Authors: G. Marmo, G. Vilasi, A. M. Vinogradov
$n$-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 $dim~g\le n+1$.
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.
To appear in J. Geom. Phys.
46 pages. LaTeX. To compile source files (.tex) one needs
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