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The local structure of n-Poisson and n-Jacobi manifolds

G. Marmo·G. Vilasi·A. Vinogradov
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Abstract

N-Lie algebra structures on smooth function algebras given by means of multi-differential operators, are studied. Necessary and sufficient conditions for the sum and the wedge product of two $n$-Poisson sructures to be again a multi-Poisson are found. It is proven that the canonical $n$-vector on the dual of an n-Lie algebra g is n-Poisson iff dim(g) are not greater than n+1. The problem of compatibility of two n-Lie algebra structures is analyzed and the compatibility relations connecting hereditary structures of a given n-Lie algebra are obtained. (n+1)-dimensional n-Lie algebras are classified and their "elementary particle-like" structure is discovered. Some simple applications to dynamics are discussed.

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G. Marmo, G. Vilasi, A. Vinogradov. 1997-09-30. The local structure of n-Poisson and n-Jacobi manifolds. https://doi.org/10.1016/s0393-0440(97)00057-0

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