×

The local structure of \(n\)-Poisson and \(n\)-Jacobi manifolds. (English) Zbl 0978.53126

An \(n\)-Lie algebra is a vector space \(V\) equipped with an \(n\)-ary skew-symmetric operation \([\cdot,\dots,\cdot]:V^n\rightarrow V\) which satisfies the generalized Jacobi identity: \[ [[u_1,\dots,u_n],v_1,\dots,v_{n-1}]=\sum_i[u_1,\dots,[u_i,v_1,\dots, v_{n-1}],\dots,u_n]. \] An \(n\)-Lie algebra structure on the associative commutative algebra \(C^\infty(M)\) of smooth functions on a manifold \(M\) is called \(n\)-Poisson structure (or Nambu-Poisson structure) if the Leibniz identity is satisfied, i.e. the \(n\)-bracket on \(C^\infty(M)\) is the bracket \(\{\cdot,\dots,\cdot\}_\Lambda\) associated with an \(n\)-vector field \(\Lambda\): \[ \{ u_1,\dots,u_n\}_\Lambda= \langle \Lambda,du_1\wedge\dots\wedge du_n\rangle. \] Similarly, \(n\)-Jacobi brackets are defined in the paper as \(n\)-Lie brackets on \(C^\infty(M)\) given by means of multi-differential operators. They turn out to be associated with the pairs \((\Lambda,\Gamma)\), where \(\Lambda\) is an \(n\)-vector field and \(\Gamma\) is an \((n-1)\)-vector field on \(M\), by \[ \{ u_1,\dots,u_n\}_{(\Lambda,\Gamma)}=\{ u_1,\dots,u_n\}_\Lambda+ \sum_i(-1)^{i+1}u_i\{ u_1,\dots,u_{i-1},u_{i+1},\dots,u_n\}_\Gamma. \] The main result of the paper is the local description of \(n\)-Poisson and \(n\)-Jacobi structures for \(n>2\): around points where the tensors do not vanish every \(n\)-Poisson tensor \(\Lambda\) can be written in the form \(\partial_1\wedge\dots\wedge\partial_n\), and every \(n\)-Jacobi structure \((\Lambda,\Gamma)\) in the form \[ (\partial_1\wedge\dots\wedge\partial_n,\partial_1\wedge\dots\wedge \partial_{n-1}) \] for some local coordinates \((x_1,\dots,x_m)\), \(\partial_i=\frac{\partial}{\partial x_i}\). Note that the result on \(n\)-Poisson structures was obtained earlier independently by P. Gautheron [Lett. Math. Phys. 37, 103-116 (1996; Zbl 0849.70014)] and D. Alekseevsky and P. Guha [Acta Math. Univ. Comen. 65, No. 1, 1-9 (1996; Zbl 0864.70012)].
The problem of compatibility of two \(n\)-Lie algebra structures is also analyzed and \((n+1)\)-dimensional \(n\)-Lie algebras are classified by analogy to the Bianchi classification for ordinary Lie algebras in dimension 3. Simple applications to dynamics are discussed.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
17B63 Poisson algebras
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
70H99 Hamiltonian and Lagrangian mechanics
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Alexeevsky, D.; Guha, On decomposability of Nambu-Poisson tensor, Acta math. univ. Comenian, 65, 1-9, (1996) · Zbl 0864.70012
[2] Cabras, A.; Vinogradov, A.M., Extensions of the Poisson bracket to differential forms and multivector fields, J. geom. phys., 9, 75-100, (1992) · Zbl 0748.58008
[3] Cariñena, J.F.; Ibort, L.A.; Marmo, G.; Perelomov, A., The geometry of Poisson manifolds and Lie algebras, J. phys. A, 27, 7425, (1994) · Zbl 0847.58025
[4] Cariñena, J.F.; Ibort, L.A.; Marmo, G.; Stern, A., The Feynman problem and the inverse problem for Poisson dynamics, Phys. rep., 263, (1995)
[5] De Filippo, S.; Marmo, G.; Salerno, M.; Vilasi, G., On the phase manifold geometry of integrable nonlinear field theory, (1982), unpublished
[6] De Filippo, S.; Marmo, G.; Salerno, M.; Vilasi, G., A new characterization of completely integrable systems, Il nuovo cimento B, 83, 97, (1984)
[7] Filippov, V.T., n-Lie algebras, Sibirsk. math. zh., 26, 6, 126, (1985) · Zbl 0585.17002
[8] Frolicher, A.; Nijenhuis, A., Theory of vector valued differential forms I, Indag. math. A, 23, 338, (1956) · Zbl 0079.37502
[9] Gnedbaye, A.V., LES algèbres k-aires et leur opèrades, C.R. acad. sci. Paris, Série I, 321, (1995)
[10] Grabowski, J., Abstract Jacobi and Poisson structures. quantization and star-products, J. geom. phys., 9, 45-73, (1992) · Zbl 0761.16012
[11] Kirillov, A.A.; Kirillov, A.A., Local Lie algebras, Usptkhi mat. nauk, Russian math. surveys, 31, 4, 55-76, (1976) · Zbl 0357.58003
[12] Krassil’shchik, I.S.; Lychagin, V.V.; Vinogradov, A.M., Geometry of jet spaces and nonlinear partial differential equations, (1986), Gordon and Breach New York · Zbl 0722.35001
[13] Landi, G.; Marmo, G.; Vilasi, G., Recursion operators: meaning and existence, J. math. phys., 35, 2, 808, (1994) · Zbl 0801.58020
[14] Levi-Civitae, T.; Amaldi, U., Lezioni di meccanica razionale zanichelli, (1929), Bologna
[15] Loday, J.L., La renaissance des operades, Sem. bourbaki 47ème annee, 792, (1994 1995)
[16] Lychagin, V.V.; Lychagin, V.V., A local classification of non-linear first order partial differential equations, Usptkhi mat. nauk, Russian math. surveys, 30, 1, 105-175, (1975) · Zbl 0315.35027
[17] Magri, F., A simple model of integrable Hamiltonian equations, J. math. phys., 19, 1156, (1978) · Zbl 0383.35065
[18] Magri, F., A geometrical approach to the nonlinear solvable equations, Lecture notes in phys., 120, 233, (1980)
[19] Michor, P.; Vinogradov, A.M.; Michor, P.; Vinogradov, A.M., n-ary Lie and associative algebras, (), (1996), to appear · Zbl 0928.17029
[20] Nambu, Y., Generalized Hamiltonian mechanics, Phys. rev. D, 7, 2405, (1973) · Zbl 1027.70503
[21] Nijenhuis, A., Trace-free differential invariants of triples of vector 1-forms indag, Math. A, 49, 2, (1987)
[22] Saletan, E.J.; Cromer, A.H., Theoretical mechanics, (1971), Wiley New York · Zbl 0249.70001
[23] Takhtajan, L.A., On foundation of generalized Nambu mechanics, Comm. math. phys., 160, 295, (1994) · Zbl 0808.70015
[24] Vinogradov, A.M., The logic algebra for the theory of linear differential operators, Soviet math. dokl., 13, 1058-1062, (1972) · Zbl 0267.58013
[25] Vinogradov, A.M.; Vinogradov, A.M., The C spectral sequence, Lagrangian formalism and conservation laws: I the linear theory: II the non-linear theory, J. math. anal. appl., J. math. anal. appl., 100, 41-129, (1984) · Zbl 0548.58015
[26] Vinogradov, A.M.; Krassil’shchik, I.S., What is the Hamiltonian formalism, Russian math. surveys, 30, 177-202, (1975) · Zbl 0327.70006
[27] A. Vinogradov and M. Vinogradov, Alternative n-Poisson manifolds, in progress. · Zbl 1074.17501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.