**DIPS 3/97**[tex source, PostScript, dvi]-
**Title:**The local structure of n-Poisson and n-Jacobi manifolds**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.