1 99 8 Nambu - Lie Groups by Izu Vaisman

We extend the Nambu bracket to 1-forms. Following the Poisson-Lie case, we define Nambu-Lie groups as Lie groups endowed with a multiplica-tive Nambu structure. A Lie group G with a Nambu structure P is a Nambu-Lie group iff P = 0 at the unit, and the Nambu bracket of left (right) invariant forms is left (right) invariant. We define a corresponding notion of a Nambu-Lie algebra. We give several examples of Nambu-Lie groups and algebras. In 1973, Nambu [14] studied a dynamical system which was defined as a Hamiltonian system with respect to a ternary, Poisson-like bracket defined by a Jacobian determinant. A few years ago, Takhtajan [15] reconsidered the subject, proposed a general, algebraic definition of a Nambu-Poisson bracket of order n which, for brevity, we call a Nambu bracket, and gave the basic properties of this operation. The Nambu bracket is an intriguing operation, in spite of its rather restrictive character, which follows from the fact conjectured in [15], and proven by several authors [6], [1], [13], [8], [11] namely, that, locally and with respect to well chosen coordinates, any nonzero Nambu bracket is just a Jacobian determinant. In this paper, we show that a Nambu bracket induces a corresponding bracket of 1-forms, and use the latter for a characterization of Nambu-Lie groups, a natural generalization of the Poisson-Lie groups (e.g, [16]). The relation with a corresponding notion of a Nambu-Lie algebra is discussed, and several examples of Nambu-Lie groups and algebras are given.