Hopf Structure in Nambu-Lie n-Algebras

We give a definition and study Hopf structures in ternary ( and n-ary) NambuLie algebra. The fundamental concepts of 3-coalgebra, 3-bialgebra and Hopf 3algebra are introduced. Some examples of Hopf structures are analyzed. Mathematics Subject Classification (1991) 70H99, 58F07 1 [email protected] 2 [email protected] In 1973 Yoishira Nambu [1] proposed a generalization of classical Hamiltonian mechanics, using ternary and higher-order brackets ( n-ary brackets or multibrackets). During the past two decades Nambu proposal has been a matter for many investigations [2] [12] and the permanent interest for this issue is related with the recognition of the feasible physical richness and the mathematical beauty of ternary and higher algebraic systems. Recently such an algebraic structure has been analyzed and reformulated by Tachtajan [7] – [9] in an invariant geometrical form. He proposed the notion of Nambu-Lie ”gebra”, which is a generalization of Lie algebras for ternary (in general n-ary) case. The ternary algebra or ”gebra” is a linear space in which conditions of antisymmetry and generalized Jacobi identity are fulfilled. In this letter we investigate some new aspects of the Nambu proposal, connected with Hopf algebra concept. That is, a ternary (and n-ary) Hopf structure is introduced as generalization of the usual Hopf algebra, and some examples are presented. Let us begin with a definition of a ternary (associative) algebra. Definition 1 . A ternary algebra with unit over a commutative ring C is a vector space A together with a way of multiplying three elements a, b, c ∈ A m : A⊗ A⊗ A = A → A, such that m(a⊗ b⊗ c) = abc (1) The unit element in A is thus defined: m(1⊗ 1⊗ a) = m(1⊗ a⊗ 1) = m(a⊗ 1⊗ 1) = a; (2) and the 3-associativity means (abc)de = a(bcd)e = ab(cde). (3) Definition 2 . A 3associator can be defined by I = 2(abc)de− a(bcd)e− ab(cde), (4) Let us give examples of an associative and non-associative 3-algebras. Example 1 . (Associative 3-algebra.) Let A = {a, b, c...;m} be a set of linear operators on a Hilbert space, such that m is the usual associative product, then I = 0. Example 2 . (Non-associative 3-algebra. ) Consider A = {a, b, c...;m} a set of analytic functions defined on R, such that m(a⊗ b⊗ c)(x) = ∂a(x) ∂x1 ∂b(x) ∂x2 ∂c(x) ∂x3 , where x = (x1, x2, x3). In this case, I (3) 6= 0. Indeed, denoting ∂i = ∂ ∂xi , ∂ij = ∂ ∂xi∂xj , i, j = 1, 2, 3,