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 the title page style files titlatex.tex.