Notes on Cohomologies of Ternary Algebras of Associative Type

The paper is dedicated to study 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 [46]), as well as a work of S. Okubo on Yang-Baxter equation (see [47]) gave impulse to a significant 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 [28]) and Sokolov in 1972 (see [52]). 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 [1,29,30,31,32,33]). The concept of ternary algebras was introduced first by Jacobson (see [27]). In connection with problems from Jordan theory and quantum mechanics, he defined 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] = xy-yx. 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 [41]). 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 S 5 (resp. S 2n-1 ) instead of S 3 (see [23] [45]), 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 first in Nambu mechanics (see [46]), see also [53] for the algebraic formulation of the Nambu mechanics. The abstract definitions 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 [14,15,24,36,54,54,55]. In another hand, ternary algebras or more generally n-ary algebras of associative type were studied by Carlsson, Lister, Loos (see [6,38,40]). The typical and founding example of totally associative ternary algebra was introduced by Hestenes (see [25]) defined on the linear space of rectangular matrices A, B,C ∈ M m,n 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 M m,n as it is linear on the first 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 [18,19]), see also [20,26]. 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], see [42] for a review. We built a 1-cohomology and 2-cohomology of partially associative ternary algebras fitting with the deformation theory. The generalized Poisson structures and n-ary Poisson brackets were discussed in [9,10,22,45]. While the quantization problem was considered in [11,12]. Further generalizations and related works could be found in [4,5,21,48,49,50]. In this paper we summarize in the first Section the definitions of ternary algebras of associative type and Lie type with examples, and recall the basic settings of homological algebra. Section 2 is devoted to study the cohomology of partially associative ternary algebras with values in the algebra. We provide the first and the second coboundary operators and show that their extension to a 3-coboundary does not exist. This shows that the operad of partially ternary algebras is not a Koszul operad. In Section 3, we consider weak totally associative ternary algebras for which we construct a p-coboundary operator extending, to any p, the 2-coboundary operators already defined by Takhtajan (see [54]). In Section 4, 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 algeb

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