This paper studies the properties of Nambu-Poisson geometry from the(n-l,k)-Dirac structure on a smooth manifold M.Firstly,we examine the automorphism group and infinitesimal on higher order Courant algebroid,to prove...This paper studies the properties of Nambu-Poisson geometry from the(n-l,k)-Dirac structure on a smooth manifold M.Firstly,we examine the automorphism group and infinitesimal on higher order Courant algebroid,to prove the integrability of infinitesimal Courant automorphism.Under the transversal smooth morphismΦ:N-→M and anchor mapping of M on(n-1,k)-Dirac structure,it's holds that the pullback(n-1,k)-Dirac structure on M turns out an(n-1,k)-Dirac structure on N.Then,given that the graph of Nambu-Poisson structure takes the form of(n-1,n-2)-Dirac structure,it follows that the single parameter variety of Nambu-Poisson structure is related to one variety closed n-symplectic form under gauge transformation.WhenΦ:N-→M is taken as the immersion mapping of(n-1)-cosymplectic submanifold,the pullback Nambu-Poisson structure on M turns out the Nambu-Poisson structure on N.Finally,we discuss the(n-1,O)-Dirac structure on M can be integrated into a problem of(n-1)-presymplectic groupoid.Under the mapping II:M-→M/H,the corresponding(n-1,O)-Dirac structure is F and E respectively.If E can be integrated into(n-1)-presymplectic groupoid(g,2),then there exists the only,such that the corresponding integral of F is(n-1)-presymplectic groupoid(g,).展开更多
In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM ⊕∧nT*M for an m-dimensional manifold. As an application, we revisit Nambu-Poisson ...In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM ⊕∧nT*M for an m-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an (n + 1)-vector field π is closed under the higher-order Dorfman bracket iff π is a Nambu-Poisson structure. Consequently, there is an induced Leibniz algebroid structure on ∧nT*M. The graph of an (n+1)-form ω is closed under the higher-order Dorfman bracket iff ω is a premultisymplectic structure of order n, i.e., dω = 0. Furthermore, there is a Lie algebroid structure on the admissible bundle A ∧nT*M. In particular, for a 2-plectic structure, it induces the Lie 2-algebra structure given in (Baez, Hoffnung and Rogers, 2010).展开更多
文摘This paper studies the properties of Nambu-Poisson geometry from the(n-l,k)-Dirac structure on a smooth manifold M.Firstly,we examine the automorphism group and infinitesimal on higher order Courant algebroid,to prove the integrability of infinitesimal Courant automorphism.Under the transversal smooth morphismΦ:N-→M and anchor mapping of M on(n-1,k)-Dirac structure,it's holds that the pullback(n-1,k)-Dirac structure on M turns out an(n-1,k)-Dirac structure on N.Then,given that the graph of Nambu-Poisson structure takes the form of(n-1,n-2)-Dirac structure,it follows that the single parameter variety of Nambu-Poisson structure is related to one variety closed n-symplectic form under gauge transformation.WhenΦ:N-→M is taken as the immersion mapping of(n-1)-cosymplectic submanifold,the pullback Nambu-Poisson structure on M turns out the Nambu-Poisson structure on N.Finally,we discuss the(n-1,O)-Dirac structure on M can be integrated into a problem of(n-1)-presymplectic groupoid.Under the mapping II:M-→M/H,the corresponding(n-1,O)-Dirac structure is F and E respectively.If E can be integrated into(n-1)-presymplectic groupoid(g,2),then there exists the only,such that the corresponding integral of F is(n-1)-presymplectic groupoid(g,).
基金supported by National Natural Science Foundation of China(Grant No. 10871007)US-China CMR Noncommutative Geometry (Grant No. 10911120391/A0109)+1 种基金China Postdoctoral Science Foundation (Grant No. 20090451267)Science Research Foundation for Excellent Young Teachers of Mathematics School at Jilin University
文摘In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM ⊕∧nT*M for an m-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an (n + 1)-vector field π is closed under the higher-order Dorfman bracket iff π is a Nambu-Poisson structure. Consequently, there is an induced Leibniz algebroid structure on ∧nT*M. The graph of an (n+1)-form ω is closed under the higher-order Dorfman bracket iff ω is a premultisymplectic structure of order n, i.e., dω = 0. Furthermore, there is a Lie algebroid structure on the admissible bundle A ∧nT*M. In particular, for a 2-plectic structure, it induces the Lie 2-algebra structure given in (Baez, Hoffnung and Rogers, 2010).
