deformation.*

**Remark 4.4.** A partially associative ternary algebra such that any deformation is equivalent
to trivial deformation is called rigid. The previous result gives a sufficient condition for the
rigidity. Recall that rigid ternary algebras has a great interest in the study of algebraic varieties
of ternary algebras. The Zariski closure of the orbit of a rigid ternary algebra gives an irreducible
component.

The concept of degeneration (also called contraction) of algebras appeared first in the physics literature (Segal 1951, In¨on¨u and Wigner 1953) [23] to describe the classical mechanics as a degeneration of quantum mechanics. Later in 1961, Saletan [37], generalized this concept and gave a necessary and sufficient condition for the existence of Lie algebra contractions. The concepts of degenerations and deformations give more information about the structure of the object and helps, in general, to construct new algebras. They are also used to study algebraic varieties of algebras (associative, Lie, . . . ) see (Gabriel, Gerstenhaber, Mazzola, Richardson, Makhlouf, Nijenhuis, Goze, In¨on¨u, Saletan, . . . ) and in the theory of quantum groups by Celegheni, Giachetti, Sorace et Tarlini [7,8].

The aim of this section is to introduce the concept of degeneration of ternary algebras. Let = (V,m) be a ternary algebra and Isom(V ) be the set of invertible maps of End(V ). Recall that the action of Isom(V ) on the ternary algebras denoted f. is defined by

**Definition 5.1.** Let be two ternary algebras on a -vector space V and ft be a one
parameter continuous family of endomorphism of V such that

One supposes that ft is invertible for One supposes that f_{t} is invertible for . If the limit exists when t → 0 and is equal to τ_{0}, where τ_{0} belongs to the Zariski closure of the orbit of τ_{1}, we say that τ_{0} is a degeneration of τ_{1}.

The following proposition gives the connection between degeneration and deformation.

**Proposition 5.1. ***Let τ _{0} be a formal degeneration of τ_{1}, then τ_{1} is a formal deformation of τ_{0}.*

**Proof.** In fact, if is the degeneration of is a deformation of
*τ _{0}*.

**Remark 5.1. **The converse is in general false.

In this section, we establish the classification up to isomorphism of 2-dimensional partially associative ternary algebras and totally associative ternary algebras. Ternary Lie algebras with dimension lower than 5 correspond to antisymmetric ternary operations.

**Proposition 6.1. ***Any 2-dimensional partially associative ternary algebra is trivial or isomorphic
to the ternary algebras defined by the following non trivial product*

**Proposition 6.2. **Any nontrivial 2-dimensional totally associative ternary algebra is isomorphic
to one of the following totally associative ternary algebras:

In the case of ternary Lie algebras, we have

**Proposition 6.3.** Any n-dimensional antisymmetric ternary operation with n ≤ 4 is a ternary
Lie algebra.

These results of classification are obtained either by a direct reasoning or using a formal computation software [29].

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