Nambu-Poisson manifolds and associated \(n\)-ary Lie algebroids (Q2716584)

This paper is an attempt to construct an \(n\)-ary analog of a Lie algebroid defined by \textit{J. Grabowski} and \textit{G. Marmo} [Differ. Geom. Appl. 12, 35-50 (2000; Zbl 1026.17006)] canonically associated with an \(n\)-ary Nambu-Poisson bracket. The binary model for this construction is the Lie algebroid structure on the cotangent bundle \(T^*M\) associated with a Poisson structure on \(M\). The author proves an \(n\)-ary analog of the theorem by \textit{J. L. Koszul} [Astérisque, Hors série, 257-271 (1985; Zbl 0615.58029)] describing Lie algebroid brackets on differential forms in terms of generating operators. NEWLINENEWLINENEWLINEHowever, the main result of this paper, i.e., a construction of an \(n\)-ary Lie algebroid canonically associated with a Nambu-Poisson structure, is wrong. This is because checking the corresponding condition only for functions and exact forms is not enough. The obtained bracket satisfies the generalized Jacobi identity for closed forms, but not for arbitrary forms. This phenomenon occurs for \(n>2\). NEWLINENEWLINENEWLINEIn a Corrigendum [34, 9753 (2001)] the author points out that Theorem 9 is incorrect and gives some related comments.