Deformations of ternary algebras

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. 2000 MSC: 17A40, 20N10, 16E40, 14D15, 55N20 Introduction 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. 1 Definitions Let V be a vector space over k, an algebraically closed field of characteristic zero. A ternary operation on V is a linear map m : V ⊗ V ⊗ V −→ V . Assume that V is n-dimensional with n finite and let B = {e1, ..., en} be a basis of V , the ternary operation m is completely determined by its structure constants Cs ijk defined by 42 H. Ataguema and A. Makhlouf m(ei ⊗ ej ⊗ ek) = n ∑


Introduction
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.

Definitions
Let V be a vector space over k, an algebraically closed field of characteristic zero. A ternary operation on V is a linear map m : V ⊗ V ⊗ V −→ V . Assume that V is n-dimensional with n finite and let B = {e 1 , ..., e n } be a basis of V , the ternary operation m is completely determined by its structure constants C s ijk defined by m(e i ⊗ e j ⊗ e k ) = n s=1 C s ijk e s (1.1)

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 The space T <2> (V ) provided with the ternary operation induced by triple concatenation given by 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 S 2n+1 on V ⊗2n+1 by (V ⊗2n+1 ) S 2n+1 . 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.
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: 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 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.

Algebraic varieties of ternary algebras
Let V be n-dimensional k-vector space with n finite and {e 1 , . . . , e n } 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 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 i 1 , . . . , i 6 = 1, . . . , n. This set forms a quadratic algebraic variety embedded in k n 4 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 k n 4 and is determined by the following polynomial system: where i 1 , . . . , i 6 = 1, . . . , n.

Algebraic varieties of ternary Lie algebras
The set of n-dimensional Lie ternary algebras denoted by £ 3 n is an algebraic variety embedded in k n 2 (n−1)(n−2) . 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 GL n (k) on varieties of ternary algebras
The action of the group GL n (k) on an algebraic variety of ternary algebras Υ, where Υ indicates tAss n , pAss n or £ 3 n , is defined as follows: Let m ∈ Υ, the orbit of m is denoted by θ(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, The orbit θ(m) is provided, when k = C (a complex field), with the structure of a differentiable manifold. In fact, θ(m) is image through the action of the Lie group GL n (k) of the point m, considered as a point of The Zariski tangent space to Υ 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.

Operads of ternary 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 k-vector spaces denoted V ect k is an associative algebra (resp. unital associative algebra) in the monoidal category (S − mod, ⊗). More explicitly, a k-linear operad is a collection of n-dimensional vector spaces P(n) provided with actions of the symmetric groups S n and a distinguished element I for P(1) (the unit) and compositions such that for µ ∈ P(k), ν ∈ P(n), and for all k, n ∈ N and i = 1, . . . , k one has satisfying the following axioms.
• Equivariance: compatibility of the symmetric group action with compositions. By setting the permutations π ∈ S k and ρ ∈ S n define σ = π • i ρ ∈ S k+n−1 . The equivariance is can be expressed by • Associativity of the compositions: for all λ ∈ P(l), µ ∈ P(k), ν ∈ P(n), we have • The unit I is defined by for all µ ∈ P(k) and i = 1, . . . , k. In this case the operad is unital.
The ternary operations lead to the following operads: • pAss ( [17] generalized this concept for the algebraic operads. Let P 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].

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. In this theory the scalar's field is extended to the power series ring. A more general approach was developed by Fialowski and her collaborators, following Schlessinger where a commutative algebra is taken instead of power series ring in one variable. The fundamental results of Gerstenhaber's theory connect deformation theory with the suitable cohomology groups. There is no general cohomology theory. Every structure has a cohomology, a Hochschild cohomology for associative algebras, Chevalley-Eilenberg cohomology for Lie algebras etc. In the following we define the concept of deformation for any ternary operation and define the suitable 1 and 2 cohomology groups for partially associative ternary algebras adapted to formal deformation.
Let V be a vector space over a field k and m 0 be a ternary operation on V .
may be considered by using the k[[t]]-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 where m i ∈ Hom(V ⊗ V ⊗ V, V ).

Deformations of partially associative ternary algebras
The deformation of a partially associative ternary algebra is determined by the deformation of the ternary operation.
defined by satisfying the following condition We call the condition (4.4) the deformation equation of partially associative ternary algebra τ 0 .

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 m t = i≥0 m i t i and m t = i≥0 m i t i of m 0 (m 0 = m 0 ), we say that they are equivalent if there is a formal iso- ]-linear map that may be written in the form A deformation m t of m 0 is said to be trivial if and only if m t is equivalent to m 0 .
The condition (4.9) may be written By identification of coefficients, one obtains that the constant coefficients are identical m 0 = m 0 because Φ 0 = Id 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 τ = (V, m 0 ) be a partially associative ternary algebra on a k-vector space V.
Definition 4.5. We call ternary p-cochain a linear map ϕ : 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 τ the map Definition 4.7. We call ternary 2-coboundary operator of partially associative ternary algebra τ the map 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 δ 2 • δ 1 = 0.
The cohomology spaces relative to these coboundary operators are The space of 2-coboundaries of τ is The space of 2-cocycles of τ is Remark 4.3. One has B 2 (τ, τ )⊂Z 2 (τ, τ ), because δ 2 • δ 1 = 0. Note also that Z 1 (τ, τ ) gives the space of derivations of a ternary algebra τ , denoted Der(τ ). 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 m t be a deformation of a partially associative ternary algebra τ 0 = (V, m 0 ).
By using the definition of 2-coboundaries and by gathering the first and the last term, the deformation equation (4.4) may be written  Proof. We saw in the previous section that if two deformations m and m are equivalent then m 1 = m 1 + δΦ 1 . Recall that two elements are cohomologous if there difference is a coboundary. Thus, δm 1 = 0 =⇒ δm 1 = δ(m 1 + δΦ 1 ) = δm 1 + δ(δΦ 1 ) = 0 which end the proof.

Proposition 4.2.
There is, over k[[t]]/t 2 , a one to one correspondence between the elements of H 2 (τ 0 , τ 0 ) and the infinitesimal deformation Proof. The proof follows from a direct calculation.
The previous theorem leads to the following fundamental corollary. 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.

Degeneration of ternary algebras
The concept of degeneration (also called contraction) of algebras appeared first in the physics literature (Segal 1951, Inönü 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, Inönü, 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 τ = (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.τ is defined by Definition 5.1. Let τ 1 and τ 0 be two ternary algebras on a k-vector space V and f t be a one parameter continuous family of endomorphism of V such that One supposes that f t is invertible for t = 0. If the limit f t .τ 1 exists when t → 0 and is equal to τ 0 , where τ 0 belongs to the Zariski closure θ(τ 1 ) 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.
Remark 5.1. The converse is in general false.

Classifications
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 m(e 1 ⊗ e 1 ⊗ e 1 ) = e 2 (6.1) 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].