zbMATH — the first resource for mathematics

A survey on Nambu-Poisson brackets. (English) Zbl 0953.53023
This paper is an excellent survey of the main geometrical results related to Nambu-Poisson brackets. The concept of a Nambu-Poisson structure was introduced by L. Takhtajan [Commun. Math. Phys. 160, No. 2, 295–315 (1994; Zbl 0808.70015)] in order to find an axiomatic formalism for an \(n\)-bracket operation proposed by Y. Nambu [Phys. Rev. D (3) 7, 2405–2412 (1973; Zbl 1027.70503)] to generalize Hamiltonian mechanics. A Nambu-Poisson manifold of order \(n\) is a manifold \(M\) endowed with a skew-symmetric \(n\)-bracket of functions \(\{ , \dots , \}\) satisfying the Leibniz rule and the fundamental identity (a generalization of the Jacobi identity). The Poisson manifolds are just the Nambu-Poisson manifolds of order \(2\). If \(M\) is a Nambu-Poisson manifold of order \(n\), one can introduce the Nambu-Poisson \(n\)-vector \(P\) which is characterized by the relation \(\{f_{1}, \dots , f_{n}\} = P (df_{1}, \dots , df_{n})\), for \(f_{1}, \dots , f_{n} \in C^{\infty}(M, R)\).
The definition of Hamiltonian vector fields and alternative geometric characterization and structure of Nambu-Poisson tensors are reviewed in Section 2.
Section 3 is devoted to Nambu-Poisson-Lie groups and last section is related with some quantization aspects.

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
70H99 Hamiltonian and Lagrangian mechanics
81S10 Geometry and quantization, symplectic methods
Full Text: EMIS EuDML