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
In this paper,we study the structure of nonabelian omni-Lie algebroids.Firstly,taking Lie algebroid(E,[·,·]_(E,ρE))as the starting point,a nonabelian omni-Lie algebroid is defined on direct sum bundle DE⊕J...In this paper,we study the structure of nonabelian omni-Lie algebroids.Firstly,taking Lie algebroid(E,[·,·]_(E,ρE))as the starting point,a nonabelian omni-Lie algebroid is defined on direct sum bundle DE⊕JE,where DE and JE are,respectively,the gauge Lie algebroid and the jet bundle of vector bundle E,and study its properties.Furthermore,it is concluded that the nonabelian omni-Lie algebroid is a trivial deformation of the omni-Lie algebroid,and the nonabelian omni-Lie algebroid is a matched pair of Leibniz algebroids.展开更多
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).
基金supported by the National Natural Science Foundation of China(Grant Nos.11961049,11601219).
文摘In this paper,we study the structure of nonabelian omni-Lie algebroids.Firstly,taking Lie algebroid(E,[·,·]_(E,ρE))as the starting point,a nonabelian omni-Lie algebroid is defined on direct sum bundle DE⊕JE,where DE and JE are,respectively,the gauge Lie algebroid and the jet bundle of vector bundle E,and study its properties.Furthermore,it is concluded that the nonabelian omni-Lie algebroid is a trivial deformation of the omni-Lie algebroid,and the nonabelian omni-Lie algebroid is a matched pair of Leibniz algebroids.
