A Note on n - ary Poisson BracketsbyPeter
نویسندگان
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, 11, 8, 10, 14]. 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, e.g., 11, 6, 1, 4], and the generalized Poisson brackets 2, 3], etc. 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 15]), 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.
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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