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.


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
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