A Note on n - ary Poisson
نویسنده
چکیده
A class of n-ary Poisson structures of constant rank is indicated. Then, one proves that the ternary Poisson brackets are exactly those which are deened by a decomposable 3-vector eld. The key point is the proof of a lemma which tells that an n-vector (n 3) is decomposable ii all its contractions with up to n ? 2 covectors are decomposable. In the last years, several authors have studied generalizations of Lie algebras to various types of n-ary algebras, e.g., 5, 12, 9, 11, 15]. In the same time, and intended to physical applications, the new types of algebraic structures were considered in the case of the algebra C 1 (M) of functions on a C 1 manifold M, under the assumption that the operation is a derivation of each entry separately. In this way one got the Nambu-Poisson brackets, In this note, we write down the characteristic conditions of the n-ary generalized Poisson structures in a new form, and give an example of an n-ary structure of constant rank 2n, for any n even or odd. Then, we prove that the ternary Poisson brackets are exactly the brackets deened by the decomposable 3-vector elds. The key point in the proof of this result is a lemma (that seems to appear also in 16]), which tells that an n-vector P is decomposable ii 1991 Mathematics Subject Classiication 58 F 05. International Institute for Mathematical Physics in Vienna, Austria, and he expresses here his thanks to ESI for invitation and support. This paper is in nal form and no version of it will appear elsewhere.
منابع مشابه
Ja n 19 99 A Note on n - ary Poisson Brackets
A class of n-ary Poisson structures of constant rank is indicated. Then, one proves that the ternary Poisson brackets are exactly those which are defined by a decomposable 3-vector field. The key point is the proof of a lemma which tells that an n-vector (n ≥ 3) is decomposable iff all its contractions with up to n − 2 covectors are decomposable. In the last years, several authors have studied ...
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