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
Received date: December 5, 2006; Revised date: January 26, 2007
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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  . 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
Definition 1.1. A totally associative ternary algebra is given by a -vector space V and a ternary operation m satisfying
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
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 . The free totally associative ternary algebra is given by the following
Proposition 1.1. Let V be a finite-dimensional vector space. We set
The space T<2>(V ) provided with the ternary operation induced by triple concatenation given by
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
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
Definition 1.4. A ternary operation is said commutative if
Remark 1.2. A symmetric ternary operation is commutative.
Definition 1.5. An antisymmetric ternary algebra is characterized by the relation
Definition 1.6. A ternary Lie algebra is an antisymmetric ternary operation satisfying the generalized Jacobi condition:
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
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 ei, ej , ek one has
Consequently, the ternary operation is completely determined by the set of structure constants:
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
where This set forms a quadratic algebraic variety embedded in and is denoted by tAssn.
Algebraic varieties of partially associative ternary algebras
The set of n-dimensional partially associative ternary algebras is denoted by pAssn., it forms an algebraic variety embedded in and is determined by the following polynomial system:
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 i1, . . . , i6 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
Let the orbit of m is denoted by (m) and defined by
In other words,
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 Z2(m,m) and the tangent space to the orbit corresponds to B2(m,m), the cohomology group Z2(m,m) and B2(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 . 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).
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 Sn 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
• The unit I is defined by
• 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  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 .
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  for associative algebras, and in 1967 by Nijenhuis and Richardson for Lie algebras . 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 m0 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
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 m0 on V is given by a ternary operation mt 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
satisfying the following condition
We call the condition (4.4) the deformation equation of partially associative ternary algebra .
In the following, we study the equation (4.4) and thus characterize the deformations of partially associative ternary algebras. The equation may be written
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
This equation is equivalent to the following infinite system:
• 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
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.
which is equivalent to
By identification of coefficients, one obtains that the constant coefficients are identical
and for coefficients of t one has
from which it follows
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
Definition 4.7. We call ternary 2-coboundary operator of partially associative ternary algebra τ the map
Remark 4.2. The operator δ2 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 pth 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 m1 in the deformation mt is a 2-cocycle of the cohomology of the partially associative ternary algebra τ0.
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 m1 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
Proof. The proof follows from a direct calculation.
Assume now that
The deformation equation implies which means If further with then using a formal morphism we obtain that the deformation mt is equivalent to the deformation given for all by
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)  to describe the classical mechanics as a degeneration of quantum mechanics. Later in 1961, Saletan , 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 ft is invertible for . If the limit exists when t → 0 and is equal to τ0, where τ0 belongs to the Zariski closure of the orbit of τ1, we say that τ0 is a degeneration of τ1.
The following proposition gives the connection between degeneration and deformation.
Proposition 5.1. Let τ0 be a formal degeneration of τ1, then τ1 is a formal deformation of τ0.
Proof. In fact, if is the degeneration of is a deformation of τ0.
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 .