## The local structure of $$n$$-Poisson and $$n$$-Jacobi manifolds.(English)Zbl 0978.53126

An $$n$$-Lie algebra is a vector space $$V$$ equipped with an $$n$$-ary skew-symmetric operation $$[\cdot,\dots,\cdot]:V^n\rightarrow V$$ which satisfies the generalized Jacobi identity: $[[u_1,\dots,u_n],v_1,\dots,v_{n-1}]=\sum_i[u_1,\dots,[u_i,v_1,\dots, v_{n-1}],\dots,u_n].$ An $$n$$-Lie algebra structure on the associative commutative algebra $$C^\infty(M)$$ of smooth functions on a manifold $$M$$ is called $$n$$-Poisson structure (or Nambu-Poisson structure) if the Leibniz identity is satisfied, i.e. the $$n$$-bracket on $$C^\infty(M)$$ is the bracket $$\{\cdot,\dots,\cdot\}_\Lambda$$ associated with an $$n$$-vector field $$\Lambda$$: $\{ u_1,\dots,u_n\}_\Lambda= \langle \Lambda,du_1\wedge\dots\wedge du_n\rangle.$ Similarly, $$n$$-Jacobi brackets are defined in the paper as $$n$$-Lie brackets on $$C^\infty(M)$$ given by means of multi-differential operators. They turn out to be associated with the pairs $$(\Lambda,\Gamma)$$, where $$\Lambda$$ is an $$n$$-vector field and $$\Gamma$$ is an $$(n-1)$$-vector field on $$M$$, by $\{ u_1,\dots,u_n\}_{(\Lambda,\Gamma)}=\{ u_1,\dots,u_n\}_\Lambda+ \sum_i(-1)^{i+1}u_i\{ u_1,\dots,u_{i-1},u_{i+1},\dots,u_n\}_\Gamma.$ The main result of the paper is the local description of $$n$$-Poisson and $$n$$-Jacobi structures for $$n>2$$: around points where the tensors do not vanish every $$n$$-Poisson tensor $$\Lambda$$ can be written in the form $$\partial_1\wedge\dots\wedge\partial_n$$, and every $$n$$-Jacobi structure $$(\Lambda,\Gamma)$$ in the form $(\partial_1\wedge\dots\wedge\partial_n,\partial_1\wedge\dots\wedge \partial_{n-1})$ for some local coordinates $$(x_1,\dots,x_m)$$, $$\partial_i=\frac{\partial}{\partial x_i}$$. Note that the result on $$n$$-Poisson structures was obtained earlier independently by P. Gautheron [Lett. Math. Phys. 37, 103-116 (1996; Zbl 0849.70014)] and D. Alekseevsky and P. Guha [Acta Math. Univ. Comen. 65, No. 1, 1-9 (1996; Zbl 0864.70012)].
The problem of compatibility of two $$n$$-Lie algebra structures is also analyzed and $$(n+1)$$-dimensional $$n$$-Lie algebras are classified by analogy to the Bianchi classification for ordinary Lie algebras in dimension 3. Simple applications to dynamics are discussed.

### MSC:

 53D17 Poisson manifolds; Poisson groupoids and algebroids 17B63 Poisson algebras 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 70H99 Hamiltonian and Lagrangian mechanics

### Keywords:

$$n$$-Lie algebra; Nambu bracket; Jacobi manifold

### Citations:

Zbl 0864.70012; Zbl 0849.70014
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