Abstract
We study partially and totally associative ternary algebras of first and second kind. Assuming the vector space underlying a ternary algebra to be a topological space and a triple product to be continuous mapping we consider the trivial vector bundle over a ternary algebra and show that a triple product induces a structure of binary algebra in each fiber of this vector bundle. We find the sufficient and necessary condition for a ternary multiplication to induce a structure of associative binary algebra in each fiber of this vector bundle. Given two modules over the algebras with involutions we construct a ternary algebra which is used as a building block for a Lie algebra. We construct ternary algebras of cubic matrices and find four different totally associative ternary multiplications of second kind of cubic matrices. It is proved that these are the only totally associative ternary multiplications of second kind in the case of cubic matrices. We describe a ternary analog of Lie algebra of cubic matrices of second order which is based on a notion of j-commutator and find all commutation relations of generators of this algebra.
Highlights
Assuming the vector space underlying a ternary algebra to be a topological space and a triple product to be continuous mapping, we consider the trivial vector bundle over a ternary algebra and show that a triple product induces a structure of binary algebra in each fiber of this vector bundle
A ternary algebra or triple system is a vector space A endowed with a ternary law of composition τ : A × A × A → A which is a linear mapping with respect to each of its arguments, and we will call this mapping a ternary multiplication or triple product of a ternary algebra A
This method was extended to super Lie algebras in [7] and later was applied by the same authors in [8] to construct a gauge field theory by introducing fundamental fields associated with the elements of a ternary algebra
Summary
A ternary algebra or triple system is a vector space A endowed with a ternary law of composition τ : A × A × A → A which is a linear mapping with respect to each of its arguments, and we will call this mapping a ternary multiplication or triple product of a ternary algebra A. The notion of a j-skew-symmetric form can be assumed as a basis for a ternary analog of Grassmann, Clifford, and Lie algebras These ternary structures were developed in [1, 2, 13, 15] and applied to construct a ternary analog of supersymmetry algebra in [3, 11, 12, 14]. The sufficient and necessary condition a ternary multiplication must satisfy in order to induce a structure of associative binary algebra in each fiber is given in terms of the vector bundle over a ternary algebra. Unital associative algebras and homomorphisms, we show that this structure allows to construct a large class of ternary algebras including a ternary algebra of rectangular matrices and ternary algebras of sections of a vector bundle over a smooth finite-dimensional manifold.
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