Abstract
n-Lie algebra structures on smooth function algebras given by means of multi-differential operators, are studied and their canonical forms are obtained. Necessary and sufficient conditions for the sum and the wedge product of two n-Poisson structures 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 ⩽ 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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